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
May 7th, 2012, 01:45 AM   #1
Newbie
 
Joined: May 2012

Posts: 2
Thanks: 0

Conditions for convergence of real integral

Hi Everybody,

I've been pondering this question for a while now, and have realised (after far too long...) that I really need some help. Here's the question:

Consider a real function f(x). How do the conditions for the convergence
of the real integral of f(x) relate to the conditions under which
the integral of f(z) goes to zero on C_R? (Here C_R is the simple closed contour described counterclockwise consisting of: the segment of the real axis from z=-R to z=R, and the top half of the circle abs(z)=R.)


I know the question is somewhat ambiguous (it makes no suggestion as to the type of real integral of f(x): improper? definite? indefinite?), so I have only been considering the case of improper real integrals of f(x) so far. For this particular case, I have considered: 1. the cauchy principle value of an improper real integral of f(x), 2. using residue theory to re-write a real integral of f(x) as int[f(x), -R, R]+int[f(z), C_R] = i2pi*Res(f(z), z=z_k), 3. showing that if R is made large enough, the contour mentioned above will contain the singularities of f(z) that lie in the upper half of the complex plane (y>=0), and 4. the value of int[f(z), C_r] will go to zero as R goes to infinity, and so int[f(x), -R, R] = i2pi*Res(f(z), z=z_k). From this, I have so far concluded that if the improper real integral of f(x) converges, then: 1. f(z) has no singularities that lie on the real axis, and 2. the value of the integral int[f(z), C_R] must go to zero as R goes to infinity. However, I do not know if this approach is even answering the question (in the case of improper real integrals of f(x)). Am I on the right track?? Once again, so far I have only considered improper real integrals of f(x). Any help would be very, very much appreciated, thanks everybody!!
Blomster is offline  
 
May 7th, 2012, 03:20 AM   #2
Newbie
 
Joined: May 2012

Posts: 2
Thanks: 0

Re: Conditions for convergence of real integral

Edit to previous post: C_R is the simple closed contour described counterclockwise consisting of the top half of the circle abs(z)=R. It does not include the segment of the real axis from z=-R to z=R. My apologies,
Blomster is offline  
Reply

  My Math Forum > College Math Forum > Complex Analysis

Tags
conditions, convergence, integral, real



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
Real integral Math1234 Real Analysis 1 January 21st, 2014 12:14 PM
Integral Convergence veronicak5678 Real Analysis 5 April 9th, 2012 01:19 PM
Convergence of Integral jams Complex Analysis 1 March 5th, 2012 04:02 PM
Integral with Difficult Conditions uniquesailor Real Analysis 5 February 24th, 2012 11:13 AM
Integral - Real in C Complex Analysis 13 December 4th, 2010 06:39 AM





Copyright © 2019 My Math Forum. All rights reserved.