\text{Homework 2}

\text{Show that:}

\displaystyle \sum_{n=1}^{\infty} \frac{\operatorname{coth}(\pi n)}{n^7}=\frac{19 \pi^7}{56700}

\text{Problem 1}

\displaystyle S(a, b)=\sum_{n=0}^{\infty} \frac{1}{n^2+a^2} \frac{1}{n^2+b^2},

\text{1) Evaluate the following sum:}

\text{Problem 2:}

\text{2) Find the limit when } b\rightarrow a

\text{2) Find the limit when } (b^2+a^2) \rightarrow 0

\displaystyle I=\int_1^{\infty} \frac{d x}{x \sqrt{x^2-1}}

\text{Calculate the following integral:}

\text{Problem 3:}

\text { Use calculus of residues to evaluate the integral: }\\ \displaystyle \int_0^{\infty} \frac{x^{p-1}}{1+x} d x\\

\text{Problem 4:}

\text { where } p<1 \text{ and p is positive }\text {. }