MHB How to Evaluate the Integral of z(z+1)cosh(1/z) Over a Unit Circle?

[brunette15]

Hey everyone,

I am trying to evaluate the following integral: \int z(z+1)cosh(1/z) dz with a C of |z| = 1. Can someone please guide me with how to start? I have tried to parametrise the integral in terms of t so that z(t) = e^it however the algebra doesn't seem to work...

 

[alyafey22]

Hey everyone,

I am trying to evaluate the following integral: \int z(z+1)cosh(1/z) dz with a C of |z| = 1. Can someone please guide me with how to start? I have tried to parametrise the integral in terms of t so that z(t) = e^it however the algebra doesn't seem to work...

Please write your initial attempt.

 

[Prove It]

Hey everyone,

I am trying to evaluate the following integral: \int z(z+1)cosh(1/z) dz with a C of |z| = 1. Can someone please guide me with how to start? I have tried to parametrise the integral in terms of t so that z(t) = e^it however the algebra doesn't seem to work...

If you parameterise the contour $\displaystyle \begin{align*} \left| z \right| = 1 \end{align*}$ with $\displaystyle \begin{align*} z = \mathrm{e}^{\mathrm{i}\,t} , \, 0 \leq t \leq 2\,\pi \end{align*}$, then $\displaystyle \begin{align*} \mathrm{d}z = \mathrm{i}\,\mathrm{e}^{\mathrm{i}\,t}\,\mathrm{d}t \end{align*}$ and we get the integral

$\displaystyle \begin{align*} \oint_C{ z\,\left( z + 1 \right) \cosh{ \left( \frac{1}{z} \right) } \,\mathrm{d}z } &= \int_0^{2\,\pi}{ \mathrm{e}^{\mathrm{i}\,t}\,\left( \mathrm{e}^{\mathrm{i}\,t} + 1 \right) \cosh{ \left( \mathrm{e}^{-\mathrm{i}\,t} \right) } \, \mathrm{i}\,\mathrm{e}^{\mathrm{i}\,t}\,\mathrm{d}t } \end{align*}$

How do you think you can go from here?