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ENDTERM COMPLEX FUNCTIONS JUNE 30 2015, 9:00-12:00

• Put your name and student number on every sheet you hand in.

• When you use a theorem, show that the conditions are met.

• Include your partial solutions, even if you were unable to complete an exercise.

Exercise 1 (10 pt ): Let α, β, γ be three different complex numbers satis- fying

β − α

γ − α = α − γ β − γ .

Prove that the triangle with vertices {α, β, γ} is equilateral, i.e.

|β − α| = |γ − α| = |β − γ|.

Exercise 2 (10 pt ): Find all entire functions f such that |f0(z)| < |f (z)|

for all z ∈ C.

Exercise 3 (15 pt ): Consider the polynomial equation anzn+ an−1zn−1+ · · · + a1z + a0 = 0 with real coefficients ak∈ R satisfying

a0 ≥ a1 ≥ a2 ≥ · · · ≥ an > 0 . Prove that this equation has no roots with |z| < 1.

Exercise 4 (20 pt ): Let f be a meromorphic function on C. Suppose there exist C, R > 0 and integer n ≥ 1 such that |f (z)| ≤ C|z|n for all z ∈ C with

|z| ≥ R.

a. (10 pt ) Prove that the number of poles of f in C is finite.

b. (10 pt ) Prove that f is a rational function, i.e. it can be written as a

ratio of two polynomials. Turn the page!

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Exercise 5 (25 pt ): Let a > 0. By integrating the function f (z) = 1

z

1

cos(2πia) − cos(2πz) over a suitable closed path, show that

X

n=−∞

1

a2+ n2 = π a

e2πa− e−2πa e2πa+ e−2πa− 2. Hint : Use a square path.

Bonus Exercise (20 pt ): Find all entire functions f such that f (z2) = (f (z))2

for all z ∈ C.

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