Page 63 - Calculus with Complex Numbers

6 . 2 T h e m e th o d
W e illustrate the method for evaluating inlinite real integrals using complex
calculus by applying it to the integral

O dx
. x2 + 1

considered in Section 6. 1.
Let y = n + yz be the D-shaped contour consisting of the real interval
g- A, .P1 = n together with the upper semicircle yz having g- A, .P1 as diameter
(Figure 6. 1). And consider the contotlr integral

dz
2
.

1
+
y
L
j dz o j dz 1 '
z

J .12 + 1 .l : ' y 2
z
+
.

l
v
.
On n
J dz . /R dx
. :2 + 1 )-R .x2 + 1 '
/1
Inside y For all R > 1 the integrand has one singularity at z = i where the
residue is 1/2ï . Therefore

On yz we have IzI= R therefore

Iz2 + 11k: Rl - 1
(see lnequality 5 of Section 1. 1 1) from which it follows that
1 1
:f z
. :2 + 1 R - 1

Ffgure 8. /