UNIT3
Complex Integration

In case of a complex function f(z) the path of the definite integral can be along any curve from z = a to z = b.
In case the initial point and final point coincide so that c is a closed curve, then this integral called contour integral and is denoted by
If f(z) = u(x, y) + iv(x, y), then since dz = dx + i dy
We have
It shows that the evaluation of the line integral of a complex function can be reduced to the evaluation of two line integrals of real function.
Properties of line integral
2. Sense reversal
3. Partitioning of path 4. ML – inequality 
Example: Evaluate
along the path y = x.
Sol.
Along the line y = x, dy = dx that dz = dx + i dy dz = dx + i dx = (1 + i) dx On putting y = x and dz = (1 + i)dx

Example: Evaluate
where c is the circle with center a and r. What is n = 1.
Sol.
The equation of a circle C is z  a = r or z – a = Where varies from 0 to 2π dz = Which is the required value. When n = 1

Example: Evaluate
where c is the upper half of the circle z – 2 = 3.
Find the value of the integral if c is the lower half of the above circle.
Sol.
The equation of the circle is Or
Now for the lower semi circle

Key takeaways
3. Sense reversal
4. Partitioning of path 5. ML – inequality

A function f(z) is analytic and its derivative f’(z) continuous at all points inside and on a closed curve c, then
Proof:
Suppose the region is R which is closed by curve c and let By using Green’s theorem Replace by and by 
So that 
Example1: Evaluate
where C is z + 3i = 2 
Sol.
Here we have Hence the poles of f(z), Note put determine equal to zero to find the poles. Here pole z = 3i lies in the given circle C. So that

Example 2:
where C =

Sol.
where f(z) = cosz = by cauchy’s integral formula =

Example 3:
Solve the following by cauchy’s integral method:
Solution:
Given, = = = 
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.
Example1: Evaluate
by using Cauchy’s integral formula.
Here c is the circle z  2 = 1/2
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 
Example2: 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

Example3: 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

Taylor’s series
If f(z) is analytic inside a circle C with centre ‘a’ then for z inside C,
Laurent’s series
If f(z) is analytic in the ring shaped region R bounded by two concentric circles C and of radii ‘r’ and where r is greater and with centre at’a’, then for all z in R
Where 
Example: Expand sin z in a Taylor’s series about z = 0.
Sol.
It is given that Now We know that, Taylor’s series So that
Hence 
Example: Expand
f(z) = 1/ [(z  1) (z  2)] 
in the region z < 1.
Sol.
By using partial fractions Now for z<1, both z/2 and z are < 1, Hence we get from second equation Which is a Taylor’s series. 
Example: Find the Laurent’s expansion of
In the region 1 < z + 1< 3.
Sol.
Let z + 1 = u, we get Here since 1 < u < 3 or 1/u < 1 and u/3 < 1, Now expanding by Binomial theorem Hence Which is valid in the region 1 < z + 1 < 3 
Key takeaways
2. Laurent’s series Where 
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 nonisolated.
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 Step1: If exists and it is finite then z = a is a removable singular point. Step2: If does not exists then z = a is an essential singular point. Step3: 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 
Cauchy’s residue theorem
If f(z) is analytic in a closed curve C, except at a finite number of poles within C, then
;DOProof:
Suppose be the nonintersecting circles with centres at respectively.
Redii so small that they lie within the closed curve C. then f(z) is analytic in the multipUYle connected region 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
 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 
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 McgrawHill, New Delhi, 2010
 Higher engineering mathematics, HK Dass