6.3 Integration of Analytic Functions: Cauchy’s Integral
Formula
We can define the complex integration as a natural extension of Riemann integral
with regard to a real variable. Suppose that there is a curve C in the complex plane.
Both ends of the curve are fixed and located at z a and z b . Suppose also that individual
points of the curve C are described using a parameter t such that
z ¼ z t
ð Þ ¼ x t
ð Þ þ iy t
ð Þ t 0 t a t t n t b
ð
Þ ,
ð6:71Þ
where z 0 z a ¼ z(t a ) and z n z b ¼ z(t b ) together with z i ¼ z(t i ) (0 i n). For this
parametrization, we assume that C is subdivided into n pieces designated by z 0 , z 1 ,
Á Á Á, z n . Let f (z) be a complex function defined in a region containing C. In this
situation, let us assume the following summation S n :
S n ¼
X n
i¼1
f ζ i
ð Þ z i À z iÀ1
ð
Þ ,
ð6:72Þ
where ζ i lies between z i À 1 and z i (1
i
n). Taking the limit n ! 1 and
concomitantly jz i À z i À 1 j ! 0, we have
lim
n!1
S n ¼ lim
n!1
X n
i¼1
f ζ i
ð Þ z i À z iÀ1
ð
Þ :
ð6:73Þ
If the constant limit exists independent of choice of z i (1
i
n À 1) and
ζ i (1 i n), this limit is called a contour integral of f (z) along the curve C. We
write it as
I ¼ lim
n!1
S n
Z
C
f z
ð Þdz ¼
Z z b
z a
f z
ð Þdz:
ð6:74Þ
Using (6.48) and z i ¼ x i + iy i , we rewrite (6.72) as
S n ¼
X n
i¼1
u ζ i
ð Þ x i À x iÀ1
ð
ÞÀv ζ i
ð Þ y i À y iÀ1
ð
Þ
½
þ
X n
i¼1
i v ζ i
ð Þ x i À x iÀ1
ð
Þþu ζ i
ð Þ y i À y iÀ1
ð
Þ
½
:
Taking the limit of n ! 1 as well as jx i À x i À 1 j ! 0 and jy i À y i À 1 j ! 0
(0 i n), we have
I ¼
Z
C
u x, y
ð Þdx À v x, y
ð Þdy
½
þ i
Z
C
v x, y
ð Þdx þ u x, y
ð Þdy
½
:
ð6:75Þ
Further rewriting (6.75), we get
6.3 Integration of Analytic Functions: Cauchy’s Integral Formula
207
Formula
We can define the complex integration as a natural extension of Riemann integral
with regard to a real variable. Suppose that there is a curve C in the complex plane.
Both ends of the curve are fixed and located at z a and z b . Suppose also that individual
points of the curve C are described using a parameter t such that
z ¼ z t
ð Þ ¼ x t
ð Þ þ iy t
ð Þ t 0 t a t t n t b
ð
Þ ,
ð6:71Þ
where z 0 z a ¼ z(t a ) and z n z b ¼ z(t b ) together with z i ¼ z(t i ) (0 i n). For this
parametrization, we assume that C is subdivided into n pieces designated by z 0 , z 1 ,
Á Á Á, z n . Let f (z) be a complex function defined in a region containing C. In this
situation, let us assume the following summation S n :
S n ¼
X n
i¼1
f ζ i
ð Þ z i À z iÀ1
ð
Þ ,
ð6:72Þ
where ζ i lies between z i À 1 and z i (1
i
n). Taking the limit n ! 1 and
concomitantly jz i À z i À 1 j ! 0, we have
lim
n!1
S n ¼ lim
n!1
X n
i¼1
f ζ i
ð Þ z i À z iÀ1
ð
Þ :
ð6:73Þ
If the constant limit exists independent of choice of z i (1
i
n À 1) and
ζ i (1 i n), this limit is called a contour integral of f (z) along the curve C. We
write it as
I ¼ lim
n!1
S n
Z
C
f z
ð Þdz ¼
Z z b
z a
f z
ð Þdz:
ð6:74Þ
Using (6.48) and z i ¼ x i + iy i , we rewrite (6.72) as
S n ¼
X n
i¼1
u ζ i
ð Þ x i À x iÀ1
ð
ÞÀv ζ i
ð Þ y i À y iÀ1
ð
Þ
½
þ
X n
i¼1
i v ζ i
ð Þ x i À x iÀ1
ð
Þþu ζ i
ð Þ y i À y iÀ1
ð
Þ
½
:
Taking the limit of n ! 1 as well as jx i À x i À 1 j ! 0 and jy i À y i À 1 j ! 0
(0 i n), we have
I ¼
Z
C
u x, y
ð Þdx À v x, y
ð Þdy
½
þ i
Z
C
v x, y
ð Þdx þ u x, y
ð Þdy
½
:
ð6:75Þ
Further rewriting (6.75), we get
6.3 Integration of Analytic Functions: Cauchy’s Integral Formula
207