8
Sequences and Series of Functions.
Series Representations. Some Special
Functions
In this chapter we proceed further with the study of sequences and series
of functions that was undertaken in Chapter 4. For the main definitions
and some elementary properties of such sequences and series, we refer the
reader back to Sections 4.10 and 4.11.
8.1 INTEGRATION AND DIFFERENTIATION OF
SEQUENCES AND SERIES OF FUNCTIONS
Theorem 8.1 Suppose that:
1. The :functions Fn(z) are continuous on a contour C (n = 1, 2, ... ).
- Fn(z) ~ F(z) on the contour C.
Thenlim J Fn(z) dz= J lim Fn(z) dz= J F(z) dz
n-+oo n-+oo
G G G
Proof Hypothesis 1 implies either that the functions Fn ( z )'are defined and
continuous on some open set that contains C, or that the fun«tions are
merely defined and continuous along C. In either case, as follows from
the note to Corollary 4.4, the function F(z) is continuous along C. Hence
J 0 F( z) dz exists.
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