Reference no: EM132580801

Questions -

Q1. For each of the following, give an example of a function with the stated properties, or else argue that there is no such function.

(a) p + 1 distinct simple zeros and a pole of order q + 1,

(b) a removable singularity at z = p and an essential singularity at z = q.

Q2. TRUE or FALSE? State and justify.

(a) The function z sin(1/z) has a removable singularity at 0.

(b) If f has a zero of order 2 at 0, then f' has a simple zero at 0.

(c) If f has a pole of order 2 at 0, then f' has a simple pole at 0.

(d) If f has an essential singularity at 0, then 1/f has a removable singularity at 0.

Q3. Let f (z) = 1/z+z^-1, z ≠ 0.

(a) Find the Laurent expansion of f(z) in powers of z for |z| > 1 and for |z| < 1.

(b) Find the Laurent expansion of f(z) about z = i and its domain of convergence.

Q4. Is there an entire function f which satisfies f(z)² = g(z) for all z ∈ C, where

(a) g(z) = z,

(b) g(z) = e^z,

(c) g(z) = sin z?

Justify your answer. (Hint: You may find it useful to consider what happens at special z values.)

Q5. Find the minimum and the maximum values of |z² + (p + 1)i| on the closed disc {z ∈ C : |z| q + 1}.

Q6. Evaluate

∫_C1/(z(z- π)(z-2π)) dz

where C is the circle of radius p + 1 centered at 0, oriented positively.

Q7. Evaluate

₀∫^∞(1+cosx)/(1+x²) dx

using contour integration and a contour of your choice. Show and justify all of your steps.