Keyhole contour integration method - Revision history

Vipul: 2 revisions

2008-05-18T19:14:15Z

2 revisions

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Vipul: /* Computing the integral over keyhole contours */

2008-05-01T20:18:20Z

Computing the integral over keyhole contours

← Older revision		Revision as of 20:18, 1 May 2008
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	* Send the radius of the outer circle to <math>\infty</math>		* Send the radius of the outer circle to <math>\infty</math>
	* Send the ~~radois~~ of the inner circle to <math>0</math>		* Send the radius of the inner circle to <math>0</math>
	* Send the angle of parting that separates the two ends of the inner circle, to <math>0</math>		* Send the angle of parting that separates the two ends of the inner circle, to <math>0</math>

	We then use various limiting techniques to compute the integral on the outer circle, the inner circle, and the two lines.		We then use various limiting techniques to compute the integral on the outer circle, the inner circle, and the two lines.

Vipul: New page: The '''keyhole contour integration method''' is a method for computing Cauchy principal values for integrals of real-analytic functions over the positive real line. Specifically, it he...

2008-05-01T18:37:15Z

New page: The '''keyhole contour integration method''' is a method for computing Cauchy principal values for integrals of real-analytic functions over the positive real line. Specifically, it he...

New page

The '''keyhole contour integration method''' is a method for computing [[Cauchy principal value]]s for integrals of real-analytic functions over the positive real line. Specifically, it helps to solve problems of the form:

<math>\int_0^\infty f(x) \, dx</math>

There are two motivations for using keyhole contours:

* The function branches. For instance, the logarithm function, or a fractional power map.
* The function has poles on the negative real axis, so we cannot use the [[semicircular contour integration method]] or the [[mousehole contour integration method]].

==Setting up the complex-valued function==

We first choose a meromorphic function <math>g</math> on the [[generalized slit plane|plane slit at the positive real axis]], such that its limit from one direction, has real or imaginary part equal to <math>f</math>

==Computing the integral over keyhole contours==

The keyhole contour is defined as follows: it has an outer, almost complete circle and an inner, almost complete circle, both centered at the origin. The two circles are joined by straight lines parallel to the real axis. This gives the shape of a keyhole. We take a limit in three ways:

* Send the radius of the outer circle to <math>\infty</math>
* Send the radois of the inner circle to <math>0</math>
* Send the angle of parting that separates the two ends of the inner circle, to <math>0</math>

We then use various limiting techniques to compute the integral on the outer circle, the inner circle, and the two lines.