
Complex Analysis Complex Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
March 12th, 2015, 09:39 AM  #1 
Senior Member Joined: Aug 2014 From: United States Posts: 137 Thanks: 21 Math Focus: Learning  Application of Cauchy generalized integral theorem
I am asked to find $\displaystyle\oint_\gamma \frac{z^2 + e^z} {(zi\pi)^2}dz$. Using Cauchy's Generalized Integral Formula. I am getting $\boxed{  1+2\pi i}$. Is this correct? (I don't have an answer key) Also I need to find $\displaystyle\int_0^{2\pi} e^{\alpha\cos \theta} \sin(\alpha\cos \theta)d\theta$. Using Cauchy's integral formula. I am not quite sure how to do this one. My attempt was to apply Euler's formula and then go from there. How do I use Cauchy's integral formula? Thanks 
March 19th, 2015, 05:27 AM  #2 
Senior Member Joined: Sep 2007 From: USA Posts: 349 Thanks: 67 Math Focus: Calculus 
1. Assuming $\displaystyle z=\pi i$ lies within the closed contour of the first problem, by the Cauchy integral formula, $\displaystyle \oint_{\gamma}\frac{z^2+e^z}{(z\pi i)^2}=2\pi i f'(\pi i)$ where $\displaystyle f(\alpha)=\alpha^2+e^\alpha$. Therefore, the answer is, $\displaystyle 2\pi i f'(\pi i)=2\pi i(2(\pi i)+e^{\pi i})=4\pi^22\pi i$. 2. For the second problem replace cosine with its exponential form. $\displaystyle \int_{0}^{2\pi}e^{\alpha\cos\theta}\sin(\alpha\cos \theta)d\theta$ $\displaystyle \int_{0}^{2\pi}e^{\frac{\alpha}{2}(e^{i\theta}+e^{i\theta})}\sin(\frac{\alpha}{2}(e^{i\theta}+e^{i\theta}))d\theta$ A substution is now in order, $\displaystyle z=e^{i\theta}$, $\displaystyle d\theta=\frac{i dz}{z}$. This now becomes a closed contour around the unit circle in the complex plane. $\displaystyle i\oint_{z=1}\frac{e^{\frac{\alpha}{2}(z+z^{1})}\sin(\frac{\alpha}{2}(z+z^{1}))}{z}dz$ $\displaystyle i\oint_{z=1}\frac{e^{\frac{\alpha(z^2+1)}{2z}} \sin\left(\frac{\alpha(z^2+1)}{2z}\right)}{z}dz$ Now the Cauchy integral formula can be used. However, by analyzing the original real integral, it can be shown that the integral is zero. That is the same result of the Cauchy integral formula. 

Tags 
application, cauchy, generalized, integral, theorem 
Search tags for this page 
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
generalized gelfondschneide theorem  raul11  Number Theory  0  April 25th, 2014 03:36 PM 
Using Cauchy's residue theorem  WWRtelescoping  Complex Analysis  2  April 22nd, 2014 02:51 AM 
Generalized integral  bellum753  Real Analysis  1  May 28th, 2013 05:30 AM 
Cauchy's integral theorem is mistaken  stone  Number Theory  0  July 13th, 2010 01:12 PM 
Applying CauchyLipschitz theorem  Seng Peter Thao  Applied Math  0  June 30th, 2007 10:44 AM 