Power Series
This chapter is devoted to an important tool for constructing holomorphic functions: convergent power series. It is the basis for the introduction of new non-algebraic holomorphic functions, the elementary transcendental functions. It turns out that power
- PDF / 397,939 Bytes
- 42 Pages / 439.36 x 666.15 pts Page_size
- 13 Downloads / 272 Views
Power Series
This chapter is devoted to an important tool for constructing holomorphic functions: convergent power series. It is the basis for the introduction of new non-algebraic holomorphic functions, the elementary transcendental functions. It turns out that power series play an even more central role in the theory of holomorphic functions, a role beyond enabling the construction of complex transcendental functions that are the extension of the real transcendental functions. A much stronger result holds. All holomorphic functions are (at least locally) convergent power series. This will be proven in the next chapter. The first section of this chapter is devoted to a discussion of elementary properties of complex power series. Some material from real analysis, not usually treated in books or courses on that subject, is studied. The concept of a convergent power series is extended from series with real coefficients to complex power series, and tests for convergence are established. In the second section, we show that convergent power series define holomorphic functions. Section 3.3 introduces important complex-valued functions of a complex variable including the exponential function, the trigonometric functions, and the logarithm. This is followed by Sects. 3.4 and 3.5, which describe an identity principle and introduce the new class of meromorphic functions; these functions are holomorphic on a domain except that they “assume the value 1” (in a controlled way) at certain isolated points, known as the poles of the function. Meromorphic functions are defined locally as ratios of functions having power series expansions. It will hence follow subsequently that these are locally ratios of holomorphic functions. After some more work we will be able to replace “locally” by “globally.” We develop the fundamental identity principle and its corollary, the principle of analytic continuation, for functions that locally have convergent power series expansions, and discuss the zeros and poles of a meromorphic function. The principle of analytic continuation is one of the most powerful results in complex function theory. Once we show that every holomorphic function is locally defined by a power series, we will see that the principle of analytic continuation says that a holomorphic function defined on an open connected set is
R.E. Rodr´ıguez et al., Complex Analysis: In the Spirit of Lipman Bers, Graduate Texts in Mathematics 245, DOI 10.1007/978-1-4419-7323-8 3, © Springer Science+Business Media New York 2013
39
40
3 Power Series
remarkably rigid: its behavior at a single point in the set determines its behavior at all other points of the set. Holomorphicity at a point is an extremely strong concept.
3.1 Complex Power Series Let A C and let ffn g D ffn g1 nD0 be a sequence of functions defined on A (in the previous chapter sequences were indexed by Z>0 ; for convenience, in this chapter, they will be indexed by Z0 ). We form the new sequence (known as a series) fSN g, where SN .z/ D
N X
fn .z/; z 2 A;
nD0
and th
Data Loading...
