# (p.245) Appendix 8 CONTOUR INTEGRALS

# (p.245) Appendix 8 CONTOUR INTEGRALS

The Cauchy residue theorem states that the integral anticlockwise around a closed contour in the complex plane of any function *f*(*z*) which is analytic apart from poles within the contour is equal to 2πi multiplied by the sum of the residues of *f*(*z*) at these poles. We use this theorem to evaluate some of the integrals required in Chapters 5 and 6.

First we consider the integral

*W*with

*W*

_{0}, and note that

*t*in (A8.1) can be set equal to zero. We choose a contour consisting of the part of the real axis from −

*R*to +

*R*, closed by the semicircle R exp(iθ) in the lower half plane, as shown in Fig. A8.1. The lower half plane is chosen to ensure that the integrand tends to zero on the semicircle as

*R*→ ∞. The simple pole of the integrand in the lower half plane is at and the residue there is Letting

*R*→ ∞ gives

*clockwise*direction. Note that if the integrand contained exp(i ω

*t*) rather than exp(−i ω

*t*), then we could close the contour in the upper half plane and obtain the same result (A8.2).

The second integral

*R*to +

*R*, indented by constructing a small semicircle of radius ρ

*R*exp(iθ) also in the upper half plane, as shown in Fig. A8.2. The simple pole of the integrand in the upper half plane is at Δ = iγ and the residue there is . The integral around the large semicircle tends to zero as

*R*→ ∞ while the integral around the small semicircle as ρ → 0 is

The third integral we evaluate is that in (6.5.21), required to establish the completeness of the dressed states . Consider

*I*

_{3}into partial fractions, so that

*F*(ω) is defined for ω real. In order to use contour integration to evaluate

*I*

_{3}, we need to extend the definition of

*F*(ω), and hence of , to include complex ω by analytic continuation. We formally evaluate the integral (A8.7) by closing a contour in the upper half plane, as for

*I*

_{2}, giving

*S*(ω) is the sum of the residues of at the poles of

*W*

^{2}(Δ) in the upper half plane. Consider the function G(ω) defined, for values of ω in the upper half plane, by

*F*(ω) given in (A8.9), we see that

*F*(ω) becomes

*G*(ω) + πi

*W*

^{2}(ω) in the upper half plane. Similarly in the lower half plane,

*G*(ω) = 2πi

*S*(ω) and

*F*(ω) becomes

*G*(ω) = πi

*W*

^{2}(ω). The integrand of

*I*

_{3}in (A8.8) analytically continued into the upper half plane is therefore given by

*G*(ω) has no zeros there, as we now show. Writing ω =

*x*+ i

*y*and setting ω − G(ω) = 0 leads to

*y*> 0, corresponding to the upper half plane, and hence there are no solutions. We now evaluate

*I*

_{3}by contour integration closing the contour, as before, with a large semicircle of radius

*R*in the upper half plane. We add to

*I*

_{3}the integral of (A8.12) around the semicircle, this additional integral being identically zero in the limit R → ∞. The second terms of

*I*

_{3}and of the integral of (A8.12) combine to give a zero contribution because has no poles in the upper half plane. Hence

*I*

_{3}is given by the remaining integrals, as follows:

*W*

^{2}(

*R*exp(iθ)) must tend to a finite value and

*G*(

*R*exp(iθ)) must tend to zero as R → ∞ in order for

*F*(w) to be finite. Since

*I*

_{3}is real, the second term of (A8.14) must be zero, as it is purely imaginary. The first term is just

*I*

_{3}/2 while the last is simply 1/2. Hence

*I*

_{3}= 1 as required.

The Cauchy residue theorem can also be used to sum particular infinite series. Suppose *f*(z) is a function which is analytic at the integers ,… and tends to zero at least as fast as as . Then

*n*increases to allow use of (A8.15). However, this can be overcome by writing

*S*

_{1}as

*n*

^{−2}for large

*n*and therefore tend to zero sufficiently fast as

*n*→ ∞ to allow the application of (A8.15). The simple poles of are at z = ±ω/ω

_{0}. The residues of at these points are both . Hence

*n*

^{−2}for large

*n*as required. The double poles of are at . The residue of at either pole may be calculated, using the formula for double poles, as