Complex Analysis - Maths KU

Important observation about even
functions
C R
z 3
z 2
–R 0
y
z n
z 1
z 4
x
R
Figure 6.11 Semicircular contour
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6.6 Some Consequences of the Residue Theorem 355
The limit in (9), if it exists, is called the Cauchy principal value (P.V.) of
the integral and is written
� ∞
� R
P.V. f(x) dx = lim
−∞
R→∞
f(x) dx.
−R
(11)
In (10) we have shown that P.V. � ∞
xdx= 0.To summarize:
−∞
Cauchy Principal Value
When an integral ofform (2) converges, its Cauchy principal value is the
same as the value ofthe integral. Ifthe integral diverges, it may still
possess a Cauchy principal value (11).
One final point about the Cauchy principal value: Suppose f(x) is continuous
on (−∞, ∞) and is an even function, that is, f(−x) =f(x).Then
its graph is symmetric with respect to the y-axis and as a consequence
� 0
� R
f(x) dx = f(x) dx (12)
and
−R
� R
−R
0
� 0
f(x) dx = f(x) dx +
−R
� R
0
f(x) dx = 2
� R
0
f(x) dx. (13)
From (12) and (13) we conclude that if the Cauchy principal value (11) exists,
then both � ∞
0 f(x) dx and � ∞
f(x) dx converge.The values of the integrals
−∞
are
� ∞
� ∞
� ∞
f(x) dx =P.V. f(x) dx.
0
f(x) dx = 1
2 P.V.
� ∞
f(x) dx and
−∞
To evaluate an integral � ∞
f(x) dx, where the rational function f(x) =
−∞
p(x)/q(x) is continuous on (−∞, ∞), by residue theory we replace x by the
complex variable z and integrate the complex function f over a closed contour
C that consists of the interval [−R, R] on the real axis and a semicircle CR of
radius large enough to enclose all the poles of f(z) =p(z)/q(z) in the upper
half-plane Im(z) > 0. See Figure 6.11. By Theorem 6.16 of Section 6.5 we
have
� �
� R
n�
f(z) dz = f(z) dz + f(x) dx =2πi Res(f(z),zk),
C
−R
CR
where zk, k =1,2,...,n denotes poles in the upper half-plane.If we can
show that the integral �
f(z) dz → 0asR→∞, then we have
CR
� ∞
� R
P.V. f(x) dx = lim
−∞
R→∞ −R
−∞
f(x) dx =2πi
k=1
−∞
n�
Res(f(z),zk). (14)
k=1