3.1 Differentiation | unit 3 function of complex variable

M-II

UNIT-3

FUNCTION OF COMPLEX VARIABLE

In the narrow sense of the term, the theory of function of a complex variable is the theory of analytic functions (cf. Analytic function) of one or several complex variables. As an independent discipline, the theory of functions of a complex variable took shape in about the middle of the 19th century as the theory of analytic functions.

Complex function-

x + iy is a complex variable which is denoted by z

If for each value of the complex variable z = x + iy in a region R, we have one or more than one values of w = u + iv, then w is called a complex function of z.

And it is denoted as-

w = u(x , y) + iv(x , y) = f(z)

Neighbourhood of

Let a point in the complex plane and z be any positive number, then the set of points z such that-

||<ε

Is called ε- neighbourhood of

Limit of a function of a complex variable-

Suppose f(z) is a single valued function defined at all points in some neighbourhood of point -

The-

Example-1: Find-

Sol. Here we have-

Divide numerator and denominator by , we get-

Continuity- A function w = f(z) is said to be continuous at z = , if

Also if w = f(z) = u (x, y) + iv (x, y) is continuous at z = then u (x, y), v (x, y) are also continuous at z = .

Differentiability-

Let f(z) be a single valued function of the variable z, then

Provided that the limit exists and has the same value for all the different ways in which approaches to zero.

Example-2: if f(z) is a complex function given below, then discuss

Sol. If z→0 along radius vector y = mx

But along ,

In different paths we get different value of that means 0 and –i/2, in that case the function is not differentiable at z = 0.

Key takeaways-

Neighborhood of

||<ε

2. Limit of a function of a complex variable-

Suppose f(z) is a single valued function defined at all points in some neighborhood of point -

The-

3. A function w = f(z) is said to be continuous at z = , if

3.2 Cauchy-Riemann equations

In Cartesian form-

Theorem; The necessary condition for a function to be analytic at all the

points in a region R are

(ii)

Provided,

Proof:

Let be an analytic function in region R.

Along real axis

Then f’(z), becomes-

………… (1)

Along imaginary axis

From equation (1) and (2)

Equating real and imaginary parts

Therefore-

and

These are called Cauchy Riemann Equations.

C-R equation in polar from-

C-R equations in polar form are-

Proof:

As we know that-

x = r cos and u is the function of x and y

z = x + iy = r ( cos

Differentiate (1) partially with respect to r, we get-

Now differentiate (1) with respect to , we get-

Substitute the value of , we get-

Equating real and imaginary parts, we get-

Proved

Key takeaways-

The necessary condition for a function to be analytic at all the points in a region R are

(ii)

2. C-R equations in polar form are-

3.3 Analytic functions

A function is said to be analytic at a point if f is differentiable not only at but at every point of some neighborhood at .

Note-

1. A point at which the function is not differentiable is called singular point.

2. A function which is analytic everywhere is called an entire function.

3. An entire function is always analytic, differentiable and continuous function (converse is not true)

4. Analytic function is always differentiable and continuous but converse is not true.

5. A differentiable function is always continuous but converse is not true.

The necessary condition for f(z) to be analytic-

f(z) = u + i(v) is to be analytic at all the points in a region R are-

1. …………. (1)

2. ……...…. (2)

Provided exists

Equation (1) and (2) are known as Cauchy-Riemann equations.

The sufficient condition for f(z) to be analytic-

f(z) = u + i(v) is to be analytic at all the points in a region R are-

1. …………. (1)

2. ……...…. (2)

are continuous function of x and y in region R.

Important note-

1. If a function is analytic in a domain D, then u and v will satisfy Cauchy-Riemann conditions.

2. C-R conditions are necessary but not sufficient for analytic function.

3. C-R conditions are sufficient if the partial derivative are continuous.

State and prove sufficient condition for analytic functions

Statement – The sufficient condition for a function to be analytic at all points in a region R are

2 are continuous function of x and y in region R.

Proof: Let f(z) be a simple valued function having at each point in the region R. Then Cauchy-Reimann equation are satisfied by Taylor’s Theorem

Ignoring the terms of second power and higher power

We know C-R equation

Replacing

Respectively in (1) we get

Show that is analytic at

Ans The function f(z) is analytic at if the function is analytic at z=0

Since

Now is differentiable at z=0 and at all points in its neighbourhood Hence the function is analytic at z=0 and in turn f(z) is analytic at

Example-1: If w = log z, then find . Also determine where w is non-analytic.

Sol. Here we have

Therefore-

and

Again-

Hence the C-R conditions are satisfied also the partial derivatives are continuous except at (0, 0).

So that w is analytic everywhere but not at z = 0

Example-2: Prove that the function is an analytical function.

Sol. Let =u+iv

Let =u and =v

Hence C-R-Equation satisfied.

Example-3: Prove that

Sol. Given that

Since

V=2xy

Now

But

Hence

Example-4: Show that polar form of C-R equations are-

Sol. z = x + iy =

U and v are expressed in terms of r and θ.

Differentiate it partially w.r.t. r and θ, we get-

By equating real and imaginary parts, we get-

Key takeaways-

A function is said to be analytic at a point if f is differentiable not only at but at every point of some neighborhood at .
A point at which the function is not differentiable is called singular point.
A function which is analytic everywhere is called an entire function.
If a function is analytic in a domain D, then u and v will satisfy Cauchy-Riemann conditions.
C-R conditions are necessary but not sufficient for analytic function.
C-R conditions are sufficient if the partial derivative are continuous.

3.4 Harmonic functions, finding harmonic conjugate

A function which satisfies the Laplace equation is known as a harmonic function.

Theorem- if f(z) = u + iv is an analytic function, then u and v are both harmonic functions.

Proof:

Suppose f(z) = u + iv, be an analytic function, then we have

Differentiate (1) with respect to x, we get

Differentiate (2) with respect to y, we get

Add 3 and 4-

Similarly-

So that u and v are harmonic functions.

Example: Prove that and are harmonic functions of (x, y).

Sol.

We have

Now

Here it satisfies Laplace equation so that u (x, y) is harmonic.

Now-

On adding the above results-

We get-

So that v(x, y) is also a harmonic function.

Example: Find the harmonic conjugate function of the function U (x, y) = 2x (1 – y).

Sol.

We have,

U(x, y) = 2x (1 – y)

Let V is the harmonic conjugate of U.

So that by total differentiation,

Hence the harmonic conjugate of U is

3.5 Zeros of analytic functions, singularities

A point at which a function f(z) is not analytic is known as a singular point or singularity of the function.

Isolated singular point- If z = a is a singularity of f(z) and if there is no other singularity within a small circle surrounding the point z = a, then z = a is said to be an isolated singularity of the function f (z); otherwise, it is called non-isolated.

Pole of order m- Suppose a function f(z) have an isolated singular point z = a, f(z) can be expanded in a Laurent’s series around z = a, giving

…… (1)

In some cases it may happen that the coefficient , then equation (1) becomes-

Then z = a is said to be a pole of order m of the function f(z).

Note- The pole is said to be simple pole when m = 1.

In this case-

Working steps to find singularity-

Step-1: If exists and it is finite then z = a is a removable singular point.

Step-2: If does not exists then z = a is an essential singular point.

Step-3: If is infinite then f(z) has a pole at z = a. the order of the pole is same as the number of negative power terms in the series expansion of f(z).

Example: Find the singularity of the function-

Sol.

As we know that-

So that there is a number of singularity.

is not analytic at z = a

(1/z = ∞ at z = 0)

Example: Find the singularity of

Sol.

Here we have-

We find the poles by putting the denominator equals to zero.

That means-

Example: Determine the poles of the function-

Sol.

Here we have-

We find the poles by putting the denominator of the function equals to zero-

We get-

By De Moivre’s theorem-

If n = 0, then pole-

If n = 1, then pole-

If n = 2, then pole-

If n = 3, then pole-

Key takeaways-

The pole is said to be simple pole when m = 1.
Cauchy’s residue theorem-

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

3.6 Cauchy Integral formula (without proof)

Cauchy’s integral formula-

Cauchy’s integral formula can be defined as-

Where f(z) is analytic function within and on closed curve C, a is any point within C.

Example-1: Evaluate by using Cauchy’s integral formula.

Here c is the circle |z - 2| = ½

Sol. it is given that-

Find its poles by equating denominator equals to zero.

There is one pole inside the circle, z = 2,

So that-

Now by using Cauchy’s integral formula, we get-

Example-2: Evaluate the integral given below by using Cauchy’s integral formula-

Sol. Here we have-

Find its poles by equating denominator equals to zero.

We get-

There are two poles in the circle-

Z = 0 and z = 1

So that-

Example-3: Evaluate if c is circle |z - 1| = 1.

Sol. Here we have-

Find its poles by equating denominator equals to zero.

The given circle encloses a simple pole at z = 1.

So that-

Key takeaways-

Cauchy’s integral formula-

3.7 Cauchy Residue theorem (without proof)

Residue at a pole-

If z = a is an isolated singularity of f(z) then f(x) can be expressed expanded in Laurent’s series about z = a

So that-

Note- the coefficient of which is is called the residue of f(z) at z = a and it is written as-

Since-

So that-

Method of finding residues-

If (z) has a simple pole at z = a, then Res{f(z), a} = .
If f(z) has a pole of order m at z = a, then-

Example: Find the residue of f(z) = z cos (1/z) at z = 0

Sol.

Which is the Laurent’s series expansion about z = 0

So that-

By the definition of residue-

Residue of f(z) at z = 0 is -1/2.

Cauchy’s residue theorem-

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

Proof:

Suppose be the non-intersecting circles with centres at respectively.

Radii so small that they lie within the closed curve C. then f(z) is analytic in the multiple connected regions lying between the curves C and

Now applying the Cauchy’s theorem-

Example: Find the poles of the following functions and residue at each pole:

and hence evaluate-

Where c: |z| = 3.

Sol.

The poles of the function are-

The pole at z = 1 is of second order and the pole at z = -2 is simple-

Residue of f(z) (at z = 1)

Residue of f(z) ( at z = -2)

Example: Evaluate-

Where C is the circle |z| = 4.

Sol.

Here we have,

Poles are given by-

Out of these, the poles z = -πi , 0 and πi lie inside the circle |z| = 4.

The given function 1/sinh z is of the form

Its poles at z = a is

Residue (at z = -πi)

Residue (at z = 0)

Residue (at z = πi)

Hence the required integral is =

Key takeaways-

Residue at a pole-

2. Cauchy’s residue theorem-

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

References

E. Kreyszig, “Advanced Engineering Mathematics”, John Wiley & Sons, 2006.
P. G. Hoel, S. C. Port And C. J. Stone, “Introduction To Probability Theory”, Universal Book Stall, 2003.
S. Ross, “A First Course in Probability”, Pearson Education India, 2002.
W. Feller, “An Introduction To Probability Theory and Its Applications”, Vol. 1, Wiley, 1968.
N.P. Bali and M. Goyal, “A Text Book of Engineering Mathematics”, Laxmi Publications, 2010.
B.S. Grewal, “Higher Engineering Mathematics”, Khanna Publishers, 2000.
T. Veerarajan, “Engineering Mathematics”, Tata Mcgraw-Hill, New Delhi, 2010
Higher engineering mathematics, HK Dass
BV ramana, higher engineering mathematics

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