COMPLEX ANALYSIS: Maple Worksheets, 2009
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

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COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
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CHAPTER 7 TAYLOR and LAURENT SERIES

Section 7.1 Uniform Convergence

Throughout this text we have compared and contrasted properties of complex functions with functions whose domain and range lie entirely within the reals. There are many similarities, such as the standard differentiation formulas. However, there are also some surprises, and in this chapter we will encounter one of the hallmarks distinguishing complex functions from their real counterparts.

It is possible for a function defined on the real numbers to be differentiable everywhere and yet not be expressible as a power series (see Exercise 7.2.20 at the end of Section 7.2). In the complex case, however, things are much simpler! It turns out that if a complex function is analytic in the disk , its Taylor series about will converge to the function at every point in this disk. Thus, analytic functions are locally nothing more than glorified polynomials.

Definition 7.1: Uniform convergence

The sequence converges uniformly to on the set T if for every , there exists a positive integer (which depends only on ) such that if , then for all .

Example 7.1, Page 262. The sequence = converges uniformly to the function on the entire complex plane because for any >0 , is satisfied for all for > , where is any integer greater than . We leave the details for showing this as an exercise.

There is a useful procedure known as the Weierstrass M-test that can help determine whether an infinite series is uniformly convergent.

Theorem 7.1 (Weierstrass M-test)

Suppose the infinite series has the property that for each k, for all .
If converges, then converges uniformly on T.

Theorem 7.2 gives an interesting application of the Weierstrass M-test.

Theorem 7.2 Suppose the power series has radius of convergence .

Then for each , , the series converges uniformly on the closed disk .

Corollary 7.1 For each , , the geometric series converges uniformly on the closed disk .

Theorem 7.3 Suppose is a sequence of continuous functions defined on a set T containing the contour C. If converges uniformly to on the set T, then
(i) is continuous on T, and
(ii) = .

Corollary 7.2 If the series converges uniformly to on the set T, and C is a contour contained in T, then
= .

Example 7.2, Page 266. Show that , for all in : .

> f:='f': p:='p': P:='P': s:='s': t:='t': z:='z': Z:='Z':
f := z -> log(1-z):
t := taylor(f(Z), Z=0, 11):
s := subs(Z=z,t):
p := z -> subs(Z=z,convert(t, polynom)):
`f(z) ` = f(z);
`f(z) ` = s;
P[10](z) = p(z);

Or we could use Maple's "unapply" procedure.

> f:='f': p:='p': P:='P': s:='s': t:='t': z:='z': Z:='Z':
f := z -> log(1-z):
s := taylor(f(z), z=0, 11):
p:=unapply(convert(taylor(f(z),z=0,11),polynom),z):
`f(z) ` = f(z);
`f(z) ` = s;
P[10](z) = p(z);

Sum up the terms to verify that we have things right.

> f:='f': n:='n': s:='s': S:='S': z:='z':
f := z -> log(1-z):
S10 := z -> sum(-1/n*z^n, n=1..10):
S := z -> sum(-1/n*z^n, n=1..infinity):
`f(z) ` = f(z);
s[10](z),` = `,Sum(-1/n*z^n, n=1..10) = S10(z);
s[infinity](z),` = `,Sum(-1/n*z^n, n=1..infinity) = S(z);

The real variable plot of and is:

> plot({f(x),S10(x)},
x=-0.999..0.999, y=-3..0.7,
title=`y=ln(1-z) and y=s10(x)`,
labels=[` x`,`y `],
tickmarks=[5,7],
view=[-1..1,-3..0.7]);

End of Section 7.1.