example of changing variable

If one performs in the improper integral

I:=-ekx1+exdx  (0<k<1) (1)

the change of variable (http://planetmath.org/ChangeOfVariableInDefiniteIntegral)


the new lower limitMathworldPlanetmath becomes and the new upper limit 0; hence one obtains


Thus one has recurred I to the integral

0x-kx+1𝑑x, (2)

the value of which has been determined in the entry using residue theoremMathworldPlanetmath near branch pointMathworldPlanetmath.  Accordingly, we may write the result


Calculating the integral (1) directly is quite laborious:  one has to use Cauchy residue theorem to the integral


about the perimetre c of the rectangle

-aReza,0Imz 2π

and then to let  a (one cannot use the same half-disk as in determining the integral (2)).  As for using the method (http://planetmath.org/MethodsOfEvaluatingImproperIntegrals) of differentiationMathworldPlanetmath under the integral sign or taking Laplace transformMathworldPlanetmath with respect to k yields a more complicated integral.

Title example of changing variable
Canonical name ExampleOfChangingVariable
Date of creation 2013-03-22 18:45:49
Last modified on 2013-03-22 18:45:49
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Example
Classification msc 26A06
Related topic UsingResidueTheoremNearBranchPoint
Related topic MethodsOfEvaluatingImproperIntegrals