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## Contour Integration Around the Branch Cut

Lecture 37

If the poles of a Green's function coalesce into a branch cut, can one expect that the sum over the discrete eigenfunctions, Eq.(4.27), mutates into a corresponding integral? The answer is `yes', and this means that instead of representing a function as a discrete sum of eigenfunctions, one now represents a function as an integral transform. The Green's function for a semi-infinite string furnishes us with the archetypical recipe for obtaining this integral transform. It is a two step process:

1. Evaluate the contour integral of over a circle with unlimited large radius:

When one interchanges and on the right hand side. The contour path of integration is

In terms of the complex variable

this contour integral extends over a semicircle from to

The integrand is analytic in . Consequently, the semicircle can be straightened out into a line segment along the real axis. The integral becomes therefore

For a semi-infinite string the domain variables are only positive, . Therefore the first Dirac delta function vanishes. We are left with

 (471)

2. The second step also starts with the closed contour integral

 (472)

but this time the circular contour gets deformed into two linear paths on either side of the positive real axis , the branch cut of

Designate the two values of on opposite sides of the branch cut by and . The integral is therefore
 (473)

To evaluate the difference note that
the value of is

Consequently, the value of the Green's function at these locations is

The discontinuity across the branch cut is therefore
 (474)

Insert this result into Eq.(4.75), change the integration variable to and obtain the result that
 (475)

This two step procedure yields two alternative expressions, Eqs.(4.73) and (4.77) for the contour integral of the Green's function. Their equality yields the spectral representation of the Dirac delta function for a semi-infinite string,

 (476)

Next: Fourier Sine Theorem Up: Spectral Representation of the Previous: Coalescence of Poles into   Contents   Index
Ulrich Gerlach 2010-12-09