Page 403 - Schaum's Outline of Theory and Problems of Advanced Calculus

394 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16

INTEGRALS
Let f ðzÞ be defined, single-valued and continuous in a region r.We define the integral of f ðzÞ along
some path C in r from point z 1 to point z 2 , where z 1 ¼ x 1 þ iy 1 ; z 2 ¼ x 2 þ iy 2 ,as
ð ð ð ð
ðx 2 ;y 2 Þ ðx 2 ;y 2 Þ ðx 2 ;y 2 Þ
udx vdy þ i vdx þ udy
f ðzÞ dz ¼ ðu þ ivÞðdx þ idyÞ¼
C ðx 1 ;y 1 Þ ðx 1 ;y 1 Þ ðx 1 ;y 1 Þ
With this definition the integral of a function of a complex variable can be made to depend on line
integrals for real functions already considered in Chapter 10. An alternative definition based on the
limit of a sum, as for functions of a real variable, can also be formulated and turns out to be equivalent
to the one above.
The rules for complex integration are similar to those for real integrals. An important result is
ð ð ð
f ðzÞ dz @

j f ðzÞjjdzj @ M ds ¼ ML
ð4Þ
C C C
where M is an upper bound of j f ðzÞj on C, i.e., j f ðzÞj @ M, and L is the length of the path C.
Complex function integral theory is one of the most esthetically pleasing constructions in all of
mathematics. Major results are outlined below.

CAUCHY’S THEOREM
Let C be a simple closed curve. If f ðzÞ is analytic within the region bounded by C as well as on C,
then we have Cauchy’s theorem that
ð þ
f ðzÞ dz ¼ 0
f ðzÞ dz ð5Þ
C C
where the second integral emphasizes the fact that C is a simple closed curve.
ð
z 2
Expressed in another way, (5)is equivalent to the statement that f ðzÞ dz has a value independent of
z 1
the path joining z 1 and z 2 . Such integrals can be evaluated as Fðz 2 Þ Fðz 1 Þ, where F ðzÞ¼ f ðzÞ. These
0
results are similar to corresponding results for line integrals developed in Chapter 10.
EXAMPLE. Since f ðzÞ¼ 2z is analytic everywhere, we have for any simple closed curve C
þ
2zdz ¼ 0
C
1þi
ð 1þi
2 2
Also, 2zdz ¼ z 2 ¼ð1 þ iÞ ð2iÞ ¼ 2i þ 4

2i 2i
CAUCHY’S INTEGRAL FORMULAS
If f ðzÞ is analytic within and on a simple closed curve C and a is any point interior to C, then
1 þ f ðzÞ
dz
2 i C z a
f ðaÞ¼ ð6Þ
where C is traversed in the positive (counterclockwise) sense.
Also, the nth derivative of f ðzÞ at z ¼ a is given by
n! þ f ðzÞ
f ðnÞ ðaÞ¼ dz ð7Þ
2 i nþ1
C ðz aÞ
These are called Cauchy’s integral formulas. They are quite remarkable because they show that if
the function f ðzÞ is known on the closed curve C then it is also known within C, and the various