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1  Integrating Term by Term

Theorem 1 Consider the power series

f(z) = ¥
å
 n = 0 
 anzn\text, | z| < R(R ¹ 0)

Let C be a simple piecewise smooth curve which lies inside the circle of convergence. Then we can integrate the power series term by term:

ó
õ

C 
 æ
è 		¥
å
 n = 0 
 anzn ö
ø 		dz = ¥
å
 n = 0 
 an ó
õ

C 
 zndz
(1)

Proof. The function f(z) defined by the power series is continuous on C, so the integrals in (1) are well-defined. We need to show that


lim
n®¥ 
 ê
ê 		ó
õ

C 
 é
ë 		f(z)- n
å
 k = 0 
 akzk ù
û 		dz ê
ê 		= 0
(2)

Since C lies inside the circle of convergence, the series converges uniformly on C to f(z). For any e, there is an N(e) so that, for all z on C,

n ³ N(e) Þ ê
ê 		f(z) - n
å
 k = 0 
 akzk ê
ê 		< e

By the triangle inequality for integrals and the above inequalities, for n ³ N,

ê
ê 		ó
õ

C 
 é
ë 		f(z)- n
å
 k = 0 
 akzk ù
û 		dz ê
ê 		£ e·(\textlength of C)

Since e is arbitrary, the limit in (2) is zero.  

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