1550251515-Classical_Complex_Analysis__Gonzalez_

466 Chapter^7

y

-u + ia ia z = x + ia u + ia

z = -u + iy

z = u + iy

z = x

-u 0 u x

Fi.g. 7.l'T

Decomposing ('/.15-1) into four integrals each along a side of R, we
obtain

j

u e-x^2 dx+ la .e-(u+iy)^2 idy+ f,-u e~(x+ia)^2 dx+ (^10) e-(-u+iy)^2 idy=O
-u 0 u a
or
2 fu e-x^2 dx - ea^2 ju e-x\ cos 2ax - i sin 2ax) dx
lo -u

• 2e-u
  2

1a eY

2

sin2uy dy = 0

By equating to zero the real part of the left-hand side, we find that

2 2 2 2 2.

l

2 u e-x dx - ea ju e-x cos2axdx +2e-u la eY sm2uydy = 0

0 -u 0

(7.15-2)
Since

j

u e-x^2 cos2axdx = 2 {u e-x^2 cos2axdx

-u lo

equation (7.15-2) reduces to

lu e-x

2

dx - ea

2

1u e-x

2

cos 2ax dx + e-u

2

1a eY

2

sin 2uy dy = 0 (7.15-3)

We have

2 2 • 2 2

I l

e-u a I la

0

eY sm2uydy :::; e-u

0

eY dy

so the last term in (7.15-3) tends to zero as u --t oo. Also, as u --t oo
the first integral tends to %ft. Hence the integral in the middle term