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MAST30021 Complex Analysis, Semester 2 2023

Written assignment 4 and Cover Sheet

1.Mandatory Summary 10 points.

2. Mandatory Question (simple computation) 10 points.

Compute the following integrals with the help of the residue theorem. (Do not forget to say where and what type of singularities the corresponding complex functions have!)

(a) I = \〇π dt (Hint: have a look at the Fourier transform on intervals and note that

you must choose a suitable α for the identification z = eiαt ),

(b) I  =  尸一&(&) dx with 尸 the Cauchy principal value integral.   Use the

Kronig-Kramers relations! (Do not forget to check the conditions!)

3. Mandatory Question (simple proof) 10 points.

Find out how many zeros the following functions have in the given domains. Apply Rouché’s Theorem!

(a)  h(z) = cos(z)+  in the rectangle R = {x +iy}x e [0, 10], y e [ ·2, 2]( (Hint: } cos(x +iy)} = ←cosh2 (y) · sin2 (x) for all x, y e R);

(b)  h(z) =  + 5z · 1 in the closed disc (0, 1/2) where we employ the principal value of

the complex logarithm (Hint: log(2) s 0.7 and think of the Taylor series of the logarithm).

4. Optional Question (advanced computation) 10 points.

Compute the real series

S =

with the help of the residue theorem. Approach as follows:

(a) Rewrite the summands in terms of residues of a function f (z) = g(z) cot(πz). Where and of what type are the singularities?

(b) Perform an ML-bound to rewrite the series in terms of a limit of a contour integral. You can employ the bound of } cot(πz)} from the lecture. Write explicitly what S in terms of the contour integral and the residues is.

(c) Do not forget to write the result and simplify as much as possible (no decimal approximation, we want to see the exact result)!

5. Optional Question (advanced proof) 10 points. Prove the following statements:

(a) Let g(z) be a function meromorphic at z〇   e C and being not constant equal to 0 (meaning if g(z〇 ) = 0, the zero is isolated). Then, f (z) has an isolated essential singularity at z〇 if and only if f (z)g(z) has an isolated essential singularity at z〇 .

(b)  f (z) has an isolated essential singularity at z〇  and is non-vanishing in some punctured neighbour- hood of z〇 if and only if 1/f (z) has an isolated essential singularity at z〇  and is non-vanishing in some punctured neighbourhood of z〇 .

Answer also why we need the existence of a punctured neighbourhood where the function is not vanishing to have an “if and only if” statement. (Hint: A counterexample is sufficient.)