complex calculus formula sheet_3rd sem

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Calculus/Complex analysis 1

Calculus/Complex analysis

Complex analysis is the study of functions of complex variables. Complex analysis is a widely used and powerful

tool in certain areas of electrical engineering, and others.

Before we begin, you may want to review Complex numbers

Complex Numbers

Complex Numbers

Complex Functions

A function of a complex variable is a function that can take on complex values, as well as strictly real ones. For

example, suppose f(z) = z2. This function sets up a correspondence between the complex number z and its square, z

2,

just like a function of a real variable, but with complex numbers. Note that, for f(z) = z2, f(z) will be strictly real if z

is strictly real.

Generally we can write a function f(z) in the form f(z) = f(x+iy) = a(x,y) + ib(x,y), where a and b are real-valued

functions.

Limits and continuity

As with real-valued functions, we have concepts of limits and continuity with complex-valued functions also – our

usual delta-epsilon limit definition:

The limit of f( z) as z approaches w is L if for each ε > 0, there is a δ > 0 such that | f( z)-L |<ε for all z such that

0 < | z - w | < δ.

Note that ε and δ are real values. This is implicit in the use of inequalities: only real values are "greater than zero".One difference between this definition of limit and the definition for real-valued functions is the meaning of the

absolute value. Here we mean the complex absolute value instead of the real-valued one. Another difference is that

of how z approaches w. For real-valued functions, we would only be concerned about z approaching w from the left,

or from the right. In a complex setting, z can approach w from any direction in the two-dimensional complex plane:

along any line passing through w, along a spiral centered at w, etc.

For example, let f( z) = z2. Suppose we want to show that the limit of f( z) as z approaches i is -1. We can write z as

i+γ where we think of γ being a small complex quantity. Note then that z-i = γ. Then, with L in our definition being

-1, and w being i, we have

| f( z) - L | = | z2

+ 1 | = | (i + γ)2

+ 1 | = | 2i γ + γ2

|

By the triangle inequality, this last expression is less than

2 | γ | + | γ |2

In order for this to be less than ε, we can require that

| γ | < 1/2 min( ε/2, √ ε)

Thus, for any ε > 0, if δ = 1/2 min( ε/2, √ ε), and | z - i |<δ, then | f( z) - ( - 1) | < ε. Hence, the limit of f( z)= z2

as z

approaches i is -1.

http://en.wikibooks.org/w/index.php?title=Algebra/Complex_numbers

http://en.wikibooks.org/w/index.php?title=Algebra/Complex_numbers




Differentiation and Holomorphic Functions

Since we have limits defined, we can go ahead to define the derivative of a complex function, in the usual way:

provided that the limit is the same no matter how Δ z approaches zero (since we are working now in the complex

plane, we have more freedom!).

If such a limit exists for some value z, or some set of values - a region, we call the function holomorphic at that point

or region. Continuity and being single-valued are necessary for being analytic; however, continuity and being

single-valued are not sufficient for being analytic.

Many elementary functions of complex values have the same derivatives as those for real functions: for example D

z2

= 2 z.

Problem set

Given the above, answer the following questions (Answers follow to even-numbered questions).

1. Find the derivative of z3 from the limit definition.

2. Write e z

in the form a( x, y)+b( x, y)i

Answers

1.

2.

Cauchy-Riemann Equations

We might wonder which sorts of complex functions are in fact differentiable. It would appear that the criterion forholomorphicity is much stricter than that of differentiability for real functions, and this is indeed the case. Suppose

we have a complex function

,

where u and v are real functions. Assume furthermore that u and v are differentiable functions in the real sense. Then

we can let in the definition of differentiability approach 0 by varying only x or only y. Therefore f can only be

differentiable in the complex sense if

In fact, if u and v are differentiable in the real sense and satisfy these two equations, then f is holomorphic. These two

equations are known as the Cauchy-Riemann equations.




Integration

In single variable Calculus, integrals are typically evaluated between two real numbers

On the real line, there is one way to get from to . In the complex plane, however, there are infinitely many

different paths which can be taken between two points, and . For this reason, complex integration is always

done over a path, rather than between two points.

Let be a path in the complex plane parametrized by , and let be a complex-valued

function. Then the contour integral is defined analogously to the line integral from multivariable calculus:

Example Let , and let be a line from 0 to 1+i. This curve can be parametrized by ,

with ranging from 0 to 1. Now we can compute

Note that we also have

This indicates that complex antiderivatives can be used to simplify the evaluation of integrals, just as real

antiderivatives are used to evaluate real integrals.

Cauchy's Theorem

Cauchy's theorem states that if a function is holomorphic in the closure of an open set , and is a simple

closed curve in , then

This can be understood in terms of Green's theorem, though this does not readily lead to a proof, since Green's

theorem only applies under the assumption that f has continuous first partial derivatives...

Contour Integration

Contour over which to perform the integration

Cauchy's theorem allows for the evaluation of many improper real

integrals (improper here means that one of the limits of integration is

infinite). As an example, consider

Since

http://en.wikibooks.org/w/index.php?title=File:ContourSinzz.gif




we consider

We now integrate over the indented semicircle contour, pictured above. We parametrize each segment of the contour

as follows

,

,

,

,

By Cauchy's Theorem, the integral over the whole contour is zero. So,

We now handle each of these integrals separately.

Recalling the definition of the sine of a complex number,

Now we evaluate the other two integrals

As , the integrand approaches one, so

The fourth integral is equal to zero, but this is somewhat more difficult to show. Its form is similar to that of the third

segment:

This integrand is more difficult, since it need not approach zero everywhere. This difficulty can be overcome by

splitting up the integral, but here we simply assume it to be zero.

Combining everything, we now have

Hence,




Cauchy's Integral Formula

Cauchy's integral formula characterizes the behavior of holomorphics functions on a set based on their behavior on

the boundary of that set. If is an open set with a piecewise smooth boundary and is holomorphic in , then

This is a remarkable fact which has no counterpart in multivariable calculus. It says that if we know the values of a

holomorphic function along a closed curve, then we know its values everywhere in the interior of the curve.

Because , an open set, it follows that for all . Hence the integrand in Cauchy's

integral formula is infinitely differentiable with respect to z, and by repeatedly taking derivatives of both sides, we

get

This result shows that holomorphicity is a much stronger requirement than differentiability. In the complex plane, if

a function has just a single derivative in an open set, then it has infinitely many derivatives in that set.

Corollaries of Cauchy's Theorem

Cauchy's Theorem and integral formula have a number of powerful corollaries:

• Convergence of power series If a function is holomorphic in a disc, then its Taylor series converges in this disc.

• Liouville's Theorem If a function is bounded and holomorphic in all of then it is equal to a constant.

• Fundamental Theorem of Algebra All polynomials of degree greater than zero with complex coefficients have

a complex root. This is a simple corollary of Liouville's theorem.

• Equality of functions If two functions and are holomorphic on a connected, open set , and

on any disc in this set, then for all .



Article Sources and Contributors 6

Article Sources and ContributorsCalculus/Complex analysis Source: http://en.wikibooks.org/w/index.php?oldid=1566211 Contributors: Adrignola, ComplexZeta, Count Iblis, Danvk, Doctormatt, Dysprosia, Iamunknown,

Jguk, Mike.lifeguard, MikeBorkowski, Mkn, Notanut, 20 anonymous edits

Image Sources, Licenses and ContributorsImage:ContourSinzz.gif Source: http://en.wikibooks.org/w/index.php?title=File:ContourSinzz.gif License: Public Domain Contributors: Danvk

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