My Math Forum  

Go Back   My Math Forum > College Math Forum > Complex Analysis

Complex Analysis Complex Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
December 7th, 2011, 01:34 PM   #11
Math Team
 
Joined: Nov 2010
From: Greece, Thessaloniki

Posts: 1,990
Thanks: 133

Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus
Re: Nine Integrals

Quote:
Originally Posted by guynamedluis
Wow, Thanks a lot ZardoZ.

Your proof of 5 is pretty slick.
[color=#000000]Regarding 5 and 8, it is clear that complex analysis does not always offer simple ways to compute integrals (wnvl's way is better of course). I believe that it was given to you for a better understanding of the techniques of contour integration![/color]
ZardoZ is offline  
 
December 29th, 2011, 06:38 PM   #12
Senior Member
 
Joined: Dec 2011
From: Argentina

Posts: 216
Thanks: 0

Re: Nine Integrals

I'd like to contribute with 5.









Now by integrating by parts with





we get that



Thus we get that

for even

for odd

Now





So writing as or we get that.





which you can always write in terms of the Gamma function and factorials knowing that

Weiler is offline  
December 29th, 2011, 07:29 PM   #13
Senior Member
 
Joined: Dec 2011
From: Argentina

Posts: 216
Thanks: 0

Re: Nine Integrals

Why is the 3rd integral non zero if the function being integrated is even? My doubt arises since for any even function:

Weiler is offline  
December 29th, 2011, 07:30 PM   #14
Senior Member
 
Joined: Dec 2011
From: Argentina

Posts: 216
Thanks: 0

Re: Nine Integrals

I'm sorry, I meant for odd functions, but the function in question is even, so never mind.
Weiler is offline  
December 29th, 2011, 08:00 PM   #15
Senior Member
 
Joined: Dec 2011
From: Argentina

Posts: 216
Thanks: 0

Re: Nine Integrals

I'd like to provide a proof for

Theorem



Proof



Substitute to get



Differentiating under Leibniz's Rule:



Given that


and that



we have got



but again, by Lebniz's Rule:



so satisfies the differential equation



thus



and since



we have

QED

Although this proof is not really rigorous, I like it.
Weiler is offline  
December 29th, 2011, 08:03 PM   #16
Senior Member
 
Joined: Dec 2011
From: Argentina

Posts: 216
Thanks: 0

Re: Nine Integrals

And 2 follows from 3 since 2 is the negative integral of 3.
Weiler is offline  
December 30th, 2011, 06:33 PM   #17
Senior Member
 
Joined: Dec 2011
From: Argentina

Posts: 216
Thanks: 0

Re: Nine Integrals

Ok here goes another one:









Now











Thus



Then

Weiler is offline  
January 2nd, 2012, 01:00 PM   #18
Global Moderator
 
Joined: Dec 2006

Posts: 21,036
Thanks: 2274

More directly,
?
if I = ? ln(x)/(x + a) dx, where a > 0,
0
let x = a/u, dx = -a/u du.
? ? ?
I = ? -ln(a/u)/((a/u) + a) (-a/u) du = ? ln(a/u)/(a + u) du = ? ln(a)/(a + u) du - I.
0 0 0
?
Hence I = (1/2)ln(a)[(1/a)atan(u/a)] = (1/a)ln(a)?/2.
0
skipjack is offline  
January 2nd, 2012, 05:46 PM   #19
Senior Member
 
Joined: Dec 2011
From: Argentina

Posts: 216
Thanks: 0

Re: Nine Integrals

Nice one. I'm trying to polish my proof for 3.
Weiler is offline  
Reply

  My Math Forum > College Math Forum > Complex Analysis

Tags
integrals



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
integrals ungeheuer Calculus 10 July 21st, 2013 03:30 AM
integrals ungeheuer Calculus 10 January 26th, 2013 08:32 AM
integrals damar10 Calculus 1 April 25th, 2012 08:34 AM
Integrals johnnyboy20 Calculus 2 May 24th, 2011 04:36 PM
integrals hector manuel Real Analysis 0 May 4th, 2009 11:16 PM





Copyright © 2019 My Math Forum. All rights reserved.