Complex Analysis - Maths KU

These properties are important in
the evaluation of contour integrals.
They will often be used without mention.
☞
5.2 Complex Integrals 249
variable t continuous on a common interval a ≤ t ≤ b, then we define the
integral of the complex-valued function f(t) =f1(t)+if2(t) ona ≤ t ≤ b in
terms of the definite integrals of the real and imaginary parts of f:
� b � b
� b
f(t) dt = f1(t) dt + i f2(t) dt. (4)
a
a
The continuity of f1 and f2 on [a, b] guarantees that both � b
a f1(t) dt and
� b
a f2(t) dt exist.
All of the following familiar properties of integrals can be proved directly
fromthe definition given in (4). If f(t) =f1(t)+if2(t) and g(t) =g1(t)+ig2(t)
are complex-valued functions of a real variable t continuous on an interval
a ≤ t ≤ b, then
� b
� b
kf(t) dt = k f(t) dt, k a complex constant, (5)
� b
a
a
� b � b
(f(t)+g(t)) dt = f(t) dt + g(t) dt, (6)
a
� b
a
� c
a
� b
f(t) dt = f(t) dt + f(t) dt, (7)
a
� a
a
� b
c
f(t) dt = − f(t) dt. (8)
b
a
In (7) we choose to assume that the real number c is in the interval [a, b].
We now resume our discussion of contour integrals.
Evaluation of Contour Integrals To facilitate the discussion on
how to evaluate a contour integral �
f(z) dz, let us write (2) in an abbreviated
C
form. If we use u+iv for f, ∆x+i∆y for ∆z, limfor lim
||P ||→0 , � for �n k=1 and
then suppress all subscripts, (2) becomes
�
C
f(z)dz = lim � (u + iv)(∆x + i∆y)
�� � �
= lim (u∆x − v∆y)+i (v∆x + u∆y) .
The interpretation of the last line is
�
�
�
f(z) dz = udx− vdy+ i vdx+ udy. (9)
C
C
C
See Definition 5.2. In other words, the real and imaginary parts of a contour
integral �
�
�
f(z) dz are a pair of real line integrals udx− vdy and
C C
vdx+ udy. Now if x = x(t), y = y(t), a ≤ t ≤ b are parametric equations
C
of C, then dx = x ′ (t) dt, dy = y ′ (t) dt. By replacing the symbols x, y, dx,
a