Mousehole contour integration method - Revision history

Vipul: 5 revisions

2008-05-18T19:16:58Z

5 revisions

← Older revision	Revision as of 19:16, 18 May 2008
(No difference)

2008-04-28T23:59:54Z

Setting up the complex-valued function

← Older revision		Revision as of 23:59, 28 April 2008
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	If the holomorphic function we construct has a pole of order more than one at the origin, we add to it a polynomial in <math>1/z</math>, so that <math>f</math> continues to remain its imaginary part, but we now have only a simple pole.		If the holomorphic function we construct has a pole of order more than one at the origin, we add to it a polynomial in <math>1/z</math>, so that <math>f</math> continues to remain its imaginary part, but we now have only a simple pole.

	This technique is used to integrate expressions like ~~<math>(\sin^2 x)/(x^2)</math> and~~ <math>(x - \sin x)/(x^3)</math>.		This technique is used to integrate expressions like <math>(x - \sin x)/(x^3)</math>.

	===Computing integrals over mousehole contours===		===Computing integrals over mousehole contours===

2008-04-28T23:59:26Z

Setting up the complex-valued function

← Older revision		Revision as of 23:59, 28 April 2008
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	Although <math>f</math> is real-analytic at <math>0</math>, <math>g</math> has an essential singularity at <math>0</math>.		Although <math>f</math> is real-analytic at <math>0</math>, <math>g</math> has an essential singularity at <math>0</math>.

			If the holomorphic function we construct has a pole of order more than one at the origin, we add to it a polynomial in <math>1/z</math>, so that <math>f</math> continues to remain its imaginary part, but we now have only a simple pole.

			This technique is used to integrate expressions like <math>(\sin^2 x)/(x^2)</math> and <math>(x - \sin x)/(x^3)</math>.

	===Computing integrals over mousehole contours===		===Computing integrals over mousehole contours===

2008-04-28T23:27:55Z

← Older revision		Revision as of 23:27, 28 April 2008
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	The '''mousehole contour integration method''' is a method used for computing [[Cauchy principal value]]s for integrals of real-valued functions <math>f: \R^* \to \R</math>, that may blow up at zero.		The '''mousehole contour integration method''' is a method used for computing [[Cauchy principal value]]s for integrals of real-valued functions <math>f: \R^* \to \R</math>, that may blow up at zero.

			===Setting up the complex-valued function===

			We first choose a function <math>g</math>, holomorphic or meromorphic on the upper half-plane as well as on the real line, except possibly at zero, so that <math>f</math> is the real or imaginary part of <math>g</math>.

			For instance, consider the function:

			<math>f(x) = \operatorname{sinc}(x) := \frac{\sin x}{x}</math>

			We consider here the function:

			<math>g(z) = \frac{e^{iz}}{z}</math>

			Although <math>f</math> is real-analytic at <math>0</math>, <math>g</math> has an essential singularity at <math>0</math>.

			===Computing integrals over mousehole contours===

			A mousehole contour is described as follows: it is made up of semicircles in the upper half-plane of radii <math>R_1 < R_2</math>, and lines on the real axis joining their ends together. We then make <math>R_1 \to 0</math> and <math>R_2 \to \infty</math>, and use considerations like [[Jordan's lemma]] (and actual computation) to determine the integration along the semicircular parts).

			==Examples==

			* [[Computing the sine integral]]

2008-04-27T23:00:32Z

← Older revision		Revision as of 23:00, 27 April 2008
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	The '''mousehole contour integration method''' is a method used for computing [[Cauchy principal value]]s for integrals of real-valued functions <math>f: \R^* \to \R</math>, that may blow up at zero.		The '''mousehole contour integration method''' is a method used for computing [[Cauchy principal value]]s for integrals of real-valued functions <math>f: \R^* \to \R</math>, that may blow up at zero.

	~~===~~