Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $C$ be $x^2+y^2=9$, oriented counterclockwise.

Find $$\int_{C}\frac{e^{\pi z}}{\frac{(z-4i)^2}{z-i}}dz$$

It is easy to find the parameterization of $C$. However, when it comes to the integral, I don't know how to deal with $z$.

share|improve this question
2  
It appears you want to find $\int_C\frac{(z-i)e^{\pi z}}{(z-4i)^2}dz$. Check the integrand is analytic on and within $C$. –  Host-website-on-iPage Oct 28 '12 at 8:48
    
By the way, should this have a "homework" tag? :) –  Simon Hayward Oct 29 '12 at 15:16

1 Answer 1

up vote 1 down vote accepted

Check out Cauchy's differentiation formula and rearrange!

https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula

So here we have $\int_C\frac{(z-i)e^{\pi z}}{(z-4i)^2}dz$ so we differentiate $e^{\pi z}$ once and multiply by ${\pi i}$ $\left(=\frac{2\pi i}{2!}\right)$.

We do not need to worry about the pole at $4i$ in this case since this does not lie within $C$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.