Complex Integral - exponential divided by a monomial

Question

How does one solve integrals like this- $$I=\int^\beta_0 dx \frac{\exp(i\omega_nx)}{x-a}$$ where $\omega_n=\frac{\pi n}{\beta} $.

EDIT: $\beta$ is a finite, real number. I am looking for a principal value integral. Any help with choosing a good contour would be awesome!
(thanks to @Santosh Linkha for suggestions on improving the question)

Answer 1

$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ $\ds{\omega_{n} \equiv {\pi n \over \beta}\,,\quad n\in{\mathbb Z}\,, \quad\beta > 0}$.

With $\ds{t \equiv -\ic\omega_{n}x\quad\imp\quad x = {\ic \over \omega_{n}}\,t}$: \begin{align} I&\equiv\int_{0}^{\beta}{\exp\pars{\ic\omega_{n}x} \over x-a}\,\dd x =\int_{0}^{-n\pi\ic}{\expo{-t} \over \pars{\ic/\omega_{n}}t - a}\, {\ic \over \omega_{n}}\,\dd t =\int_{0}^{-n\pi\ic}{\expo{-t} \over t + a\omega_{n}\ic}\,\dd t \\[3mm]&=\expo{a\omega_{n}\ic} \int_{a\omega_{n}\ic}^{-n\pi\ic + a\omega_{n}\ic}{\expo{-t} \over t}\,\dd t =\expo{a\omega_{n}\ic}\bracks{% -\int_{\pars{a/\beta - 1}n\pi\ic}^{\infty}{\expo{-t} \over t}\,\dd t +\int_{\pars{a/\beta}n\pi\ic}^{\infty}{\expo{-t} \over t}\,\dd t} \end{align}

Those integrals are related to the Exponential Integral $\ds{{\rm E_{1}}\pars{z}}$. Indeed, it leads to a serie of Exponential integrals since, by definition, it requires $\ds{\verts{{\rm Arg}\pars{z}} < \pi}$. Moreover, you should study carefully the influence of parameter $\ds{a}$.

Can you take it from here ?.