Math Problems: Complex Analysis

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Problem 1

1. For every increasing function f, prove that one can find a power series S with infinite radius of convergence such that S(x) >= f(x).
Solution

Problem 2

2. If a is a removable singularity of f (a function analytic on an open set U, except at point a in U), prove that a cannot be a pole of z -> e^z.
Solution

Problem 3

3. f is a function analytic in a region U (0 in U) such that |f(1/n)| \leq e^(-n) for all n positive integer. Show that for all z in U, f(z)=0.
Solution

Problem 4

4. n being a positive integer, and f(z)=1+1/2+1/(2!z^2)+...+1/(n!z^n). Evaluate int(f '(z)/f(z),|z|=2).
Solution

Problem 5

5. Considering a function f analytic on the closed disc D(0,1) (centered at (0,0) with radius 1) such that |z| <= 1 => |f(z)| <= 1, show that for all a,b in C, |af(0)+bf '(0)| <= sqrt(|a|^2+|b|^2).
Solution

Problem 6

6. f is analytic on the closed disc D((0,0),1) with at most m roots there, and has a power series expansion at 0 given by sum(a_kz^k,k=0..+oo). Show that inf(|f(z)|; |z|=1) <= |a_0|+...+|a_m|.
Solution

Problem 7

7. Show that for every R>0 there is a positive integer N such that for all n >= N, the roots of the polynomial P_n(z)=1+z/1!+z^2/2!+...+z^n/n! lie outside the closed ball B((0,0),R).
Solution

Problem 8

8. Prove that it is impossible to partition the nonnegative integers N into finitely many (but more than 1) arithmetic progressions with pairwise distinct minimal differences.
Solution

Problem 9

9. Does there exist a function f analytic on the open disc D((0,0),1) and such that lim |f(z)| -> +oo as |z| -> 1 ?
Solution

Problem 10