Complex Analysis Problems

The following questions on Complex Analysis can be treated as an assignment as well as the suggestions on the upcoming exam.

Q1: Find the real and the imaginary part of $f(z)=\dfrac{\overline{z}}{z}, z \neq 0$.

Q2: Define the differentiability of a function f(z) at z=z₀. Prove that

f(z) = |z|

is not differentiable at z=0.

Q3: Show that the functions below are nowhere differentiable.

1. f(z) = Re(z)
2. f(z) = Im(z)

where Re(z) and Im(z) denote the real and the imaginary part of z.

Q4: Define an analytic function with an example.

Q5: State the Cauchy-Riemann equation. Prove that the function f(z) defined by

$f(z)= |Re(z) \, Im(z)|^{1/2}$

satisfies the Cauchy-Riemann equation at the origin z=0, but it is not differentiable at this point.

Q6: Find the integral

$\int_{|z|=1} \dfrac{1}{z} \, dz$.

Q7: Using M-L inequality, find the upper bound of

$|\int_\gamma \dfrac{1}{z^2} \, dz|$

where $\gamma$ is the directed line segment from 1+i to 1+3i.

Q8: Write down the Cauchy’s integral theorem and formula. Using this or otherwise, evaluate the following integrals:

$\int_{|z|=1} \dfrac{\cos z}{z(z-4)} \, dz$.

Q9: Evaluate ∫𝑧2(𝑧2+1)2γ𝑑𝑧 where γ is the circle |𝑧−𝑖|=1 Using Cauchy’s integral formula, evaluate the following integral

$\int_C \dfrac{z^2}{(z^2+1)^2} \, dz$

where C is the circle |z-i|=1.

Q10: Find the Taylor’s series expansion of $f(z)=e^z$ around the point $z=0$.