MathDB

For every complex number

z

different from 0 and 1 we define the following function

f(z) := \sum \frac 1 { \log^4 z }

where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials

P

and

Q

such that

f(z) = \displaystyle \frac {P(z)}{Q(z)}

for all

z\in\mathbb{C}-\{0,1\}

. b) Prove that for all

z\in \mathbb{C}-\{0,1\}

we have

f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}.

Problem Statement