dr. nirav vyas complexnumbers3.pdf
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Complex Function
N. B. Vyas
Department of Mathematics,Atmiya Institute of Tech. and Science,
Rajkot (Guj.)
N.B.V yas−Department of M athematics, AIT S − Rajkot (2)
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Curves and Regions in Complex Plane
Distance between two complex numbers
The distance between two complex numbers z 1 and z 2 is given by|z 1 − z 2| or |z 2 − z 1|
N.B.V yas−Department of M athematics, AIT S − Rajkot (3)
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Curves and Regions in Complex Plane
Distance between two complex numbers
The distance between two complex numbers z 1 and z 2 is given by|z 1 − z 2| or |z 2 − z 1|
Circles
A circle with centre z 0 = (x0, y0)C and radius pR+ isrepresented by |z − z 0| = p
N.B.V yas−Department of M athematics, AIT S − Rajkot (3)
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Curves and Regions in Complex Plane
Distance between two complex numbers
The distance between two complex numbers z 1 and z 2 is given by|z 1 − z 2| or |z 2 − z 1|
Circles
A circle with centre z 0 = (x0, y0)C and radius pR+ isrepresented by |z − z 0| = p
Interior and exterior part of the circle |z − z 0| = pThe set {zC, pR+/|z − z 0| < p} indicates the interior part of the circle |z − z 0| = p
N.B.V yas−Department of M athematics, AIT S − Rajkot (3)
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Curves and Regions in Complex Plane
Distance between two complex numbers
The distance between two complex numbers z 1 and z 2 is given by|z 1 − z 2| or |z 2 − z 1|
Circles
A circle with centre z 0 = (x0, y0)C and radius pR+ isrepresented by |z − z 0| = p
Interior and exterior part of the circle |z − z 0| = pThe set {zC, pR+/|z − z 0| < p} indicates the interior part of the circle |z − z 0| = p whereas {zC, pR+/|z − z 0| > p} indicatesexterior part of it.
N.B.V yas−Department of M athematics, AIT S − Rajkot (4)
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Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z 0 and radius p is given byzC, pR+/|z − z 0| < p.
N.B.V yas−Department of M athematics, AIT S − Rajkot (5)
C R C P
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Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z 0 and radius p is given byzC, pR+/|z − z 0| < p. The close circular disk is given by{zC, pR+/|z − z 0| ≤ p}
N.B.V yas−Department of M athematics, AIT S − Rajkot (5)
C R C P
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Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z 0 and radius p is given byzC, pR+/|z − z 0| < p. The close circular disk is given by{zC, pR+/|z − z 0| ≤ p}
NeighbourhoodAn open neighbourhood of a point z 0 is a subset of C containing an open circular disk centered at z 0 .
N.B.V yas−Department of M athematics, AIT S − Rajkot (5)
C R C P
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Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z 0 and radius p is given byzC, pR+/|z − z 0| < p. The close circular disk is given by{zC, pR+/|z − z 0| ≤ p}
NeighbourhoodAn open neighbourhood of a point z 0 is a subset of C containing an open circular disk centered at z 0 . MathematicallyN p(z 0) = {zC, pR+/|z − z 0| < p}
N.B.V yas−Department of M athematics, AIT S − Rajkot (5)
C R C P
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Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z 0 and radius p is given byzC, pR+/|z − z 0| < p. The close circular disk is given by{zC, pR+/|z − z 0| ≤ p}
NeighbourhoodAn open neighbourhood of a point z 0 is a subset of C containing an open circular disk centered at z 0 . MathematicallyN p(z 0) = {zC, pR+/|z − z 0| < p}A punctured or deleted neighbourhood of a point z 0 containall the points of a neighbourhood of z 0, excepted z 0 itself.Mathematically {zC, pR+/0
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Curves and Regions in Complex Plane
Annulus
The region between two concentric circles with centre z 0 of radii
p1 and p2(> p1) can be represented by p1
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Curves and Regions in Complex Plane
Annulus
The region between two concentric circles with centre z 0 of radii
p1 and p2(> p1) can be represented by p1
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Curves and Regions in Complex Plane
Annulus
The region between two concentric circles with centre z 0 of radii
p1 and p2(> p1) can be represented by p1
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Curves and Regions in Complex Plane
Open Set
Let S be a subset of C . It is called an open set if for eachpoints z 0S , there exists an open circular disk centered at z 0which included in S .
N.B.V yas−Department of M athematics, AIT S − Rajkot (9)
Curves and Regions in Complex Plane
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Curves and Regions in Complex Plane
Open Set
Let S be a subset of C . It is called an open set if for eachpoints z 0S , there exists an open circular disk centered at z 0which included in S .
Closed SetA set S is called closed if its complement is open.
N.B.V yas−Department of M athematics, AIT S − Rajkot (9)
Curves and Regions in Complex Plane
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Curves and Regions in Complex Plane
Open Set
Let S be a subset of C . It is called an open set if for eachpoints z 0S , there exists an open circular disk centered at z 0which included in S .
Closed SetA set S is called closed if its complement is open.
Connected Set
A set A is said to be connected if any two points of A can be joined by finitely many line segments such that each point on theline segment is a point of A
N.B.V yas−Department of M athematics, AIT S − Rajkot (10)
Curves and Regions in Complex Plane
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Curves and Regions in Complex Plane
Domain
An open connected set is called a domain.
N.B.V yas−Department of M athematics, AIT S − Rajkot (11)
Curves and Regions in Complex Plane
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Curves and Regions in Complex Plane
Domain
An open connected set is called a domain.
Region
It is a domain with some of its boundary points.
N.B.V yas−Department of M athematics, AIT S − Rajkot (11)
Curves and Regions in Complex Plane
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Curves and Regions in Complex Plane
Domain
An open connected set is called a domain.
Region
It is a domain with some of its boundary points.
Closed region
It is a region together with the boundary points (all boundarypoints included).
N.B.V yas−Department of M athematics, AIT S − Rajkot (11)
Curves and Regions in Complex Plane
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Curves and Regions in Complex Plane
Domain
An open connected set is called a domain.
Region
It is a domain with some of its boundary points.
Closed region
It is a region together with the boundary points (all boundarypoints included).
Bounded region
A region is said to be bounded if it can be enclosed in a circleof finite radius.
N.B.V yas−Department of M athematics, AIT S − Rajkot (12)
Function of a Complex Variable
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Function of a Complex Variable
If z = x + iy and w = u + iw are two complex variables andif to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z andwe write w = f (z )
N.B.V yas−Department of M athematics, AIT S − Rajkot (13)
Function of a Complex Variable
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u c o o Co V r
If z = x + iy and w = u + iw are two complex variables andif to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z andwe write w = f (z )
If for each value of z in a region R of the z -plane therecorresponds a unique value for w then w is called singlevalued function.
N.B.V yas−Department of M athematics, AIT S − Rajkot (13)
Function of a Complex Variable
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If z = x + iy and w = u + iw are two complex variables andif to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z andwe write w = f (z )
If for each value of z in a region R of the z -plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z 2 is a single valued function of z .
N.B.V yas−Department of M athematics, AIT S − Rajkot (13)
Function of a Complex Variable
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If z = x + iy and w = u + iw are two complex variables andif to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z andwe write w = f (z )
If for each value of z in a region R of the z -plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z 2 is a single valued function of z .
If for each value of z if more than one value of w exists thenw is called multi-valued function.
N.B.V yas−Department of M athematics, AIT S − Rajkot (13)
Function of a Complex Variable
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If z = x + iy and w = u + iw are two complex variables andif to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z andwe write w = f (z )
If for each value of z in a region R of the z -plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z 2 is a single valued function of z .
If for each value of z if more than one value of w exists thenw is called multi-valued function.
E.g.: w =√
Z
N.B.V yas−Department of M athematics, AIT S − Rajkot (13)
Function of a Complex Variable
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If z = x + iy and w = u + iw are two complex variables andif to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z andwe write w = f (z )
If for each value of z in a region R of the z -plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z 2 is a single valued function of z .
If for each value of z if more than one value of w exists thenw is called multi-valued function.
E.g.: w =√
Z
w = f (z ) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) areknown as real and imaginary parts of the function w.
N.B.V yas−Department of M athematics, AIT S − Rajkot (13)
Function of a Complex Variable
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If z = x + iy and w = u + iw are two complex variables andif to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z andwe write w = f (z )
If for each value of z in a region R of the z -plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z 2 is a single valued function of z .
If for each value of z if more than one value of w exists thenw is called multi-valued function.
E.g.: w =√
Z
w = f (z ) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) areknown as real and imaginary parts of the function w.
E.g.: f (z ) = z 2 = (x + iy)2 = (x2 − y2) + i(2xy)
N.B.V yas−Department of M athematics, AIT S − Rajkot (13)
Function of a Complex Variable
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If z = x + iy and w = u + iw are two complex variables andif to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z andwe write w = f (z )
If for each value of z in a region R of the z -plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z 2 is a single valued function of z .
If for each value of z if more than one value of w exists thenw is called multi-valued function.
E.g.: w =√
Z
w = f (z ) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) areknown as real and imaginary parts of the function w.
E.g.: f (z ) = z 2 = (x + iy)2 = (x2 − y2) + i(2xy)∴ u(x, y) = x2
−y2 and v(x, y) = 2xy
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (14)
Limit and Continuity of f (z )
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( )
A function w = f (z ) is said to have the limit l as z approaches a point z 0 if for given small positive number ε wecan find positive number δ such that for all z = z 0 in a disk|z − z 0| < δ we have |f (z ) − l| < ε
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (15)
Limit and Continuity of f (z )
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A function w = f (z ) is said to have the limit l as z approaches a point z 0 if for given small positive number ε wecan find positive number δ such that for all z = z 0 in a disk|z − z 0| < δ we have |f (z ) − l| < εSymbolically, we write lim
z→z0f (z ) = l
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (15)
Limit and Continuity of f (z )
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A function w = f (z ) is said to have the limit l as z approaches a point z 0 if for given small positive number ε wecan find positive number δ such that for all z = z 0 in a disk|z − z 0| < δ we have |f (z ) − l| < εSymbolically, we write lim
z→z0f (z ) = l
A function w = f (z ) = u(x, y) + iv(x, y) is said to becontinuous at z = z 0 if f (z 0) is defined andlimz→z0
f (z ) = f (z 0)
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (15)
Limit and Continuity of f (z )
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A function w = f (z ) is said to have the limit l as z approaches a point z 0 if for given small positive number ε wecan find positive number δ such that for all z = z 0 in a disk|z − z 0| < δ we have |f (z ) − l| < εSymbolically, we write lim
z→z0f (z ) = l
A function w = f (z ) = u(x, y) + iv(x, y) is said to becontinuous at z = z 0 if f (z 0) is defined andlimz→z0
f (z ) = f (z 0)
In other words if w = f (z ) = u(x, y) + iv(x, y) is continuousat z = z 0 then u(x, y) and v(x, y) both are continuous at(x0, y0)
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (15)
Limit and Continuity of f (z )
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A function w = f (z ) is said to have the limit l as z approaches a point z 0 if for given small positive number ε wecan find positive number δ such that for all z = z 0 in a disk|z − z 0| < δ we have |f (z ) − l| < εSymbolically, we write lim
z→z0f (z ) = l
A function w = f (z ) = u(x, y) + iv(x, y) is said to becontinuous at z = z 0 if f (z 0) is defined andlimz→z0
f (z ) = f (z 0)
In other words if w = f (z ) = u(x, y) + iv(x, y) is continuousat z = z 0 then u(x, y) and v(x, y) both are continuous at(x0, y0)
And conversely if u(x, y) and v(x, y) both are continuous at(x0, y0) then f (z ) is continuous at z = z 0.
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (16)
Differentiation of f (z )
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The derivative of a complex function w = f (z ) a point z 0 iswritten as f (z 0) and is defined bydw
dz = f (z 0) = lim
δz→0
f (z 0 + δz ) − f (z 0)δz
provided limit exists.
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (17)
Differentiation of f (z )
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The derivative of a complex function w = f (z ) a point z 0 iswritten as f (z 0) and is defined bydw
dz = f (z 0) = lim
δz→0
f (z 0 + δz ) − f (z 0)δz
provided limit exists.
Then f is said to be differentiable at z 0 if we write thechange δz = z − z 0 since z = z 0 + δz ∴ f (z 0) = lim
z→z0
f (z ) − f (z 0)z
−z 0
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (18)
Analytic Functions
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A single - valued complex function f (z ) is said to beanalytic at a point z 0 in the domain D of the z −plane, if f (z ) is differentiable at z 0 and at every point in someneighbourhood of z 0.
Point where function is not analytic (i.e. it is not singlevalued or not) are called singular points or singularities.From the definition of analytic function
1 To every point z of R, corresponds a definite value of f (z).2 f (z) is continuous function of z in the region R.
3 At every point of z in R, f (z) has a unique derivative.
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (19)
Cauchy-Riemann Equation
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f is analytic in domain D if and only if the first partialderivative of u and v satisfy the two equations
∂u∂x = ∂v∂y , ∂u∂y = −∂v∂x − − − − − (1)
The equation (1) are called C-R equations.
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (20)
Example
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Ex. Find domain of the following functions:
1 1
z 2
+ 1Sol. Here f (z ) =
1
z 2 + 1f (z ) is undefined if z = i and z = −i∴ Domain is a complex plane except z =
±i
2 arg
1z
Sol. Here f (z ) = arg
1
z
1
z = 1
x + iy = 1
x + iy xx
−iy
x − iy = x
−iy
x2 + y2
∴1
z is undefined for z = 0
Domain of arg 1z is a complex plane except z = 0.
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (21)
Example
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3
z
z + z̄
Sol. Here f (z ) = z
z + z̄ f (z ) is undefined if z + z̄ = 0
i.e. (x + iy) + (x − iy) = 0∴ 2x = 0
∴ x = 0
f (z ) is undefined if x = 0
Domain is complex plane except x = 0
N.B.V yas−
Department of M athematics, AIT S−
Rajkot (22)