Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
To evaluate improper integrals whose integrands have poles on the real axis, we
can use a path which avoids these singularities by following small semicircles with
centers at the singular points. We now illustrate the procedure with a simple
example.
Example 6.32
Show that
Z 1
0
sin x
x
dx ˆ
2 :
Solution: The function sin…z†=z does not behave suitably at in®nity. So we consider
e iz =z, which has a simple pole at z ˆ 0, and integrate around the contour C
or ABDEFGA (Fig. 6.19). Since e iz =z is analytic inside and on C, it follows from
Cauchy's integral theorem that
I
e iz
z dz ˆ 0
or
Z "
R
C
e ix Z
x dx ‡ e iz Z R
C 2
z dz ‡ e ix Z
x dx ‡ e iz
C 1
z
"
dz ˆ 0: …6:48†
We now prove that the value of the integral over large semicircle C 1 approaches
zero as R approaches in®nity. Setting z ˆ Re i , we have dz ˆ iRe i d; dz=z ˆ id
and therefore
e
ZC iz
Z
1
z dz
Z ˆ
e iz id
e iz d:
In the integrand on the right,
ˆje
iR…cos ‡i sin † jˆje iR cos jje R sin jˆe R sin :
e iz
0
0
Figure 6.19.
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