Mathematical Tools for Physics

14—Complex Variables 430

∫

C 1

eikzdz

a^4 +z^4

C 1

I’m going to use the same method as before, pushing the contour past some poles, but I have to be a bit more
careful this time. The exponential, not the 1 /z^4 , will play the dominant role in the behavior at infinity. Ifkis

positive then ifz=iy, the exponentialei

(^2) ky
=e−ky→ 0 asy→+∞. It will blow up in the−i∞direction. Of
course ifkis negative the reverse holds.
Assumek > 0 , then in order to push the contour into a region where I can determine that it’s zero, I have
to push it toward+i∞. That’s where the exponential drops rapidly to zero. It goes to zero faster than any
inverse power ofy, so even with the length of the contour going asπR, the combination vanishes.
C 2 C 3 C 4
As before, when you pushC 1 up toC 2 and toC 3 , nothing has changed, because the contour has crossed
no singularities. The transition toC 4 happens because the pairs of straight line segments cancel when they are
pushed together and made to coincide. The large contour is pushed to+i∞where the negative exponential kills
it. All that’s left is the sum over the two residues ataeiπ/^4 andae^3 iπ/^4.
∫
C 1

=

∫

C 4

= 2πi

∑

Res

eikz

a^4 +z^4

The denominator factors as

a^4 +z^4 = (z−aeiπ/^4 )(z−ae^3 iπ/^4 )(z−ae^5 iπ/^4 )(z−ae^7 iπ/^4 )

The residue ataeiπ/^4 =a(1 +i)/

√

2 is the coefficient of 1 /(z−aeiπ/^4 ), so it is

eika(1+i)/

√

2

(aeiπ/^4 −ae^3 iπ/^4 )(aeiπ/^4 −ae^5 iπ/^4 )(aeiπ/^4 −ae^7 iπ/^4 )

1

2 3