Which Contour: Rectangle or Semi-Circle?

• Jul 31st 2006, 12:51 AM
Jippo
Which Contour: Rectangle or Semi-Circle?
When evaluating a real integral over the real axis (i.e. -infty to + infty) using methods of contour integration, are there any hard and fast rules regarding the integrand which may help in deciding whether to use a Rectangular or a Circular contour, for example?
• Aug 3rd 2006, 04:11 AM
Jippo
Nobody can help, huh?
• Aug 3rd 2006, 04:15 AM
CaptainBlack
Quote:

Originally Posted by Jippo
Nobody can help, huh?

Symmetry is often a good guide.

RonL
• Aug 3rd 2006, 07:51 AM
ThePerfectHacker
Quote:

Originally Posted by Jippo
Rectangular or a Circular contour, for example?

I prefer using a semi-circle.
Because, if you use squares you have,
$\lim_{n\to\infty} \int^n \int^n$
In semi-circles, there is only one variable tending to infinity,
$\lim_{r\to\infty} \int_{-\pi/2}^{\pi/2} \int^r_0$
• Aug 3rd 2006, 08:34 AM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
I prefer using a semi-circle.
Because, if you use squares you have,
$\lim_{n\to\infty} \int^n \int^n$
In semi-circles, there is only one variable tending to infinity,
$\lim_{r\to\infty} \int_{-\pi/2}^{\pi/2} \int^r_0$

if you have symmetry you can cancel two sides of thesquare.

RonL