MathDB

Let be a natural number

n,

denote with

C

the square in the complex plane whose vertices are the affixes of

2n\pi\left( \pm 1\pm i \right) ,

and consider the set

\Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\}

Prove the following implications.
a) \exists \alpha\in\mathbb{R}_{>0} \forall z\in\partial C \left| \cos z \right|\ge\alpha e^{|\text{Im}(z)|} b) \forall f\in\Lambda \frac{1}{2\pi i}\int_{\partial C} \frac{f(z)}{z^2\cos z} dz=f'(0)+\frac{4}{\pi^2}\sum_{p=-2n}^{2n-1} \frac{(-1)^{p+1} f(z-p)}{(1+2p)^2} c) \forall f\in\Lambda \sum_{p\in\mathbb{Z}}\frac{(-1)^pf\left( \frac{(1+2p)\pi}{2} \right)}{(1+2p)^2} =\frac{\pi^2 f'(0)}{4}

Problem Statement