- #1

kakarukeys

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## Homework Statement

## Homework Equations

find in the limit [tex]r\rightarrow\infty[/tex]

[tex]\frac{-i}{2(2\pi)^2r}\int^\infty_{-\infty}\frac{p\exp(ipr)dp}{\sqrt{p^2+m^2}}[/tex]

the solution (or rather a hint) given by the book:

"The integrand, considered as a complex function of p, has brunch cuts on the imaginary axis starting at [tex]\pm im[/tex].

http://www.stochasticsoccer.com/contour.gif

To evaluate the integral we push the contour up to wrap around the upper branch cut. Defining [tex]\rho = - ip[/tex], we obtain

[tex]\frac{1}{4(\pi)^2r}\int^\infty_{m}\frac{\rho\exp(-\rho r)d\rho}{\sqrt{\rho^2-m^2}}[/tex]

in the limit, tends to

[tex]\exp(-mr)[/tex]

## The Attempt at a Solution

I can't find any theorem in complex analysis that permits a "push" of the contour shown in the figure, so I try the contour shown below:

http://www.stochasticsoccer.com/contour2.gif

but when I take limit R goes to infinity, the maximum modulus integral bound around the semicircle doesn't go to zero. so I'm stuck. Expert pls help me.

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