October 21st, 2017, 02:07 AM  #1 
Newbie Joined: Sep 2017 From: Sydney Posts: 17 Thanks: 0  Complex Numbers Integrals
Hey guys. I have a problem with part b of this question. I should calculate the residue. Can someone guide me how to correct this and what should the final solution be? Last edited by skipjack; October 21st, 2017 at 05:38 AM. 
October 21st, 2017, 04:43 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
First this is very difficult to read! Am I correct that the integral is $\displaystyle \oint \frac{e^{tz}}{(z 3)^2(z^2+ 4z+ 29)}$? And that the integration is around the circle $z= 7$? $z= 3$, which makes $z 3$ and so the denominator 0, is clearly inside that circle, while $\displaystyle z^2+ 4z+ 29= z^2+ 4z+ 4+ 25= (z+ 2)^2+ 25$ is never 0. So the only pole inside that circle is at $z= 3$ and has order 2. Now, the whole point of "residue" at a pole is this: we can expand a function in a power series at a pole with a finite number of negative power terms. If a function, f, has "a pole of order 2 at x= 3" then we can write $\displaystyle f(x)= \frac{a_{2}}{(x 3)^2}+ \frac{a_{1}}{x 3}+ a_0+ a_1(x 3)+ \cdot\cdot\cdot$ Last edited by skipjack; October 21st, 2017 at 05:16 AM. 
October 21st, 2017, 05:53 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,472 Thanks: 2039 
Robotboyx9 had already found the poles, but hadn't given the residues correctly.

October 21st, 2017, 06:42 AM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,138 Thanks: 872 Math Focus: Wibbly wobbly timeywimey stuff.  
October 21st, 2017, 02:40 PM  #5 
Senior Member Joined: Sep 2015 From: USA Posts: 2,404 Thanks: 1306 
You are correct Dan. Those two poles, $z = 2 \pm 5i$, both lie within $z \leq 7$ Fortunately their residues are both zero. The second order residue at $z=3$ is the only pole that contributes to the integral. 

Tags 
calc3, complex, integrals, numbers 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
sets of numbers (complex numbers)  jeho  Elementary Math  2  October 23rd, 2016 08:55 PM 
Can complex numbers (and properties of complex numbers) be..  jonas  Complex Analysis  2  October 13th, 2014 03:03 PM 
Complex Integrals  WWRtelescoping  Complex Analysis  1  April 1st, 2014 12:08 AM 
Evaluation of Real Integrals by COMPLEX ANALYSIS  jalil_fathi  Complex Analysis  31  July 14th, 2013 10:49 AM 
Line integrals of complex valued functions  rickgoz  Complex Analysis  4  November 11th, 2009 04:22 PM 