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JEEVAN ENGINEERS ACADEMY Engineering Mathematics

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INDEX

1. Solutions of Equations

2. Matrices

3. Geometry

4. Differentiation

5. Partial Differentiation

6. Integration

7. Vector Calculus

8. Infinite series

9. Fourier Series

10. Differential equations of I order

11. Linear Differential equations

12. Partial Differential equations

13. Complex variables

14. Laplace Transforms

15. Fourier Transforms

16. Z-Transforms

17. Probability & Statistics

18. Numerical methods




1. SOLUTIONS OF EQUATIONS

* f(x) = a0 xn + a1 x

n-1 +…….. + an-1 x+an (a0≠ 0)

Polynomial in x of degree n

* f (x) = 0 algebraic equation of degree „n‟

* If f (x) has function such as trigonometric, logarithmic exponential etc. then f (x) = 0 is called

transcendental equation

* value of „x‟ which satisfies f (x) = 0 is called root.

* process of finding roots of equation solution of that equation.

Properties

1. if α is a root of equation f(x) = 0 then polynomial f (x) is exactly divisible by (x- α) and conversely.

2. Every equation of nth degree has „n‟ roots (real or imaginary)

3. If f(a) and f(b) have different signs, then the equation f (x) = 0 has atleast one root between x = a & x = b.

4. In an equation with real coefficients, imaginary roots occur in conjugate pairs as α + iβ & α - iβ

Similarly we have a+ √ b & a- √ b as roots

* Every equation of odd degree has atleast one real root.

5. Descarte’s rule

The equation f(x) = 0 cannot have more +ve roots than the changes of signs in f(x) : and more -ve roots than the

changes of signs in f (-x)

* If an equation of nth degree has at the most „p‟ +ve roots & „q‟ –ve roots, then it follows that the equation has atleast

n- (p+q) imaginary roots.

If α1, α2, α3….. αn be the roots of equation

a0xn +a1x

n-1 +a2x

n-2+ ……..+ an-1x+an = 0

then

Quadratic equation

ax2+bx+c=0 has roots

Roots are equal if b

2-4ac = 0

Roots are real & distinct if b2-4ac >0

Roots are img if b2-4ac < 0

Progressions

1. Numbers a, a+d, a+2d….. are said to be in Arithmetic progression (A.P)

nth term Tn = a + (n-1) d

Sum Sn = n/2 (2a+(n-1)d)

2. Numbers a, ar, ar2…….. are said to be in Geometric progression (G.P)

nth term Tn = a rn-1




Sum

3. Numbers 1/a, 1/a+d, 1/a+2d…… are said to be in Harmonic progression (H.P) reciprocal of

A.P.

nth term

4.If a and b are two numbers, then their

Arithmetic mean =

Geometric mean =

Harmonic mean =

5. Natural numbers are 1, 2, 3, ……. n then

Transformation of equations

1. To find an equation whose roots are m times the roots of the given equation, multiply second

term by „m‟ third term by m2 and so on (all missing terms supplied with zero coefficients).

2. To find an equation whose roots are reciprocal of the roots of the given equation, change x to

1/x.

3. To diminish the roots of an equation f(x) = 0 by h, divide f (x) by (x-h) successively.

* To increase, h taken as - ve (or) divide by (x+h)

Reciprocal Equations

Equation is unaltered on changing x to 1/x

1. A reciprocal equation of an odd degree having coefficients of terms equidistant from beginning

& ending equal, it has root = -1

2. A reciprocal equation of an terms equidistant from the beginning & end equation but oppose in

sign, it has root = 1.

3. A reciprocal equation of even degree having coefficient of terms equidistant from beginning &

end equal but opposite in sign, it has roots = 1 and -1.




2.Linear Algebra: Determinants,Matrices

* Linear Algebra comprises of theory & application of linear system of equations, linear

transformations and eigen value problems.

* In linear algebra we make a systematic use of matrices and to a lesser extent determinants and

their properties.

Matrix

→ A system of „mn‟ numbers arranged in a rectangular array of „m‟ rows and „n‟ columns is

called a matrix of order m x n.

→ if m = n, it is called a square matrix of order „n‟.

Determinants

→ expression is called det of II order

det = a1 b2 –a2 b1

is called det of III order

→ det of nth order is

a1, b2, c3 ----- ln is called leading or principal diagonal

Minor

Minor of an element in a determinant is the determinant obtained by deleting the row and

the column which intersect in that element.

Eg: if

Co factor

Co factor of any element in a det is its minor with proper sign. The sign of an element in the „ith‟

row & „jth‟ column is (-1)i+j

Cofactor of b3 is B3 = (-1)3+2

x minor of b3




C2

Laplace’s expansion

A determinant can be expanded in terms of any row (or column) as follows: “Multiply

each element of the row (or column) in terms of which we intend expanding the det, by its

cofactor and then add up all these terms”.

i.e. ∆ = a1 A1 +b1B1 +c1C1

= a1 (b2 c3- b3 c2)- b1(a2 c3-a3 c2) + c1 (a2 b3-a3b2)

ai Aj+bi Bj+ci cj = ∆ when i = j

= 0 when i ≠ j

Properties of Determinants

1. A determinant remains unaltered by changing its rows into columns and columns into rows

2. If two parallel lines of a det are interchanged, the det retains its numerical value but changes in

sign.

* if any line of a det be passed over „m‟ parallel lines resulting det ∆1 = (-1)

m ∆.

3. A determinant vanishes if two parallel lines are identical.

4. If each element of a line be multiplied by the same factor, the whole determinant is multiplied

by that factor.

5. If each element of a line consists of m terms the det can be expressed as sum of m dets.

6. If to each elements of a line be added equi multiples of the corresponding elements of one or

more parallel lines, the determinants remains unaltered.

7. If the elements of a determinant ∆ are functions of x and two parallel lines become identical

when x = a then (x-a) is a factor of ∆.

Multiplication of determinants

The product of two determinants of the same order is itself a determinant of that order.

then

Special Matrices

Row and Column matrices

* A matrix having a single row is called row matrix

[ 1 3 5 7]

* A matrix having a single column is called column matrix




Square matrix

* A matrix having „n‟ rows & „n‟ columns is called a square matrix of order „n‟

leading or principal diagonal = 1, 3, 5

sum of diagonal elements is called trace of „A‟.

if |A| = 0 then matrix is said to be singular otherwise non singular.

Diagonal matrix

A square matrix all of whose elements except those in leading diagonal are zero is called

diagonal matrix.

A diagonal matrix whose all the leading diagonal elements are equal is called a scalar matrix.

Ex:

Diagonal Scalar

Unit matrix

A diagonal matrix of order „n‟ which has unity for all its diagonal elements is called a

unit matrix or an identity matrix of order „n‟ & denoted by „In‟.

Eg:

Null Matrix

If all the elements of a matrix are zero, it is called 0 null or zero matrix & is denoted by „0‟.

Symmetric

A square matrix A = [aii] is said to be symmetric when aij = aji i & j

Skew symmetric

If aij = - aji i & j so that all the leading diagonal elements are zero

Triangular Matrix

A square matrix all of whose elements below the leading diagonal are zero is called upper

triangular matrix.




A square matrix all of whose elements above the leading diagonal are zero, is called a lower

triangular matrix.

Matrix Operations

1. Equality of matrices

Two matrices A and B are said to equal if and only if

(i) they are of same order and

(ii) each element of A is equal to corresponding element of B.

2. Addition and Subtraction of matrices

If A, B be two matrices of the same order, then their sum A+B is defined as the matrix each

element of which is the sum of the corresponding elements of A and B

Similarly A-B is defined as a matrix whose elements are obtained by subtracting the elements

of B from the corresponding element of A.

* Only matrices of same order can be added or subtracted

* Addition of matrices is commutative

i.e. A+B = B+A.

* Addition & subtraction of matrices is associative

i.e. (A+B) –C = A+ (B-C) = B +(A-C)

3. Multiplication of matrix by a scalar

The product of a matrix A by a scalar k is a matrix whose each element is „K‟ times

corresponding elements of A.

* Distribution law holds for such products

K(A+B) = KA + KB

* If AB = 0 it does not necessarily imply that A or B is a null matrix

4. Multiplication of matrices

Two matrices can be multiplied only when the number of column in the first is equal to

the no. of rows in the second.

* such matrices are said to be comformable

* Multiplication of matrices is associative

(AB) C = A (BC)

* Multiplication of matrices is distributive

A(B+C) = AB+AC.

* If A be a square matrix then product AA is defined as A2.

* If A2 = A then the matrix A is called idempotent

* If A2 = 0, then the matrix A is called Nilpotent.

* If A2= I, then the matrix A is called involuntary




5. Transpose of a matrix

The matrix obtained from any given matrix A by interchanging rows and columns

denoted by A1 or A

T.

* transpose of m x n matrix is n x m matrix

* transpose of the transpose of a matrix coincides with itself i.e. (A1)

1 = A

* For a symmetric matrix A1 = A

* For a skew symmetric matrix A1 = A

* (AB)1 = B

1A

1

* every square matrix can be uniquely expressed as a sum of symmetric and skew symmetric

matrix.

6. Adjoint of a square matrix

1. Adjoint of A is the transposed matrix of cofactors of A.

7. Inverse of a matrix

If A be any matrix then a matrix B if it exists, such that AB=BA= I, is called the inverse of A.

i.e.

* both matrix and its inverse must be non singular

* Inverse of a matrix is unique

* (AB)-1

= B-1

A-1

Rank of a Matrix

A matrix is said to be of rank „r‟ when

(i) It has at least one non-zero minor of order r, and

(ii) Every minor of order higher than „r‟ vanishes

i.e.

Rank of a matrix is the largest order of any non-vanishing minor of the matrix.

If a matrix has a non-zero minor of order r, its rank is ≥ r.

If all minors of a matrix of order (r+1) are zero, its is rank is ≤ r.

Rank is denoted by ρ (A)

Elementary transformation of matrix

1. The interchange of any two rows (columns) [Rij] /[Cij]

2. The multiplication of any row (column) by a non-zero number [KRi]/[K ci]

3. The addition of a constant multiple of the elements of any row (column) to the corresponding

elements of any other row (column) [Ri+PRj]/[Ci +Pci]




* Elementary transformation do not change either the order or rank of a matrix.

* The value of minors may get changed by the transformation (1) & (2), their zero or non zero

character remains unaffected.

Equivalent matrix

Two matrices A and B are said to be equivalent if one can be obtained from the other by a

sequence of elementary transformations.

Two equivalent matrices have

same rank.

Gauss Jordan Method of finding inverse

“These elementary row transformations which reduce a given square matrix. A to the unit matrix

when applied to unit matrix I gives the inverse of A”.

Working

Write two matrices A&I side by side. Then perform the same row transformation on both.

As soon as A is reduced to I, the other matrix represents A-1

.

Normal form

Every non zero matrix A of rank „r‟ can be reduced by a sequence of elementary transformations

to the form

is called normal form.

* Rank of matrix A is r if and only if it can be reduced to normal form.

* Any quantity having n-components is called a vector of order‟n‟

Linear dependence

The vector are said to be linearly dependent, if these exists r numbers λ1, λ2--- λr not

all zero such that

λ1 + λ2 +… + λr = 0

* If no such numbers other than zero exists, the vectors are said to be linearly independent

Consistency of Linear System of Equations

Find the ranks of the coefficient matrix [A] and the augmented matrix [AK] by reducing A to the

triangular form by elementary row operations. Let the rank of [A] be „r‟ and that of [AK] be „r‟

[AX] =K

(i) If r ± r‟ the equation are in consistent, i.e. no solution.

(ii) If r = r‟ = n, the equations are consistent and there in unique solution

(iii) If r= r‟ < n, the equations are consistent and there are infinite number of solutions.

System of linear homogeneous equations

Consider the homogeneous linear equations [ AX] = 0

a11 x1+ a12 x2 +….. +a1nxn = 0

a21 x1 +a22 x2+ … +a2nxn = 0

…………………………….

am1x1 +am2 x2 +…..+amn xn = 0

let rank „r‟ of the coefficient matrix A by reducing it to the triangular form then

1. If r = n, the equations have only a trivial zero solution

x1 = x2 ----- xn = 0

* If r < n the eqns have (n-r) linearly independent solutions.




2. When m < n the solution is always other than x1 = x2 = ----- xn = 0, The no. of solutions is

infinite.

3. When m = n, the necessary & sufficient condition for solutions other than x1 = x2 = ----- xn = 0

is that the determinant of the coefficient matrix is zero.

* Equations are consistent and a solution is non-trivial solution.

Orthogonal transformation

The linear transformation y = Ax is said to be orthogonal if it transforms

Y12 +y

22 + …… +yn

2 into x

21 +x

22 +…..+xn

2

The matrix of an orthogonal transformation is called an orthogonal matrix.

* Square matrix A is said to be orthogonal if AAT =A

T A = I

* If A is orthogonal, AT and A

-1 are also orthogonal

* If A is orthogonal then |A| = ± 1

Characteristic equation

is called

the characteristics equation.

* The roots of the equation

(-1)n λ

n +K1 λ

n-1 + K2 λ

n-2 + ……Kn = 0 are called the characteristic roots or latent roots or

eigen values.

Eigen vectors

If then the

Linear transformation y = Ax carries the column

Vector X into column vector y by means of square matrix A.

X= [ x1 x2….. xn]1 is known eigen vector or latent vector.

Properties of Eigen values

1. Any square matrix A and its transpose AT and its transpose A

T have same eigen values.

2. Eigen values of a diagonal matrix are just the diagonal elements of the matrix.

3. The eigen value of an idempotent matrix are either zero or unity.

4. The sum of the eigen value of matrix is the sum of the elements of the principal diagonal.

5. The product of eigen value of a matrix A is equal to its determinant.

6. If λ is an eigen value of a matrix A, then 1/ λ is the eigen value of A-1

.

7. If λ is an eigen value of an orthogonal matrix, then 1/ λ is also its eigen value.

8. If λ1, λ2 ….. λn are eigen value of matrix A, then Am has the eigen values λ1

m, λ2

m……. λ

mn




Cayley- Hamilton Theorem

Every square matrix satisfies its own characteristic equation.

Reduction to Diagonal form

If a square matrix A of order n has „n‟ linearly independent eigen vectors, then a matrix P can be

found such that P-1

AP is a diagonal matrix.

* The matrix P which diagonalises A is called the modal matrix of A.

* Resulting diagonal matrix D is known as Spectral Matrix of A.

* The diagonal matrix has the eigen values of A as its diagonal elements.

* The matrix P, which diagonalise A, constitutes the eigen vectors of A

* A square matrix of order n is called similar to a square matrix A of order n if

= p-1

AP.

Reduction of Quadratic form to Canonical form

* A homogeneous expression of the second degree in any no. of variables is called a quadratic

form.

Procedure

Let λ1, λ2, λ3 be eigen values of matrix A and

Then where

Here the quadratic form is reduced to a canonical for (or sum of squares form or principal axes

form)

i.e. λ1 x2 + λ2 y

2 + λ3 z

2

* The number of positive terms in canonical form of a quadratic form is known as Index of the

form.

* Signature of the quadratic form is the difference of +ve terms and –ve terms in its canonical

form.

Nature of a quadratic form

A real quadratic form x1Ax in „n‟ variables is said to be

i. Positive definite if all the eigen values of A > 0

ii. negative definite if all the eigen values of A < 0

iii. Positive semidefinite if all the eigen values of A≥0 and atleast one eigen value = 0.

iv. negative semidefinite if all the eigen values of A ≤0 and atleast one eigen value = 0

v, indefinite if some of the eigen values of A are +ve and others – Ve

Complex Matrices

1. Conjugate of a matrix

If the elements of a matrix A= [ars] are complex numbers αrs + iβrs & αrs &βrs are real then




is called conjugate matrix

2. Hermitian matrix

A square matrix A such that AT= is said to be Hermitian matrix

3. Skew Hermitian Matrix

A square matrix A such that AT = is said to be a skew Hermitian matrix.

Properties

1. Any square matrix A can be written as the sum of a Hermitian and skew Hermitian matrices.

2. If A is a Hermitian matrix then (iA) is a skew Hermitian matrix.

3. The eigen values of a Hermitian matrix are real

4. The eigen value of a skew Hermitian matrix are purely imaginary or zero

4. Unitary matrix

A square matrix U such that is called a unitary matrix.

Properties

1. Inverse of a unitary matrix is unitary.

* Inverse of an orthogonal matrix is orthogonal

2. Transpose of a unitary matrix is unitary

* Transpose of an orthogonal matrix is orthogonal

3. Product of two unitary matrices is a unitary matrix

* product of two orthogonal matrices is an orthogonal

4. The eigen value of a unitary matrix has absolute value I.

* The eigen value of a orthogonal matrix has absolute value I.




3. GEOMETRY

Co-ordinate of a point

y

0

P

y

x

r

x

Cartesian (x, y) polar (r, θ)

x = r cos θ, y = r sin θ

r = √ (x2+y

2), θ = tan

-1 (y/x)

* Distance between two points

(x1y1) & (x2,y2)=

* Point of division of the line joining (x1,y1) & (x2,y2) in the ratio m1:m2 is

Triangle

A (x1y1) B (x2y2), C (x3, y3)

* Area

* Centroid (point of intersection of medians) is

* Incentre (point of intersection of internal bisectors of angle)

a,b,c, are length of sides of triangle

Circumcentre is the pt of intersection of right bisectors of sides of triangle.

Orthocentre is the pt of intersection of perpendicular drawn from vertices to the opposite sides

of triangle.

Straight line

Slope of line joining points (x1y1) & (x2 y2) =




* Slope of the line ax+by+c = 0 is –a/b i.e. -

Equations of a line

o Having slope m and cutting an intercept „C‟ on y-axis is y = mx+C.

o Cutting intercepts a & b from the axes is

o Passing through (x1y1) & having slope m is y-y1= m (x-x1)

o Passing through (x1y1) & making an <θ with x-axis is

o Through the point of intersection of lines ax+b1y+c1=0 & a2x+b2y+c2= 0 is

a1x+b1y+c1+k(a2x+b2y+c1+K(a2x+b2y+c2) = 0

Angle between two lines having slopes m1 & m2 is

o Two lines are parallel if m1 = m0

o Two lines are perpendicular if m1m2 = 1

o Any line parallel to line ax+by+c= 0 is ax+by+k = 0

o Any line perpendicular to line ax+by+ c= 0 is bx-ay+ k = 0

Length of the perpendicular from (x1y1) to line ax+by+c=0 is

Circle

Equation of the circle having centre (h, k) & radius r is (x-h)2 + (y-k)

2 = r

2

Equation x2+y

2+2yx+2fy+c = 0 represents a circle having centre (-g1 f) and radius = √ (g

2+f

2-

c)

Equation of the tangent at the point (x1y1) to the circle x2+y

2=a

2 is xx1+yy1 = a

2

Condition for the line y = mx+c to touch the circle x2+y

2= a

2 is c= √(1+m

2)

Length of tangent from the point (x1y1) to the circle x2+y

2+2gx+2fy+c = 0

is




Parabola

Standard equation of parabola is y2= 4 ax

o Parametric equations are x = at2, y = 2 at

o Latus rectum L L‟ = 4 a Focus S (a, 0)

o Directrix ZM is x+a = 0

Focal distance of any point P (x1, y1) on parabola y2= 4ax is Sp= x1+a

Equation of tangent at (x1y1) to parabola y2 = 4ax is yy1 = 2a (x+x1)

Condition for the line y= mx+c to touch parabola y2=4ax is c = a/m

Equation of normal to parabola y2= 4ax in terms of its slope m is y =mx-2am-am

3.

Ellipse

Standard equation is

o Parametric equations are x = a cos θ, y = b sin θ

o Eccentricity =

o Latus rectum LSL1 = 2b

2/a, Foci S (-ae, 0) & S

1 (a e, 0)

o Directrices ZM (x = a/e) & Z1M

1 (x = a/e)

Sum of focal distances of any point on the ellipse is equal to major axis i.e.

SP+S1P = 2a

Equation of the tangent at the point (x1 y1) to the ellipse

is

Condition for the line y = mx+c to touch ellipse

is




Hyperbola

↑

→M M

y

x

L

L

SC

Z Z

1

11

Standard equation is

o Parametric equations are

x= a sec θ, y = b tan θ

Eccentricity e =

Latus rectum

Directrices ZM (x= a/e) & z1m

1(x= -a/e)

Equation of tangent at the point (x1,y1) to hyperbola

is

Condition for the line y = mx+c to touch hyperbola

is

Asymptotes of hyperbola are &

Equation of rectangular hyperbola with asymptotes as axes is xy= c2

Its parametric equations are x = ct, y = c/t

Nature of a Conic

The equation ax2+2hxy+by

2+2gx+2fy+c = 0 represents

A pair of lines, if

A circle, if a=b, h =0, ∆≠ 0

A parabola, if ab-h2 = 0,∆≠ 0

An ellipse, if ab-h2> 0, ∆≠ 0

A hyperbola, if ab-h2 < 0, ∆≠ 0

A rectangular hyperbola if

ab-h2< 0, ∆≠ 0, a+b=0.




Volumes & Surface areas

Solid volume Curved surface area Total surface area

1. Cube (side „a‟) a3 4a

2 6a

2

2. Cuboid (length l, breadth

b, height h) lbh 2(l+b) h 2(lb+bh+hl)

3. Sphere (radius r) (4/3) πr 3 - 4πr

2

4. Cylinder

(base radius r, height h π r2h 2πrh 2πr(r+h)

5. Cone (base radius r, height

h) [ l= √ r2+h

2] (1/3) πr

2h πrl π r (r+l)




CALCULUS

[Differentiation, Integration]

4. Differentiation

Some standard Derivatives

,

,

,

,

,

,

,

,

,

,

,

D

n (a

mx)= m

n (loga)

n . a m

x

Dn (e

mx) = m

n e

mx




Dn Sin (ax+b) = a

n Sin (ax+b+nπ/2)

Dn Cos (ax+b) = a

n Cos (ax+b+ nπ/2)

Dn [e

ax Sin (bx+c)] = (a

2+b

2)

n/2 e

ax sin (bx+c+ntan

-1(b/a)

Dn [ e

ax Cos (bx+c)] = (a

2+b

2)

n/2 e

ax cos (bx+c+ntan

-1 (b/a))

Leibnitz’s theorem

If u, v be two functions of x possessing derivatives of the nth order then

(uv)n = unV +nc1 un-1 V1 + nc2 un-2 v2 +…….+ncnuVn

Rolle’s theorem

If (i) f(x) is continuous in [a,b]

(ii) f(x) exists for every value of c of x in (a,b) such that f(c) = 0

(iii) f(a) = f(b)

then there exists at least one value C of x in (a, b) such that f1 (c) = 0

Lagrange’s Mean-value Theorem

If (i) f(x) is continuous in [a,b] and

(ii) f1 (x) exists in (a, b) then there is atleast one value c of x in (a, b) such that

Cauchy’s Mean-value theorem

If (i) f(x) and G(x) be continuous in [a,b]

(ii) f1(x) and g

1(x) exist in (a,b)

(iii) g1(x) ≠ o for any value of x in (a,b) then

There is atleast one value c of x in (a,b) such that

Taylor’s Theorem

If (i) f(x) and its first (n-1) derivatives be continuous in [a, a+h] and

(ii) fn (x) exists for every value of x in (a, a+b)

Then there is atleast one number θ (0 < θ<1), such that

f(a+h) = f (a) + h f1(a) + h

2/2 f

11 (a)+…. + h

n/n! f

n (a+ θh).




Maclaurin’s series

If f (x) can be expanded as an infinite series, then

f (x)= f(0) + xf1(0) +

Indeterminate forms

To evaluate Lt [ (f(x)/φ(x)] in 0/0 form, differentiate the numerator & denominator separately as

many times as would be necessary to arrive a determinate form.

Form ∞/∞

Applying L‟ Hospital‟s rule.

i.e. Differentiating numerator & Denominator separately as many times as would be necessary.

Increasing & Decreasing Functions

In the function y = f(x), if y increases as x increase it is called an increasing function of x.

In the function y = f(x), if y decreases as x increases it is called a decreasing function of x.

Maxima & Minima

Procedure to find maxima & minima

1. Put the given function = f(x)

2. Find f1(x) and equate it to zero. Solve this equation and let its roots be a, b, c….

3. Find f11

(x) and substitute in it by turns x= a, b, c

* If f11

(a) is –ve, f (x) is maximum at x = a

* If f11

(a) is +ve, f(x) is minimum at x = a.

4. Sometime f11

(x) may be difficult to find out or f11

(x) may be zero at x= a. Then

* If f‟(x) changes sign from +ve to –ve as x passes through a,f(x) is minimum at x = a.

* If f1 (x) changes sign from – ve to +ve as x passes through a,f(x) is maximum at x = a.

* If f1 (x) does not change sign while passing through x = a, f(x) is neither maximum nor

minimum at x =a.




5. PARTIAL DIFFERENTIATION

If z= f(x, y) a function of two variables x & y then & partial derivative of z with respect to x

is denoted by

(or) (or) fx (x,y)

or Dxf and is given by

* Partial derivating of x with respect to y is given by

* Sometimes we may use the notation

Any function f (x, y) which can be expressed in the form xn φ (y/x) is called a homogenous

function of degree n in x and y

Euler‟s theorem

If u be a homogeneous function of degree n in x and y then.

If u f(x, y) where x = φ(f) & y = ψ (t) then total derivative is given by

If t = x

If f (x, y) = c be an implicit relation between x and y then

Taylor‟s expansion of f (x, y) in powers of (x-a) & (y-b) is given by

f(x, y)= f (a, b) + [(x-a) fx (a, b)+ (y-b) fy (a, b)]

+ ½! [ (x-a)2 fxx (a, b) +2(x-a) (y-b) fxy (a, b)+ (y-b)

2 fyy (a, b) ]+…..

Maclaurin‟s expansion of f (x, y) is given by f (x, y) = f (0, 0)+ [ x fx (0, 0)+ y fy (0, 0)]+ ….

f (a, b) is said to be stationary value of f(x, y) if fx (a, b) = 0 and fy (a, b) = 0 i.e., the function is stationary at

(a, b) but converse is not true.




Procedure to find maxima & minimum values of f (x, y)

1. Find and equate each to zero. Solve these as simultaneous equation in x & y.

Let (a, b) (c, d)… be pair of values.

2. Calculate the value of

for

each pair of values

3. (i) If rt-s2 > 0 & r< 0 at (a, b), f (a, b) is a max value.

(ii) If rt-s2> 0 & r> 0 at (a,b), f (a, b) is a min value.

(iii) If rt-s2 < 0 at (a, b) f (a, b) is not an extreme value. i.e. (a, b) is a saddle point.

(iv) If rt-s2= 0 at (a, b) the case is doubtful & needs further investigation.

Larange‟s method

1. Write F= f (x, y, z) + λ φ (x, y, z)

2. Obtain the equations

3. Solve the above equations together with φ (x, y, z) = 0. The values of x, y, z so obtained will give the stationary

value of f (x, y, z)




6. INTEGRATION

Some Standard Integration Formulae

Sinx dx = - Cos x, cosx dx = sin x

tanx dx = - log (cos x) cotx dx = log (sinx)

sec x dx = log (sec x+tan x), cosec xdx= log (cosecx-cot x)

,

,

,

, cos hx dx = sin hx




tan hx dx = log (cos hx) cot hx dx = log (sin hx)

sech2xdx = tan hx , cosech

2x dx = - cot hx.

Reduction Formulae

( π/2, only if

both m & n are even)




Definite Integrals

Properties

1.

2.

3.

4.

5. if f (x) is an even function

= 0 if f (x) is an odd function

6.

= 0 if f (2a-x) = - f (x)

7. f(x) g(x) dx = f (x) g(x) dx - f1(x) [ g(x) dx] dx.

Important Integrals

if n is even

= 0 if n is odd.




Improper Integrals

is said to be an improper integral of first kind if a = - ∞ or b = ∞ or both.

* is said to be an improper integral of second kind if f (x) is infinite for one or

more values of x in [a, b]

* is said to be convergent if the values of integral is finite.

7. VECTOR CALCULUS

A quantity which is completely specified by its magnitude & direction is called vector.

P

A

Q

A vector of unit magnitude is called a unit vector ( )

A vector of zero magnitude (which have no direction) is called a zero vector (0).

The vector represents the negative of i.e -

Two vectors having the same magnitude & the same (or parallel) directions are said to

be equal ( ).

Addition of vectors

Subtraction of vectors

Any three position vectors are collinear, if ,where

In vector algebra, the division of a vector by another vector is not defined.

Scalar or Dot Product

The scalar or dot product of two vectors is defined as the scalar ab cos θ, where θ is

the angle b/n




Properties

1. Scalar product of two vectors is commutative

2. The necessary & sufficient condition for two vectors to be perpendicular is that their scalar

product should be zero.

3. The square of a vector is a scalar which stands for the square of its magnitude.

4. For the mutually perpendicular unit vectors

5. Scalar product of two vectors is distributive

6. Schwarz inequality

7. Scalar product of two vectors is equal to the sum of the products of their corresponding

components.

8. Angle between two lines whose direction cosines are l, m, n & , , is

9. Angle between two lines whose direction ratios are a, b, c & , , is

10. Projection of the line joining two points (x1, y1, z1) & (x2, y2, z2) on a line whose direction

cosines are l, m, n is

l (x2- x1) + m (y2 – y1)+ n (z2 – z1)

Vector or Cross Product

The vector or cross product of two vectors

is defined as a vector such that

(i) its magnitude is ab sin θ, θ being angle between

(ii) its direction is perpendicular to the plane of

(iii) it forms with a right handed system.

Properties

1. Vector product of two vectors is not commutative

2. The necessary and sufficient condition for two non-zero vectors to be parallel is that their

vector product should be zero.

3. For the orthonormal vector triad

4. Relation b/n Scalar & vector products is

5. Vector product of two vectors is distributive




6. If then

Scalar Product of three vectors

If be any three vectors then the scalar or dot product with is called the scalar

product of three vectors and is written as

Properties

1. The condition for three vectors to be coplanar is that their scalar triple product should vanish.

i.e.

2. If any two vectors of a scalar triple product are equal the product vanishes i.e.

3. Every scalar triple product

(i) is independent of the position of the dot or cross and

(ii) depends upon the cyclic order of vectors

4. Scalar triple product is distributive

i.e.

5. If ,

then

Vector Product of three vectors

If be any three vectors, then the vector or cross product of with is called

the vector product of these vectors and is written as

*

*

If be any four vectors then

*

*

Vector differential operator Del, written as is defined by




The vector function f is defined as the gradient of the scalar point function f and is written

as grad f.

* grad f =

The divergence of a continuously differentiable vector point function is denoted by div .

*

The curl of a continuously differentiable vector point function is given by

is solenoidal.

is called Laplace‟s equation. is Laplacian operator

is called irrotational.

Green’s theorem

If φ (x, y) , ψ (x, y) φy & ψx be continuous in a region E of the xy-plane bounded by a

closed curve C, then

Stoke’s theorem

If S be an open surface bounded by a closed curve C and be any

continuously differentiable vector- point function, then

Where is a unit external normal at any point

of s.

Gauss divergence theorem

If is a continuously differentiable vector function in the region E bounded by the closed

surface S, then

Where is the unit external normal vector.




8. INFINITE SERIES

An ordered set of real numbers a1, a2, a3…..an is called a sequence and is denoted by (an).

If the number of terms is unlimited, then the sequence is said to be an infinite sequence

Limit

A sequence is said to tend to a limit l, if for every ε > 0, a value N of n can be found such

that for n ≥ N.

Convergence

If a sequence (an) has a finite limit, it is called a convergent sequence.

* If (an) is not convergent it is said to be divergent.

Bounded Sequence

A sequence (an ) is said to be bounded, if there exists a number K such that an < k for every

n.

Monotonic sequence

The sequence (an) is said to increase steadily or to decrease steadily according as an+1 ≥ an or

an+1 ≤ an for all values of n. Both increasing and decreasing sequences are called monotonic

sequences.

A monotonic sequence always tends to a limit, finite or infinite.

A sequence which is monotonic and bounded is convergent

If u1, u2, u3… un…. be an infinite sequence of real numbers, then u1+u2+u3+…. +un+…….∞ &

is called an infinite series.

* Denoted by ∑un & sum its first n terms by Sn

If ∑un = u1+u2+u3 + …. +un + ……∞ &

Sn = u1+u2+u3+ …. +un then

1. If „Sn‟ tends to a finite limit as n → ∞ , the series ∑un is said to be convergent.

2. If „Sn‟ tends to ± ∞ as n→∞, the series ∑un is said to be divergent

3. If „Sn‟ does not tend to a unique limit as n → ∞ then the series ∑un is said to be oscillatory or

non-convergent

Properties of series

1. The convergence or divergence of an infinite series remains unaffected by the addition or

removal of a finite number of its terms.

2. If a series in which all the terms are positive is convergent, the series remains convergent even

when some or all of its terms are negative.

3. The convergence or divergence of an infinite series remains unaffected by multiplying each

term by a finite number.

An infinite series in which all the terms after some particular term are positive term series.

* A series of positive terms either converges or diverges to +∞.

* If , the series ∑un is convergent.




* If series ∑un is divergent

Tests for convergence of a series

Comparison tests

1. If two positive term series ∑un & ∑un be such that

a. ∑un converges.

b. un ≤ vn for all values of n, then ∑un also converges.

2. If two positive term series ∑un & ∑vn be such that

a. ∑vn diverges

b. un ≥ vn for all values of n, then ∑un also diverges.

3. Limit form

If two positive term series ∑un & ∑vn be such that = finite quantity (≠ 0) then

∑un & ∑vn converge or diverge together.

Integral test

A positive term series f(1)+ f(2) + ……..+ f(n) + ….. where f (n) decreases as n increases,

converges or diverges according as the integral,

is finite or in finite

Comparison of ratios

If ∑un & ∑vn be two positive term series, the ∑un converges if (i) ∑vn converges and (ii) from

and after some particular term

D‟Alembert‟s ratio test

In a positive term series ∑un, if , then the series converges for λ < 1

And divergence for λ > 1

Cauchy‟s root test

In a positive series ∑un, if , then the series converges for λ < 1

And divergence for λ > 1

A series in which the terms are alternately +ve or – ve is called alternating series.

An alternating series u1 – u2 +u3- u4 +…..converges if

(i) Each term is numerically less than its preceding term and

(ii)

* If the given series is oscillatory

If the series of arbitrary terms u1+u2+u3+…… un+….. be such that the series |u1|+ |u2|+ |u3| +

…..+ |un| + …. is convergent, then the series ∑un is said to be absolutely convergent.

* If ∑|un| is divergent but ∑un is convergent, then ∑un is said to be conditionally convergent.




9. FOURIER SERIES

The fourier series for the function f (x) in the interval α < x< α+ 2π is given by

Where

f(x) cos nx dx

f (x) sin nx dx

a0, an, bn values are known as Euler‟s formulae

Dirichlet’s conditions

Any function f (x) can be developed as a Fourier series

where a0,an,bn constants provided:

1. f(x) is periodic, single valued and finite

2. f(x) has a finite number of discontinuities in any one period.

3. f (x) has at the most a finite number of maxima and minima.

A function f (x) is said to be even if f(-x) = f(x)

A function f (x) is said to be odd if f(-x) = - f(x)

Fourier series for the even function f(x) in the interval (-c, c) is

where

bn = 0

* If a periodic function f (x) is even, its Fourier expansion contains only cosine terms.

Fourier series for the odd function f (x) in the interval (-c, c) is

Where a0 = 0, an = 0

If a periodic function f (x) is odd, its Fourier expansion contains only sine terms.




10. DIFFERENTIAL EQUATIONS OF FIRST ORDER

A differential equation is an equation which involves differential coefficients or differentials.

An ordinary differential equation is that in which all the differential coefficients have reference

to a single independent variable.

A partial differential equation is that in which there are two or more independent variables and

partial differential coefficients with respect to any of them.

The order of a differential equation is the order of the highest derivative appearing in it.

The degree of a differential equation is the degree of the highest derivative occurring in it, after

the equation has been expressed in a form free from radicals and fractions as far as the derivatives

are concerned.

A solution (or integral) of a differential equation is a relation between the variables which

satisfies the given differential equation.

The general (or complete) solution of a differential equation is that in which the number of

arbitrary constants is equal to the order of the differential equation.

A particular solution is that which can be obtained from the general solution by giving

particular values to the arbitrary constants.

Solutions of I order and first degree differential equations

1. Variables- Separable Method:

If in an equation it is possible to collect all functions of x and dx on one side and all the functions

of y and dy on the other side, then the variables are said to be separable. Thus the general form of

such an equation is f (y) dy = φ (x) dx.

Integrating both sides, we get ∫ f (y) dy = ∫φ (x) dx+c as its solution.

2. Homogeneous Equations

Are of the form

Where f (x, y) and φ (x, y) are homogenous functions of the same degree in x and y.

To solve a homogenous equation (i) put y = vx then

(iii) Separate the variables v and x and integrate

3. Equations reducible to homogenous form

The equations of the form

Can be reduced to the homogenous form as follows:

Case (i) when

Putting x = X+h, y = Y+k (h, k being constants)

Where




solved by putting Y = Ux.

Case (ii) when

Let

Put ax+by= t and solve by variable separable method

4. Linear equations

A differential equation is said to be linear if the dependent variable and its differential

coefficients occur only in the first order degree and not multiplied together.

The standard form of a linear equation of the first order commonly known as Leibnitz‟s linear

equation is

where P, Q are functions of x,

Solution is

Integrating Factor (IF) =

Solution is y (I.F) =

5. Bernoulli‟s Equations

The equation where P, Q are functions of x, is reducible to the Leibniqz‟s

linear equation and is usually called the Bernoulli‟s equation.

To solve, divide both side by yn so that

Which is Leibnitz‟s equation in z & can be

solved easily.

6. Exact Differential equations

A differential equation of the form M (x, y) dx+ N (x, y) dy = 0 is said to be exact if its left

hand member is the exact differential of some function u (x, y) i.e. du = Mdx+Ndy = 0

Solution is u (x, y) = C

The necessary and sufficient condition for the differential equation Mdx+Ndy = 0 is

The solution of Mdx+Ndy = 0 is




(terms of N not containing x) dy = C

(y constant)

7. Equations reducible to exact equations

Sometimes a differential equation which is not exact, can be made so on multiplication by a

suitable factor called an integrating factor (I.F)

(i) I.F. Found by inspection:-

The I.F. can be found after regrouping the terms of the equation and recognizing each

group as being a part of an exact differential

xdy+ydx = d (xy)

(ii) I.F. of a homogenous Equation

If Mdx+Ndy = 0 be a homogeneous equation in x & y,

Then is an I.F. (Mx+Ny≠ 0)

(iii) I.F. for an equation of type f1 (x y) ydx + f2 (xy ) xdy = 0

If the equation Mdx+ Ndy = 0 be of this form, then is an I.F. (Mx-Ny ≠ 0)

iv. In the equation Mdx+Ndy = 0

a. If be a function of x only = f (x) say, then is an integrating factor.

b. If be a function of y only = F (y) say, then is an integrating

factor.




11. LINEAR DIFFERENTIAL EQUATIONS

Linear Differential equations are those in which the dependent variable and its derivatives

occur only in the first degree and are not multiplied together

General form

Where P1 P2….. Pn and x are functions of x only.

Linear differential equations with constant coefficients are of the form

Where K1, K2…… Kn are constant

Procedure to solve the equation

Of which the symbolic form is

(Dn+ K1D

n-1 +……+ Kn-1 D+Kn) y = x

Step – I to find the complementary function (CF)

(i) Write the Auxiliary equation

i.e. Dn+ K1 D

n-1 +……+ Kn-1 D+ Kn = 0 and

solve it for D.

(ii) Write the C.F. as follows:

Roots of A.E. C.F

1. m1, m2, m3 … (real & different roots) c1em1x

+ c2 em2x

+ c3 em3x

+ ….

2. m1, m1, m3….(two real & equal roots) (c1+c2 x) em1x

+ C3 em3x

+….

3. m1, m1, m1, m4… (three real & equal roots) (c1+c2x+c3x2)e

m1x +c4 em

4x +….

4. α+iβ, α- iβ, m3 … (a pair of imaginary roots) eax

(c1cosβx+c2 sinβx) +c3 em3x

+…

5. α±β, α±β, m5 … (2 pairs of equal imaginary roots) eax

[(c1+c2x)cos βx+(c3+c4x)sin βx] +c5

em5x

+ ….

Step – II To find the particular Integral (P.I)

From Symbolic from

(i) When X = eax

Put D = a [ f(a)≠ 0]

Put D = a [f (a) = 0; f1(a) ≠ 0]

Put D= a [ f1 (a) = 0, f

11 (a) ≠ 0]

And so on.

(ii) when x = sin (ax+b) or cos (ax+b)




Sin (ax+b) [or cos (ax+b)], Put D2 = -a

2 [ φ (-a

2) ≠ 0]

Sin (ax+b) [ or cos (ax+b)] Put D2 = -a

2

[φ (-a2)= 0, φ

1 (-a

2) ≠ 0]

And so on.

(iii) When X = xm, m being a +ve integer

P.I.

(iv) When X = eax

V, where V is a function of x

P.I.

(V) when x is any function of x

P.I.

* Resolve into partial fractions & operate each partial fraction on x remembering that

Step III To find complete solution (C.S.)

C.S. is y = C.F. + P.I.

Method of Variation of Parameters

This method is quite general & applies to equations of the form

y11

+Py1+qy = X

where P, q & X are functions of x.

It gives P.I. =

Where y1 and y2 are the solutions of y

11+Py

1+qy = 0 &

is called Wronskian of y1, y2.

Cauchy‟s homogenous Linear equation

Where X is function of x.

Such equations can be reduced to linear differential equations with constant coefficient by

putting




X = et or t = log x, then D= d/dt

Legendre‟s Linear equation




12. PARTIAL DIFFERENTIAL EQUATIONS

x and y are independent variable and z is dependent variable

Z = f (x, y) then

A linear partial differential equation of the first order, commonly known as Lagrange‟s linear

equation is of form

Pp+Qq = R

When P, Q, R are functions of x,y, z. This equation is called quasi-linear equation.

To solve equation PP +Qq = R

(i) form the subsidiary equations

(ii) solve these simultaneous equations, giving u = a & v= b as its solutions.

(iii) write the complete solution as φ (u, v) = 0 or u = f (v)

Procedure to solve the equation

Its symbolic form is (D

n+ K1 D

n-1 D

1+ …,+Kn D

n) z = F (x, y)

Or f (D, D1) z = F (x, y)

Step- I to find C.F.

(i) Write the A.E.

mn +K1 m

n-1 + ….+ Kn = 0 & solve it for „m‟.

(ii) Write the C.F. as follows

Roots of A.E. C.F.

1. m1, m2, m3…. (distinct roots) f1(y+m1 x) +f2(y+m2 x) + f3 (y+m3 x)+ ….

2. m1, m2, m3… (two equal roots) f1(y+m1 x)+ x f2 (y+m2 x)+f3 (y+m3 x)+….

3. m1,m1, m1… (three equal roots) f1 (y+m1 x)+ x f2 (y+m1x)+x2 f3 (y+m3 x) + ….

Step- II To find P.I.

From the symbolic form

(i) when F (x, y) =

(ii) [Put D = a & D1 =b]

(iii) when F (x, y) = sin (mx+ny) or cos (mx+ny)

Sin or cos (mx+ny) [ Put D2 = - m

2, DD

1= - mn, D

‟2 = - n

2]

(iii) when F (x, y) = xm y

n




(iv) when F (x, y) is any function of x & y

Step- III

C.S. = C.F. + P.I. = Z

Non-linear equations of the first order

Form I. f (p, q)= 0 i.e. equations containing p & q only its complete solution is z = ax+by+c

where a & b are connected by relation f (a, b) = 0.

z = ax+ φ (a) y+c, a & c are arbitrary constants.

From- II f (z, p, q) = 0, i.e. equations not containing x & y.

(i) assume u = x+ay & substitute P = dz/du, q= a dz/du in equation.

(ii) solve the resulting ordinary differential equation in z & u.

(iii) replace u by x+ay.

Form- III f (x, p) = F (y, q) i.e. equations in which z is absent and the terms containing x & p can

be separated from those containing y & q.

Solution is z = ∫ φ (x) dx+ ∫ψ (y) dy +b.

Form-IV Z = Px+qy+f (P, q)

Its complete solution is z = ax+by+f (a, b) which is obtained by writing a for P & b for q in the

given equation.




13. COMPLEX VARIABLES

A number of the form x+iy, where x and y are real numbers and i=√(-1) is called a complex

number.

X is called the real part of x+iy and is written as R (x+iy) and y is called the imaginary part

and is written as I (x+iy)

A pair of complex numbers x+iy and x-iy are said to be conjugate of each other.

Properties

1.If x1+iy1= x2+iy2 then x1-iy1 = x2- iy2

2. Two complex numbers x1+iy1 and x2 +iy2 are said to be equal when

R (x1+iy1)= R (x2+ iy2) i.e. x1 = x2

And I (x1+ iy1) = I (x2 + iy2) i.e. y1 = y2

3. Sum, difference product and quotient of any two complex numbers is itself a complex number.

If x1+iy1 and x2+iy2 be two given complex number.

Then (i) their sum = (x1 +x2) + i (y1+y2)

(ii) Their difference = (x1- x2) + i(y1- y2)

(iv) Their product = x1x2- y1 y2 + i (x1y2+x2y1)

(v) Their quotient

4. Every complex number x+iy can always be expanded in the form r (cos θ + i sin θ)

The number r = √ x2+y

2 is called the modulus of x+iy and is written as mod (x+iy) or |x+ iy|

The angle θ is called the amplitude or argument of x+iy and is written as amp (x+iy) or arg

(x+iy). θ = tan-1

(y/x)

Cos θ + i sin θ is briefly written as c is θ

De Moivre‟s Theorem

If n be (i) an integer, +ve or - ve

(cos θ + i sin θ)n = cos n θ + i sin n θ

(ii) a fraction + ve or – ve one of the values of (cos θ+ i sin θ)n is cos n

θ + isin n θ.

Cis θ1 .. Cis θ2 … Cis θn = C is (θ1+ θ2+ …..+ θn)

(Cos θ- i Sin θ)n = Cosnθ- isin n θ = (cos θ + i sin θ)

-n

(Cis m θ)n = Cis mn θ = (Cis n θ)

m

Complex Function

If for each value of the complex variable z (= x+iy) in a given region R, we have one or more

values of w (= u+iv), then w is said to be a complex function of z and we write w = u (x, y) + I

v(x, y) = f (z) where u, v are real functions of x & y.

If to each value of z, there corresponds one & only one values of w, then w is said to be a

single valued function of z otherwise a multivalued function

Eg: w = 1/z is a single valued function

W = √ z is a multivalued function

Exponential function of a complex variable

The exponential function of the complex variable z = x+iy is




Properties

1. Exponential form of Z = reiθ

2. ez is periodic function having imaginary period 2 πi,

3. ez is not zero for any value of z

4.

Circular function of a complex variable

The circular functions of the complex variable z is given as

Properties

1. Sin z, cos z are periodic with period 2 π, tan z, cot z are periodic with π.

2. cos z, sec z are even functions while sinz, cosec z are odd functions

3. Zeros of Sin z are given by z = ± 2 n π & zeroes of cos z are given by z = ± ½ (2n+1) π, n = 0,

1, 2…..

Euler‟s theorem

eiθ = cos θ + i sin θ, where θ is real or complex

Hyperbolic functions

If x be real or complex

(i) is defined as hyperbolic sine of x [ sin hx]

(ii) is defined as hyperbolic cosine of x [ cos hx ]

Properties

1.Sin hz & cos hz are periodic with period 2 π i

2. cos hz is an even function & sin hz is an odd function

3. Sin ho= 0 , cos ho = 1, tan ho = 0.

4. Relations b/n hyperbolic & circular functions

Sin i x = i sin hx

Cos i x = cos h x

Tanix = i tan hx

5. Formulae of hyperbolic functions

cosh2 x- sin h

2 x = 1

Sec h2 x+ tan h

2 x = 1

Cot h2 x- Cosech

2 x = 1

Sin h (x±y) Sin hx cos hy ± cos hx sin hy

Cos h ( x ±y) = cos hx cos hy± sin hx sin hy

Tan h (x ±y) =

Sin h2x = 2 Sin hx cos hx

Cos h2x = cos h2 x + sin h

2 x = 2 cos h

2 x -1 = 1+2 sin h

2 x




Sin h 3x = 3Sin hx+ 4 Sin h3 x

Cos h3x = 4 cosh3 x- 3 Cos hx

Tan h 3x =

Sin hx + Sin hy =

Sin hx- sin hy

Cos h x+ coshy

Cos hx- Cos hy

Inverse hyperbolic functions

Sin h-1

z = log [ Z+ √z2+1]

Cos h-1

z = log [ z+ √ z2-1]

Tan h-1

z = ½ log [ (1+z)/(1-z)]

Logarithmic function of a complex variable

If z ( = x + I y) and w ( = u + I v) be so related that ew = z, then w is said to be a logarithm of z to

the base „e‟ and is written as w = logez.

Logarithm of a complex number has an infinite no. of values and is therefore a multi-valued

function.

Log Z = log (x+ iy) = 2 in π + log (x+ iy )

Log (x+ iy) = log ( √ x2 + y

2) + i [ 2n π + tan

-1 (y/x)].

Exponential series ex =

Sine, Cosine, Sin h or Cosh series

Logarithmic series




Gregory‟s series

Binomial Series

Geometric series

a+ar+ar2 + ….to n terms =

a+ar+ar2+ ….. ∞ =

Derivative of f(z)

The derivative of w = f (z) is defined to be

provided the limit exists and has the same value for all the different ways in which δz approaches

zero.

The necessary and sufficient conditions for the derivative of the function w = u(x, y)+ i

v(x, y) = f (z) to exist for all values of z in a region R, are

(i) are continuous functions of x & y in R.

(ii) Cauchy- Riemann equations

Analytic functions

A function f (z) which is single valued & possess a unique derivative with respect to z at all

points of a region R, is called an analytic function of z in that region.




A function which is analytic everywhere in the complex plane is known as an entire function.

A point at which an analytic ceases to possess a derivative is called a singular point of the

function.

If a complex function is one known to be analytic it can be differentiated just in the ordinary

way.

The real & imaginary parts of an analytic function are called conjugate functions.

Cauchy‟s theorem

If f (z) is an analytic function and f1 (z) is continuous at each point within and on a closed curve C,

then ∫c f (z) d z= 0

Cauchy‟s integral formula

If f (z) is analytic within and on a closed curve and if a is any point within C, then

Taylor‟s series

If f (z) is analytic inside a circle C with centre at a, then for z inside C,

f (z) = f (a) + f1 (a) (z-a) +

Laurent‟s Series

If f (z) is analytic in the ring- shaped region R bounded by two concentric circles C and C1 of

radii r & r1 ( r > r1) and with centre at a, then for all z in R

f (z) = a0 + a1 (z-a) + a2 (z-a)2 + ….+ a-1 (z-a)

-1 + a-2 (z-a)

-2 + ….

Where

Γ being any curve in R, encircling C1.

A zero of an analytic function f (z) is that value of z for which f (z) = 0

Singularities of analytic functions

(i) Isolated singularity

If z = a is a singularity of f (z) such that f (z) is analytic at each point in its

neighbourhood, then z = a is called isolated singularity.

(ii) Removable singularity

If exists finitely, then z = a is a removable singularity.

(iii) Poles

If all the negative powers of (z-a) in f (z) after the nth are missing then the singularity at z

= a is called a pole of order n

f (z) = a0+ a1(z-a) + a2 (z-a)2+ …. + a-1 (z-1)

-1 + ….

A pole of first order is called a simple pole.

(iv) Essential Singularity




If the number of negative powers of (z-a) in f (z) is infinite then z = a is called an

essential singularity

In this case, does not exist.

Residues

The coefficient of (z-a)-1

in the expansion of f (z) around an isolated singularity is called the

residue of f(z) at that point

Residue theorem

If f (z) is analytic in a closed curve C except at a finite number of singular points within C, then

x (sum of the residues at the singular points within C).

Calculation of residues

1. If f (z) has a simple pole at z = a then

Res f (a) =

2. Res f (a )

Where ψ (z) = (z-a) F(z), F(a) ≠ 0.

3. If f (z) has a pole of order n at z = a then

Res f (a) =




14. LAPLACE TRANSFORMS

Let f (t) be a function of t defined for all positive values of t. Then the Laplace transforms of f

(t) denoted by L { f (t)} is defined by

L { f (t)} =

Provided that the integral exists.

If f (t) = L-1

{ f (s)} then f (t) is called the inverse Laplace transform of f (s).

Important Laplace transforms

L (1) =

L (tn) =

L (eat) =

L (Sin at ) =

L (cos at) =

L (sin h at)=

L (cos h at ) =

L (e at t

n) =

L (eat Sin bt) =

L (eat Cos bt) =

L (eat Sinh bt) =

L (eat Cosh bt) =

Properties

1. Linearity property

If a, b, c be any constants and f, g, h any functions of t then

L [ a f (t) + b g (t) – c h (t)] = aL { f (t)} + bL { g (t)} – c L [ h (+1)}




2. First shifting property

If L { f (t)} = then

L { eat f (t)} =

3. Change of scale property

If L {f (t)} = then

L { f(at)} =

If f (t) is continuous and is finite, then the Laplace transform of f(t) i.e.

exists for s > a

If f1 (t) be continuous and L {f (t)}= f(s) then L {f

1(t)}

If f1 (t) and its first (n-1) derivatives be continuous then L {f

n (t)}

If L { f (t)} =

If L {f (t)} = then

L { tn f (t) } where n = 1, 2, 3, ….

If L { f (t)} = then

ds provided integral exists.

Inverse transforms




Convolution theorem

If L-1

{ } = f (t) and L-1

{ } = g (t)

Then L-1

{ } =

If f (t) is a periodic function with period T, then




15. FOURIER TRANSFORMS

The fourier transform of f(x) is given by

F(s) =

The inverse fourier transform of F (s) is given by

The fourier sine transform of f(x) is 0 < x < ∞ is

o Inverse Fourier sine transform of Fs(s) is

The Fourier cosine transform of f (x) is 0 < x < ∞ is

Fc(s)=

o Inverse Fourier cosine transform of Fc(s) is

The finite fouirier sine transform of f(x) in 0 <x < c, is defined as

o The inverse finite fourier sine transform of Fs(n) is given by

The finite fourier cosine transform of f(x) in 0< x< c is defined as

o The inverse finite fourier cosine transform of Fc(n) is

Properties

1. Linear property

If F(s) & G(s) are fourier transforms of f(x) & g(x) respectively then

F[ a f(x) + b g(x)] = a F (s) + b G(s)

Where a & b are constants

2. Change of scale property

If F(s) is the complex Fourier transform of f(x), then

F { f (ax)}




3. Shifting property

If F(s) is the complex fourier transform of f(x), then

F { f(x-a)} = eisa

F(s)

4. Modulation theorem

If f(s) is the complex fourier transform of f(x) then

F{f(x) cos ax} =1/2 [ F(s+a) + F (s-a)]

Fs{ f(x) cos ax} = ½ [ Fs (s+a) + Fs (s-a)]

Fc{ f(x)sin ax} = ½ [ Fs (s+a) - Fs (s-a)]

Fs{ f(x) sin ax} = ½ [ Fs (s-a) – Fc (s+a)]

Convolution

The convolution of two functions f(x) and g(x) over the interval (- ∞, ∞) is defined as f * g =

The fourier transform of the convolution f(x) & g(x) is the product of their Fourier transforms

i.e. F {f(x) * g (x)} = F {f(x)} . F {g (x)}

Parseval’s Identity

If the fourier transforms of f(x) & g(x) are F(s) & G(s) then

i)

ii) Where bar implies the complex conjugate

Parseval‟s identities for fourier sine & cosine transforms are

(i)

(ii)

(iii)

(iv)

Relation between Fourier & Laplace transforms

If f (t) = e-xt

g (t), t > 0 then

= 0, t < 0

F {f (t)} = L { g (t)}

The Fourier transform of the nth derivative of f(x) is




Inverse Laplace transforms by method of Residues

= Sum of the residues of at the poles of f(s)

6. Z-TRANSFORMS

If the function un is defined for discrete values (n = 0, 1, 2, ….) and un = 0 for n < 0, then its z-

transform is defined to be

whenever the infinite series converges

The inverse Z-transform is written as

Z-1

[ U(z)] = un

Standard Z- transforms

Sequence un (n≥0) Z- transform U(z)

1

-1

K

n

n2

np

δ (n) = 1, n = 0 1

= 0, n ≠ 0

u (n)= 0, n <0 Z/ Z-1

= 1 n ≥ 0

an

n an

n2 a

n

Sin n θ




Cos nθ

an sin nθ

an cos n θ

Sin hnθ

Cos hnθ

an sinhnθ

an cos hnθ

an un U (Z/a)

un+1 z [ U (z) – u0]

un+2 Z2 [ U (z)- u0- u1z

-1]

un+3 z3 [ U(z) – u0-u1 z

-1-u2 z

-2]

un-k z-k

U (z)

n un - z d/dz [ U (z)]

u0

Properties

1. Linearity property

If a, b, c be any constants and un, vn, wn be any discrete functions, then

Z(a un+bv n-cwn) = aZ(un) + bZ (vn) – cZ(wn)

2. Damping rule

If z(un) = U(z) then z(a-n

un) = U (az)

Z (an un)= U (z/a)

3. Shifting un to the right

If z (un) = U (z) then Z (un-k) = z-k

U(z), (k > 0]

4. Shifting un to the left

If z(un) = U(z) then

Z (un+k) = zk [ U(z)– u0-u1 z

-1- u2 z

-2 …. -uk-1 z

-(k-1)]

5. Multiplication by n

If z(un) = U(z) then Z (n un) = - z dU(z)/dz.




Initial value theorem:

If z(un) = U(z), then u0 =

Final value theorem:-

If z(un) = U(z) then

Some standard inverse Z transforms

U(z) Inverse z- transform un

an u (n)

(n+1)an u (n)

½! (n+1) (n+2) an u (n)

an-1

u (n-1)

(n-1) an-2

u (n-2)

½ (n-1) (n-2) an-3

u (n-3)

Convolution theorem

If Z-1

[ U (z)] = un and z-1

[ V(z)] = vn then

= un *vn

The region of the z-plane in which U(z) converges absolutely is known as region of

convergence (ROC) of U(z).

For a right sided sequence the ROC is |z| > |a|

.0 z =a

→

For a left handed sequence the ROC is |z| < |b|




b

0 →

For a two-sided sequence ROC is |b| < |z| < |c|

0

b c




17. PROBABILITY & STATISTICS

Permutations

The number of permutations of n different things taken r at a time is

n (n-1) (n-2)… (n-r+1)= nPr =

The no. of circular permutations formed with „n‟ objects is (n-1)!

If the direction is not specified or considered then no. of circular permutations is

The no. of permutations of n objects of which n1 are alike, n2 are alike and n3 are alike is

Combinations (or) Selections

The number of combinations of n different objects taken r at a time is

ncn-r = ncr

2n objects can be divided into two equal groups in

(m+n+p) objects can be divided into three groups of m objects, n objects, p objects in

ways.

The number of straight lines drawn through „n‟ points on a circle is nc2.

The no. of diagonal of a polygon with n vertices is

The no. of triangles formed by joining vertices of a polygon with n vertices is nc3 =

Basic Terminology

Exhaustive events;

A set of events is said to be exhaustive if it includes all the possible events.

Mutually Exclusive events:

If the occurrence of one of the events procludes the occurrence of all others, then such a set of

events is said to be mutually exclusive

Equally Likely events:

If one of the events cannot be expected to happen in preference to another then such events are

said to be equally likely.

Odds in favour of an event:




If the number of ways favourable to an event A is m and the no. of ways not favourbale to A is n

then odds in favour of A = m/n

Odds against A = n/m

Probability

If there are n exhaustive, mutually exclusive & equally likely cases of which m are favourable to

an event A, then probability (P) of happening of A is

P (A) = m/n

Chance of A not happening is q or P (A1)

P (A) + P (A

1) = 1 always

If an event is certain to happen then its probability is unity.

If an event is certain not to happen then its probability is zero.

Statistical (or empirical) definition

If in n trials, an event A happens m times, then the probability (p) of happening of A is given by

P = p (A) =

Random experiment:

Experiments which are performed essentially under the same conditions & whose results cannot be

predicted are known as random experiments.

Sample Space:

The set of all possible outcomes of a random experiment is called sample space for that

experiment(s).

The elements of the sample space S are called the sample points.

Event

The outcome of a random experiment is called an event.

Every subset of a sample space S is an event.

The null set φ is also an event & is called an impossible event.

Probability of an impossible event is zero i.e. P (φ) = 0.

Axioms

(i) The numerical value of probability lies between 0 & 1.

i.e. for any event A of S, 0≤ p(A) ≤1.

(ii) The sum of probabilities of all sample events is unity i.e. p (s) = 1

(iii) Probability of an event made of two or more sample events is the sum of their probabilities

Notations

(i) Probability of happening of events A or B is written as P (A+B) or P (AUB)

(ii) Probability of happening of both the events A & B is written as P (AB) or P (A∩B).

(iii) „Event A implies ( ) event B‟ is expressed as A B.

(iv) „Event A & B are mutually exclusive‟ is expressed as A∩B = φ




For any two events A & B

P (A∩B1) = P(A)- P (A ∩B)

P (A1∩B) = P (B) – P (A ∩B)

Addition Law of probability (Theorem of Total Probability)

If the probability of an event A happening as a result of a trial is P (A) and the probability of a

mutually exclusive event B happening is P (B), then the probability of either of the events

happening as a result of the trial is P(A+B) or P ( A U B) = P (A) + P (B)

If A, B are any two events (not mutually exclusive) then

P (A+B) = P (A) + P (B)- P(AB)

i.e. P(A UB) = P(A) + P(B) – P(A∩B)

Independent events

Two events are said to be independent, if happening or failure of one does not affect the

happening or failure of the other.

Otherwise the events are said to be dependent.

Conditional probability

For two dependent events A & B, the symbol P (B/A) denotes the probability of occurrence of B,

when A has already occurred. It is known as the conditional probability and is read as a probability

of B given A`.

Multiplication Law of Probability (Theorem of compound probability)

If the probability of an event A happening as a result of trial is P (A) and after A has happened the

probability of an event B happening as a result of another trial (i.e. conditional probability of B

given A) is P (B/A) then the probability of both the events A & B happening as a result of two

trials is P (AB) or P(A ∩B) = P (A). P (B/A)

The conditional probability of A given B is P (A/B) then

P (A ∩B) = P (B). P (A/B)

If the events A& B are independent i.e. if the happening of B does not depend on whether A

has happened or not, then P (B/A) = P (B) & P(A/B)= P (A).

Therefore P (AB) or P (A∩B) = P (A) . P (B)

If P1, P2 be the probabilities of happening of two independent events, then

(i) The probability that the first event happens & the second fails is P1 (1-P2)

(ii) The probability that both events fail to happen is (1-P1) (1-P2)

(iii) The probability that atleast one of the events happens is 1- (1-P1) (1-P2). This is commonly

known as their cumulative probability

If P1, P2, P3… Pn be the chances of happening of n independent events, then their cumulative

probability is

1- (1-p1) (1-p2) (1-p3)… (1-pn)

Baye’s theorem

An event A corresponds to a number of exhaustive events B1, B2….Bn If P (Bi) and P (A/Bi) are

given then




Random Variable

If a real variable x be associated with the outcome of a random experiment, then since the values

which x takes depend on chance, it is called a random variable or stochastic variable or simply a

variate.

If a random variables takes a finite set of values, it is called a discrete variate.

If it assumes an infinite number of uncountable values, it is called a continuous variate.

Discrete Probability Distribution

If the probability that X takes the values xi, is pi then

P (x= xi)= Pi or p (xi) for i = 1, 2

Where (i) p (xi) ≥ 0 for all values of i,

(ii) ∑ p(xi) = 1

The set of values xi with their probability Pi constitutes a discrete probability distribution of

the discrete variable X.

The distribution function F(x) of the discrete variate X is defined by

F(x) = P (X ≤x) = P (xi) where x is any integer.

The distribution function is also sometimes called cumulative distribution function.

Continuous Probability Distribution

The probability distribution of a continuous variate x is defined by a function f(x) such that the

probability of the variate x falling in the small interval x-1/2 dx to x+1/2 dx is f(x) dx

i.e. P (x- ½ dx ≤ x ≤ x+1/2 dx) = f (x) dx

* f (x) is called the probability density function

* the continuous curve y = f (x) is called probability curve

* the density function f (x) is always positive &

If F(x) = P (X ≤x) = then F(x) is defined as the cumulative distribution

function or distribution function of continuous variate X.

F1(x) = f(x) ≥ 0, F(x) is a non-decreasing function.

F(- ∞) = 0 ;

F (∞) = 1

= F(b) – F(a)

Expectation

The mean value (μ) of the probability distribution of a variable X is known as its expectation

, discrete distribution

, continuous distribution




Variance

=

ζ is the standard deviation of the distribution

The rth moment about mean (denoted by μr) is defined by

μr = ∑ (xi- μ)r f (xi), discrete

=

Mean deviation from the mean is given by

∑ |xi- μ| f(xi), discrete

The probability of r success is ncr P

r q

n-r

The probability of atleast „r‟ successes in n trials

= ncr Pr q

n-r + ncr+1 P

r+1 q

n-r-1 + …. + ncn P

n

Binomial Distribution

If we perform a series of independent trials such that for each trial P is the probability of a

success and q that of a failure, then the probability of r successes in a series of n trials is given by P (x

=r) =ncr Pr q

n-r

The probability of the number of successes obtained is called the binomial distribution

Mean = np

Standard deviation = √(npq)

Variance = npq

Skewness =

Poisson Distribution

It is a distribution related to the probabilities of events which are extremely rare, but

which have a large number of independent opportunities for occurrence.

Mean = λ

Standard deviation = √ λ

Variance = λ

Skewness = 1/ λ

Normal Distribution

The normal curve is of form




where

Mean deviation from the mean (μ)

Moments about mean

μ2n+1 = 0

μ2n = (2n-1) (2n-3) … 3.1ζ2n

Coefficient skewness is zero

Mean

If x1, x2, x3…. Xn are a set of n values of a variate, then the arithmetic mean (or simply mean) is

given by

In a frequency distribution if x1, x2, …. xn be the mid values of the class intervals having

frequencies f1, f2….. fn respectively we have

Median

If the values of a variable are arranged in the ascending order of magnitude, the median is the

middle item if the number is odd and is the mean of the two middle items if the number is even

Median =

Where L = Lower limit of the median class.

N= total frequency

f= frequency of the median class

h= width of the median class

C= cumulative frequency upto the class preceding the median class

Mode

The mode is defined as that value of the variable which occurs most frequently, i.e, the value of

the maximum frequency

Mode =

Where L = Lower limit of class containing mode.

∆1 = excess of modal frequency over freq of the preceding class.

∆2 = excess of modal freq over following class.

h= size of modal class.

Curves having a single mode are termed as unimodal, those having two modes as bi-modal

and those having more than two modes as multi-modal.




Mean-Mode = 3 (Mean-Median)

Geometric mean

If x1 x2, …. xn are a set of n observations then the geometric mean is given by

G.M= (x1 x2… xn) 1/n

Harmonic mean

If x1, x2, …. Xn be a set of n observations, then the harmonic mean is defined as the reciprocal of

the (arithmetic) mean of the reciprocal of the quantities

Standard Deviation (ζ)

Correlation

When the changes is one variable are associated or followed by changes in the other, is called

correlation.

Data connecting such two variables is called bivariate population.

If an increase (or decrease) in the values of one variable corresponds to an increase (or

decrease) in the other, the correlation is said to be positive.

If the increase (or decrease) in one corresponds to the decrease (or increase) in the other, the

correlation is said to be negative.

If there is no relationship indicated between two variables they are said to be independent or

uncorrelated.

The numerical measure of correlation is called the coefficient of correlation and is defined by

the relation

Where x = deviation from the mean

Y = deviation from the mean

ζx = S.D. of x. series

ζy = S.D. of y series

n = no. of values of two variables

Lines of Regression

A line of best fit for the given distribution of dots is called the line of regression.

If there are two lines, such that one giving the best possible mean values of y for each

specified value of x and the other giving the best possible values of x for given values of y.

The former is known as the line of regression of y on x and the latter as the line of regression

of x on y.

The line of regression of y on x is




o Slope is called regression coefficient = r ζy/ζx

The line of regression of x on y is

o Regression coefficient = r ζx/ζy

The correlation coefficient r is the geometric mean between the two regression co-efficients

i.e. we have

* The acute angle between two regression lines is

When (1) r= 0, θ = π/2, the two lines of regression are perpendicular to each other.

(2) r = ±1, θ= 0 or π, the lines of regression coincide and there is perfect correlation between

variables x and y.

Rank Correlation

A group of n individuals may be arranged in order of merit with respect to some characteristic.

The same group would give different orders for different characteristics. Considering the orders

corresponding to two characteristics A and B, the correlation between these n pairs of ranks is

called the rank correlation in the characteristics A and B for that group of individuals

Rank correlation coefficient

Where di = xi-yi




18. NUMERICAL METHODS

Solution of Algebraic and Transcendental equations

To find the roots of an equation f (x) = 0, we start with a known approximate solution and apply

any of the following methods.

1. Bisection method:

If f(x) is continuous between a & b, and f(a) & f(b) are of opposite signs then there is a root

between a & b.

Let f (a) be-ve & f(b) be +ve, then the first approximation to the root is x1 = ½ (a+b)

If f(x1)= 0, then x1 is a root of f(x) = 0.

Otherwise, the root lies between a & x1 or x1 & b according as f (x1) is positive or negative.

Then we bisect the interval as before and continue the process until the root is found to desired

accuracy.

2. Method of false position (or) Regular-falsi method (i)




Here we choose two points x0 & x1 such that f(x0) and f(x1) are of opposite of signs i.e., the graph

of y = f(x) crosses the x-axis between these points.

This indicates that a root lies between x0 & x1 consequently f (x0) f (x1) < 0

Equation of the chord joining the points A[x0, f(x0)] & B[ x1, f(x1)] is

The method consisting in replacing the curve AB by means of the chord AB and taking the

point of intersection of the chord with the x-axis as an approximation to the root.

3. Newton Raphson Method

(Order of convergence = 2)

Secant method (1.62)

Iterative formula to find 1/N is xn+1 = xn (2- Nxn)

Iterative formula to find√ N is xn+1 = ½ (xn + N/xn)

Iterative formula to find 1/√ N is xn+1 = ½ (xn + 1/N xn)

Iterative formula to find is

Solution of Non linear simultaneous equations = Newton Raphson method

Consider the equations f(x,y) = 0, g (x, y) = 0. If an initial approximation (x0 y0) to a solution has

been found by graphical method or otherwise, then a better approximation (x1, y1) as

X1 = x0+h, y1 = y0+K so that

F(x0 + h, y0+k) = 0 & g(x0 + h, y0 +K)= 0

Finite differences

Suppose we are given the following values of y = f(x) for a set of values of x:

X: x0 x1 x2 … xn

Y: y0 y1 y2 … yn

Then the process of finding the values of y corresponding to any value of x = xi between x0 & xn

is called interpolation

Interpolation is the technique of estimating the value of a function for any intermediate value

of the independent variable.

The process of computing the value of the function outside the given range is called extra

polation.




1. Forward differences

The difference y1-y0, y2-y1, …yn-yn-1 when denoted by ∆y0, ∆y1, … ∆yn-1 respectively are

called the first forward differences where ∆ is the forward difference operator.

i.e. first forward differences are

∆yr = yr+1 - yr

∆2 yr = ∆yr+1 - ∆yr

∆p yr = ∆

P-1 yr+1 - ∆

P-1 yr defined p

th forward difference.

Forward Difference Table

Value of x Value of y 1st diff 2

nd diff 3

rd diff

X0 y0

∆y0

X0+h y1 ∆2y0

∆y1 ∆3y0

X0+2h y2 ∆y2 ∆2y1

X0+3h y3 ∆3y1

∆2 y2

∆y3

X0 +4h y4

2. Backward differences

The differences y1-y0, y2- y1, ….. yn – yn-1

When denoted by y1, y2, … yn respectively, are called the first backward difference

where is the backward difference operator

yr = yr – yr-1

2 yr = yr - yr-1,

3 yr =

2 yr-

2 yr-1 etc.

Backward difference table


nd diff 3

rd diff

X0 y0

y1

2y2

X0+h y1 3y3

y2

2y3

3y4

X0+2h y2

y3

2y4

X0+3h y3

y4

X0+ 4h y4




3. Central differences

The central difference operator δ is defined by y1-y0= δy1/2 , y2-y1 = δy3/2 …. yn- yn-1

= δyn-1/2

δy3/2 – δy1/2 = δ2 y1, δy5/2 – δy3/2 = δ

2 y2, …..

δ2y2 – δ

2y1 = δ

3y3/2 and so on.

Central difference table


nd diff 3

rd diff

X0 y0

δy1/2

δ2y1

X0+h y1 δ3y3/2

δy3/2

X0+2h y2 δ2y2

δy5/2 δ3y5/2

δ2y3

X0+3h y3

δy7/2

X0+ 4h y4

Newton‟s forward interpolation formula

Where

Newton‟s backward interpolation formula

Where

Central difference Interpolation Formulae

x y 1st diff 2

nd diff 3

rd diff

x0-2h y-2

∆y-2 (=δy-3/2)

∆2y-2(=δ

2y-1)

∆3y-2 (=δ

3y-1/2)

X0-h y-1

∆y-1 (=δy-1/2)

X0 y0 ∆2y-1 (=δ

2y0)

∆y0 (=δy1/2) ∆3y-1(=δ

3y1/2)

X0+h y1 ∆2y0(= δ

2y1)

∆y1 (=δy3/2)

X0 +2h y2




1. Gauss‟s forward interpolation formula

….

In the central difference notation

2. Gauss‟s backward interpolation formula

+…..

3. Stirling‟s formula

4. Bessel‟s Formula

5. Everett‟s formula

Where P=1-q

Interpolation with Unequal intervals

1. Lagrange‟s formula

If y = f(x) takes the values y0, y1, …. Yn corresponding to x = x0, x1, … xn then

F(x)= (x-x1) (x-x2)… (x-xn) y0 + (x-x0) (x-x2)… (x-xn)

(x0-x1) (x0-x2).. (x0 –xn) (x1-x0) (x1-x2)… (x1-xn)

y1+ …+ (x-x0) (x-x1)… (x-xn-1) yn

(xn-x0) (xn-x1)… (xn-xn-1)

2. Divided difference

If (x0, y0), (x1, y1), (x2, y2)… be given points, then the first divided differences for the arguments

x0, x1 is defined by the relation




3. Newton‟s divided difference formula

Y = f(x) = y0 + (x-x0) [ x0, x1] + (x-x0) (x-x1) [ x0, x1, x2] +….

Where [x0, x1] = and so on

Numerical Integration

The process of evaluating a definite integral from a set of tabulated values of the integral f(x) is

called numerical integration

This process when applied to a function of a single variable is known as quadrature.

1. Newton- Cote‟s quadrature formula

n = 1, 2, 3,

2. Trapezoidal rule

Putting n = 1 is Newton-cote‟s quadrature formula

[ (y0+yn)+2 (y1+y2+…+yn-1)]

3. Simpson‟s one-third rule

Putting n = 2

[ (y0+yn)+4 (y1+y3+ …+yn-1) +2 (y2+y4+…+yn-2)]

4. Simpson‟s three eighth rule

Putting n = 3

[ (y0+yn)+3 (y1+y2+y4+y3+…..+yn-1)+2 (y3+y6+…+yn-3)]

5. Weddle‟s rule

Putting n = 6




(y0+5y1+y2+6y3+y4+5y5+2y6+5y7+y8+….)

Numerical Solution of Ordinary Differential equations

Methods for finding the solution of first order differential equations of form

dy/dx = f (x, y), given y (x0) = y0 are as follows

1. Picard‟s Method

First approximation y1 to the solution is

y=y0 +

y1= y0 +

Second approximation y2 = y0 +

Third approximation y3= y0 + and so on.

2. Taylor series Method

y(x) = y0 + (x-x0) (y1)0 +

3. Euler‟s Method

dy/dx = f (x, y), y (x0) = y0

y1 = y0 + h f (x0, y0)

y2 = y1 + hf (x0+h y1)

:

:

yn = yn-1 +h f (x0 + h, yn-1)

4. Modified Euler‟s Method

Y1(1)

= y0 + h/2 [ f (x0, y0)+ f (x0 +h, y1)]

Y1(2)

= y0 + h/2 [ f (x0, y0) + f (x0+h, y1(1)

)]

Y2(1)

= y1+h/2[ f (x0+h, y1)+ f (x0+2h, y2)]

5. Runge‟s method

Procedure

Calculate successively

K1 = h f (x0, y0)

K2 = h f (x0+ ½ h, y0+ ½ K1)

K1 = h f (x0 + h, y0+K1)

K3 = h f (x0 + h, y0 + K1)

Finally K = 1/6 (K1+4K2+ K3)




y = y0 +K is the solution

6. Runge-Kutta method

Procedure

Calculate successively

K1 = h f (x0,y0)

K2 = h f (x0+ ½ h, y0+ ½ K1)

K3 = h f (x0 +1/2 h, y0+1/2 K2)

K4 = h f (x0 + h, y0 + K3)

Finally K = 1/6 (K1+2K2+ 2K3+K4)

y1 = y0 +K is the solution

7. Predictor-corrector methods

(i) Milne‟s method

Given dy/dx = f (x,y) and y = y0 when x = x0; to find an approximate value of y for x =

x0+ nh by Milne‟s method, proceed as follows

y0 = y(x0) being given, we compute

y1 = y (x0+h), y2 =(x0+2h), y3 = y (x0+3h) by picard‟s or Taylor‟s series method

Next we calculate

f0 = f(x0 y0), f1 = f (x0+h, y1), f2 = f (x0 + 2h, y2), f3 = f (x0 +3h, y3),

then y4 = y0 + 4h/3 (2f1- f2+2f3) called predictor

y4 = y2 + h/3 (f2+4f3+f4) called corrector

y5 = y (x0 +5h) = y1+ 4h/3 (2f2-f3+2f4) Predictor

y5 = y3+ h/3 (f3+4f4+ f5) and so on

(ii) Adams- Bashforth method:

Given dy/dx = f (x, y) & y0 = y (x0) we compute

y-1= y (x0-h), y-2 = y (x0- 2h), y-3= y (x0-3h)

by Taylor‟s series or Euler‟s method or Runge- Kutta method

Next we calculate

f-1 = f (x0-h, y-1), f-2 = f (x0-2h, y-2) f-3 f(x0-3h, y-3)

y1 = y0 + h/24 (55f0- 59f-1 +37f-2 -9f-3)

Adams- Bashforth predictor formula

y1= y0+ h/24 [ 9f1+19f0-5f-1+f-2)

Correctors‟ formula.




GATE OLD QUESTION PAPERS

GATE-2005 One Mark Questions

1. The following differential equation has

a. degree = 2, order = 1 b. degree= 1, order = 2

c. degree= 4, order = 3 d. degree= 2, order = 3

2. A fair dice is rolled twice. The probability that an odd number will follow an even number is

a. b.

c. d.

3. A solution of the following differential equation is given by

a. y = e2x

+ e-3x

b. y = e2x

+ e3x

c. y = e-2x

+ e3x

d. y = e-2x

+ e-3x

GATE- 2005 Two Marks Questions

4. In what range should Re(s) remain so that the Laplace transform of the function e(a+2)t+5

exits.

a. Re (s) > a+2 b. Re (s) > a+7

c. Re (s) < 2 d. Re (s) > a + 5

5. Given the matrix , the eigen vector is

a. b.

c. d.

6. Let and Then (a+b) =




a. b.

c. d.

7. The value of the integral is

a. 1 b. π

c. 2 d. 2 π

8. The derivative of the symmetric function drawn in given figure will look like

→

↑

a. →

↑

b. →

↑

c.

→

↑

d.

→

↑

9. Match the following and choose the correct combination

Group-I Group-II

E. Newton-Raphson method 1. Solving nonlinear equations

F. Rung-kutta method 2. Solving linear simultaneous equations

G. Simpson‟s Rule 3. Solving ordinary differential equations

H. Gauss elimination 4. Numerical integration

5. Interpolation




6. Calculation of Eigen values

a. E-6, F-1, G-5, H-3 b. E-1, F-6, G-4, H-3

c. E-1, F-3, G-4, H-2 d. E-5, F-3, G-4, H-1

10. Given an orthogonal matrix

a. b.

c. d.


11. The rank of the matrix is

a. 0 b. 1

c. 2 d. 3

12. x x P, where P is a vector is equal to

a. P x x P - 2 P b.

2 P + ( x P)

c. 2 P + x P d. ( .P)-

2 P

13. ,where P is a vector, is equal to

a. P. d b. x xP. d

c. x P. d d. ∫∫∫ .P dv

14. A probability density function is of the form

p(x) = Ke-α|x|

, x (-∞,∞)




The value of K is

a. 0.5 b. 1

c, 0.5α d. α

15. A solution for the differential equation (t) + 2x(t) = δ (t) with initial condition x(0-) = 0 is

a. e-2t

u(t) b. e2tu(t)

c. e-t u(t) d.e

t u(t)

16. A low-pass filter having a frequency response H(jω) = A (ω) ejφ(ω)

does not produce any phase

distortion if

a. A (ω) = Cω2, φ (ω) = kω

3

b. A (ω) = Cω2, φ (ω) = kω

3

c. A (ω) = Cω, φ (ω) = kω2

d. A (ω) = C, φ (ω) = kω-1

GATE-2006 Two Marks Questions

17. The eigen values and the corresponding eigen vectors of a 2 x 2 matrix are given by

Eigen value Eigen vector

λ1 = 8

λ2 = 4

The matrix is

a. b.

c. d.

18. For the function of a complex variable W = In z (where, W = u + jv and Z = x + jy, the u =

constant lines get mapped in Z-plane as

a. set of radial straight lines b. set of concentric circles

c. set of confocal hyperbolas d. set of confocal ellipses

19. The value of the contour integral |z-j|=2 dz in positive sense is

a. j π/2 b. - π/2




c. -jπ/2 d. π/2

20. The integral sin3 θ d θ is given by

a. 1/2 b. 2/3

c. 4/3 d. 8/3

21. Three companies X, Y and Z supply computers to a university. The percentage of computers

supplied by them and the probability of those being defective are tabulated below

Company % of computers Probability of being

Supplied defective

X 60% 0 .01

Y 30% 0.02

Z 10% 0.03

Given that a computer is defective, the probability that it was supplied by Y is

a. 0.1 b. 0.2

c. 0.3 d. 0.4

22. For the matrix the eigen value corresponding to the eigen vector

a. 2 b. 4

c. 6 d. 8

23. For the differential equation the boundary conditions are

(i) y = 0 for x = 0 and (ii) y = 0 for x = a

The form of non-zero solutions of y (where m varies over all integers) are

a. b.

c. d.

24. As x increased from - ∞ to ∞, the function f (x) =

a. monotonically increases




b. monotonically decreases

c. increases to a maximum value and then decreases

d. decreases to a minimum value and then increases


25. The following plot shows a function y which varies linearly with x. The value of the integral I

= y dx is

X

Y

1 2 3

1

2

3

-1

a. 1.0 b. 2.5

c. 4.0 d. 5.0

26. For |x| << 1, coth (x) can be approximated as

a. x b. x2

c. 1/x d. 1/x2

27.

a. 0.5 b. 1

c. 2 d. not defined

28. Which one of the following is stricity bounded?

a. 1/x2 b. e

x

c. x2 d. e

-x2

29. For the function e-x

the linear approximation around x = 2 is

a. (3-x) e-2

b. 1-x

c. [3+2√ 2- (1+ √ 2) x] e-2

d. e-2





30. It is given that X1, X2…. XM are M non-zero orthogonal vectors. The dimension of the vector

space spanned by the 2M vectors X1, X2 … XM, – X1, - X2,… - XM is

a. 2M

b. M+1

c. M

d. dependent on the choice of X1, X2… XM

31. Consider the function f(x) = x2 – x-2. The maximum value of f(x) in the closed interval [-4, 4]

is

a. 18 b. 10

c. -2.25 d. indeterminate

32. An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in

Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability

of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is

a. 0.5 b. 0.18

c. 0.12 d. 0.06

33. The solution of the differential equation under the boundary conditions

(i) y = y1 at x= 0 and

(ii) y = y2 at x = ∞, where k, y1 and y2 are constant is

a. y = (y1 – y2) exp(-x/k2) + y2

b. y = (y2 – y1) exp (-x/k) + y1

c. y = (y1 – y2) sin h(x/k) + y1

d. y = (y1-y2) exp (-x/k) + y2

34. The equation x3- x

2 + 4x -4 = 0 is to be solved using the Newton- Raphson method. If x = 2 is

taken as the initial approximation of the solution, then the next approximation using this method

will be

a. 2/3 b. 4/3

c. 1 d. 3/2

35. Three functions f1 (t), f2(t) and f3(t) which are zero outside the interval [0, T] are shown in the

figure. Which of the following statements is correct?




a. f1(t) and f2 (t) are orthogonal

b. f1(t) and f3(t) are orthogonal

c. f2(t) and f3(t) are orthogonal

d. f1(t) and f2(t) are orthonormal

36. If the semi-circular contour D of radius 2 is as shown in the figure. Then the value of the

integral is

→

↑

0

→

↑↓

↓

D

j

02

-j2

a. j π b. – j π

c. – π d. π





37. All the four entries of the 2 x 2 matrix are nonzero, and one of its eigen

values is zero. Which of the following statements is true?

a. p11 p22 – p12 p21 = 1 b. p11 p22 – p12 p21 = -1

c. p11 p22 – p12 p21 = 0 d. p11 p22 + p12 p21 = 0

38. The system of linear equations

4x+ 2y = 7

2x+y = 6

has

a. a unique solution

b. no solution

c. an infinite number of solutions

d. exactly two distinct solutions

39. The equation sin (z) = 10 has

a. no real or complex solution

b. exactly two distinct complex solutions

c. a unique solution

d. an infinite number of complex solutions

40. For real values of x, the minimum value of the function f(x)= exp (x) + exp (-x) is

a. 2 b. 1

c. 0.5 d. 0

41. Which of the following functions would have only odd powers of x in its Taylor series

expansion about the point x = 0?

a. sin (x3) b. sin (x

2)

c. cos (x3) d. cos (x

2)

42. Which of the following is a solution to the differential equation

a. x(t) = 3e-t b. x(t) = 2e

-3t

c. x(t) = (-3/2) t2 d. x(t) = 3t

2





43. The recursion relation to solve x = e-x

using Newton Raphson method is

a. b. c.

d.

44. The residue of the function f(z) =

a. b.

c. d.

45. Consider the matrix The value of eP is

a.

b.

c.

d.

46. In the Taylor series expansion of exp(x) + sin (x) about the point x = π, the coefficient of (x-

π)2 is

a. exp (π) b. 0.5 exp (π)

c, exp (π) + 1 d. exp (π) -1




47. The value of the integral of the function g(x,y) = 4x3 + 10y

4 along the straight line segment

from the point (0,0) to the point (1,2) in the x-y plane is

a. 33 b. 35

c. 40 d. 56

48. Consider points P and Q in the x-y plane, with P = (1, 0) and Q= (0,1). The line integral

along the semicircle with the line segment PQ as its diameter

a. is-1

b. is 0

c. is 1

d depends on the direction (clockwise or anticlockwise) of the semicircle

GATE-2009 One Mark questions

49. The order of the differential equation + y 4 = e

-t is

a. 1 b. 2

c. 3 d. 4

50. A fair coin is tossed 10 times. What is the probability that Only the first two tosses will yield

heads?

a. b.

c. d.

51. If f(z) = c0 + c1z-1

, then dz is given by

a. 2 πC1 b. 2π (1+C0)

c. 2 πjC1 d. 2 πj (1+C0)


52. The Taylor series expansion of at x = π is given by

a. b.




c. d.

53. Match each differential equation in Group I to its family of solution curves from Group II

Group I Group II

a. 1. Circles

b. 2. Straight lines

c. 3. Hyperbolas

d.

a. A-2, B-3, C-3, D-1 b. A-1, B-3, C-2, D-1

c. A-2, B-1, C-3, D-3 d. A-3, B-2, C-1, D-2

54. The eigen values of the following matrix are

a. 3, 3 + 5j, 6-j b. -6, + 5j, 3 + j, 3-j

c. 3 + j, 3-j, 5 + j d. 3, -1 + 3j, -1-3j




ANSWERS & EXPLANTIONS

1. (b)

Order is highest derivative term. Degree is power of highest derivative term.

2. (d)

Since both events are independent of each other.

P(odd/even) =

3. (b)

A.E. D2 – 5D + 6 = 0

(D-2) (D-3) = 0

D = 2, 3

Therefore y = e2x

+ e3x

4. (a)

f(t)= e(a+2) t+5

= e5.e

(a+2)t

Therefore for L.T. to exists, Re (s) > a + 2

5. (c)

A- λI = 0

(-4- λ) (3- λ) -8 = 0

-12 + 4 λ-3 λ + λ2- 8 = 0

λ2 + λ – 20 = 0




(λ +5) (λ-4) = 0

Therefore λ = -5, 4

Let eigen vector be

Putting λ1 = - 5

m1 + 2m2 = 0

Taking the value m1 = 2 & m2 = -1

Thus the eigen vector correspondence to eigen value λ1 = -5 is

Again, eigen vector X2 corresponding to eigen value λ2 = 4 is

-8m1 + 2m2 = 0

4m1 – m2 = 0

- 8m1 = - 2m2

m1 = - 4; m2 = - 1 m1 = -2, m2 = - 8

When λ = -5,




6. (a)

[AA-1

]= I

- 2a-0.1b=0, 3b = 1 b = 1/3

7. (a)

Comparing with

Here µ = 0

8. (c)

Given function has negative slope in +ve half and +ve slope in –ve half. So its differentiation

curve is satisfied by (c).




9. (C)

10. (c)

[AAT]

-1 = I

For orthogonal matrix AAT = I i.e. unity matrix

Inverse of I = I

11. (c)

R3 → R1- R3

Therefore Rank = 2

12. (d)

From vector triple product.

A x (B x C) = B (A.C)- C(A. B)

A = , B = , C=P

x x P = ( .P) – P ( . )= ( .P) - 2 P

13. (a)

∫∫ (∆ X P) ds = P.d (strokes Theorem)

14. (c)




2K= α

K = 0.5 α

15. (a)

(t) + 2x(t) = δ (t)

Taking L.T. on both sides

sX(s) –x(0) + 2X(s) = 1

X(s) [s+2] = 1

X(t) = e-2t

u(t)

16. (b)

For distortion less transmission

Phase response should be linear φ (ω) = Kω

17. (a)

[λI-A] = 0

(λ-6)2 – 4 = 0

λ = 8, 4

* By property of eigen matrix sum of diagonal elements should be equal to sum of values of λ.

18. (b)

W= nZ = loge z

u + jv= loge (x+jy) = ½ log (x2 + y

2) + itan

-1 (y/x)

u is constant




½ log (x2 + y

2) = C

X2 + Y

2 = C [equation of circle having same centre (0, 0)]

19. (d)

Polo (0,2) lies enside the circle |z-j| = 2

By Cauchy‟s integral formula.

| z-j| = 2

20. (c)

, Let cos θ = t

-sin θ d θ = dt, θ = 0, cos 0 = 1 = t

θ = π, cos π=-1 = t

=

I = (2/3) + (2/3) = 4/3

21. (d)

S→ supply by y d → defective




Probability that the computer was supplied by y, if the product is defective

P(s ∩d) = 0.3 x 0.02 = 0.006

P (d) = 0.6 x .01 + 0.3 x 0.02 + 0.1 x 0.03 = 0.015

22. (c)

Given eigen vector

(4- λ) (101) + 2(101) = 0 4- λ + 2 = 0

λ = 6

23. (a)

D2 + K

2 = 0

D = ± jK

Y = A cos Kx + jB sin kx

At x= 0, y = 0 A = 0, y = jB sin kx

x=a, y =0 0 = B sinka

B ≠ 0 else y = 0 always

Sinka = 0




24. (a)

At x → ∞, functions value increases.

25. (b)

Y = X + 1

= ½ (9-4) = 2.5

26. (c)

27. (a)

28.(d)

→

↑

x

y

→

2y= 1/x x

y= e

→x

y

→

↑

y=

↑y

→x→

↑

→

y

x →

y= 1/x2




→

↑

x

y

→

y= x -x2y= e

→x

y

→

↑

y=

↑y

→x→

↑

→

y

x →

y= 2

29. (a)

30. (c)

31. (a)

f (x) = x2 – x – 2

= (x+1) (x-2)

f(-4) = 18

f (+4) = 10

f(x) is maximum in interval [-4, 4] at x = -4

32. (c)

P (A) = 0.2 A→ failing in paper 1

P (B) = 0.3 B → failing in paper 2

P (A/B) = 0.6




Prop. of failing in both P (A∩B) = P(A/B) x P(B)

= 0.6 x 0.2 = 0.12

33. (d)

Complementary function

y = C1ex/k

+ C2 –x/k

……. (1)

X = 0; y = y1 = C1 + C2… (2)

X = ∞; y = y2

34. (b)

f (x) = x3 – x

2 + 4x-4

= 3x2 – 2x+4

f(2) = 8

f(2) = 12

35. (c)

Two functions f(x) & g(x) are said to be orthogonal if




=

= 0

36. (a)

= 2πj x (sum of residne)

Residue at pole s = -1 is 0

Residue at pole s = 1 is

37. (c)

Eigen values are the roots of the determinant formed by matrix [si-P]

[sI-P] = 0 (s-p11) (s-p22) – p12p21 = 0

S2 – (p11 + p22) s+p11p22 – p11 p22 – p12p21 = 0

Since, one of the its eigen values is zero, therefore, putting s = 0

P11p22 – p12p21 = 0

Which is the desired condition

38. (b)

The system can be written in matrix from as

The Augmented matrix [A | B] is given by

2

2 1

4

6

7

Performing Gauss elimination on this [A|B] as follows:




2

2 1

4

6

7

_R

2

_

_R2

2

2_4

R1

1R1=

2

0 0

4

5/2

7

Now Rank [A|B]= 2

(The number of non-zero rows in [A|B]

Rank [A] =1

(The number of non-zero rows in [A])

Since Rank [A|B] ≠ Rank [A]

The system has no solution

39. (a)

Sin (z) = 10

Since maximum value of sin (z) = 1,

Therefore, the above equation has no real or complex solutions

40. (a)

f(x) = ex + e

-x

Arithmetic mean of ex and 1/e

x is

Geometric mean of ex and 1/e

x is

It is known that A.M. ≥ G.M.

Therefore, (ex + e

-x) min = 2

41. (a)




42. (b)

(D+3) X(t) = 0

D = - 3

So, x(t) = CeDt

= Ce-3t

43. (c)

The given equation to be solved is x = e-x

Which can be rewritten as

f (x) = x- e-x

= 0

= 1 + e-x

The Newton-raphson iterative formula is

44. (a)

Since is finite and non-zero, f(z) has a pole of order two at z = 2

The residue at z = a is given for a pole of order n as

Res f(a) =

Here n = 2 (pole of order 2) & a = 2

Res f (2)




= [-2 (z+2)-3

] z=2

46. (b)

f (x) = ex + sin x

we wish to expand about x = π

Taylor‟s series expansion about X = a is

Now about x = π

f (x) = f(π) + (x- π) (π) +

The coefficient of (x- π)2 is

Here f(x) = ex + sinx

(x) = ex + cosx

(x) = ex _sinx

(π) = eπ – sin π

= e

π – 0 = e

π

The coefficient of (x- π)2 is

47. (a)

Equation of straight line from point (0,0) to (1,2) is

Or

Y = 2x

G (x,y) = 4x3 + 10 y

4

= 4x3 + 10 (2x)

4

= 4x3 + 160x

4




= 1 + 32 = 33

48. (b)

=

=

49. (b)

Hieghest derivative of differential equation is 2

51. (d)

f(z)= c0 + c1 z-1

Unit circle

It has one pole at origine

So

52. (b)




Now

f‟ (π) = 0

(π) = -1/6

so the expansion is

f(x) = -1 + (-1/6) (x- π)2 +…..

53. (a)

A.

log y = log x + log c

= log cx

Y = cx….. Equation of straight line

B.

log y = -log x + log c

log yx = log c

yx = c

y = c/x….. Equation of hyperbola.

C.




y2 – x

2= c

2

D.

x2 + y

2 = c

2…. Equation afcircle

54. (d)

Sum of the eigen values are the sum of the principle diagonal element of the matrix

Sum of the diagonal current

= 3-1-1

=1

Sum of the eigen values

= 3-1 + 3j-1-3j

= 3-1-1

=1

Hence (d) option is correct.

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