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Laplace integrals


The improper integrals

โˆซโˆž-โˆžacosxx2+a2๐‘‘xโ€ƒandโ€ƒโˆซโˆž-โˆžxsinxx2+a2๐‘‘x,

where a is a positive , are called Laplace integrals.โ€‰ Both of them have the same value ฯ€e-a.

The evaluation of the Laplace integrals can be performed by first determining the integrals

โˆซโˆž-โˆžeixx-ia๐‘‘xโ€ƒandโ€ƒโˆซโˆž-โˆžeixx+ia๐‘‘x

where one integrates along the real axisMathworldPlanetmath.โ€‰ Therefore one has to determine the integrals

โˆฎeizz-ia๐‘‘zโ€ƒandโ€ƒโˆฎeizz+ia๐‘‘z

around the perimeter of the half-disk with the arc in the upper half-plane, centered in the origin and with the diameter โ€‰(-R,+R).โ€‰ The residue theoremMathworldPlanetmath yields the values

โˆฎeizz-ia๐‘‘z=โ€„2iฯ€e-aโ€ƒandโ€ƒโˆฎeizz+ia๐‘‘z=โ€‰0.

As in the entry example of using residue theorem, the parts of these contour integrals along the half-circle tend to zero whenโ€‰ Rโ†’โˆž.โ€‰ Consequently,

โˆซโˆž-โˆžeixx-ia๐‘‘x=โ€„2iฯ€e-aโ€ƒandโ€ƒโˆซโˆž-โˆžeixx+ia๐‘‘x=โ€„0.

These equations imply by adding and subtracting and then taking the real (http://planetmath.org/RealPart) and the imaginary partsDlmfMathworld, the

โˆซโˆž-โˆžacosxx2+a2๐‘‘x=โˆซโˆž-โˆžxsinxx2+a2๐‘‘x=ฯ€e-a.

References

  • 1 R. Nevanlinna & V. Paatero: Funktioteoria.โ€‰ Kustannusosakeyhtiรถ Otava. Helsinki (1963).
Title Laplace integrals
Canonical name LaplaceIntegrals
Date of creation 2013-03-22 18:43:17
Last modified on 2013-03-22 18:43:17
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Definition
Classification msc 40A10