COMPLEX ANALYSIS: Maple Worksheets, 2001
(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 7.5 Applications of Taylor and Laurent Series

In this section we show how Taylor and Laurent series can be used to derive important properties of analytic functions. We begin by showing that the zeros of an analytic function must be "isolated" unless the function is identically zero. A point of a set is called isolated if there exists a disk about that does not contain any other points of .

Theorem 7.13 Suppose is analytic at and that . If is
not identically zero, then there exists a punctured disk in which has no zeros.

Corollary 7.11 (L'Hopital's rule) Suppose and are analytic at .

If and but , then = .

Theorem 7.14 (Division of power series)
Suppose and are analytic at with power series representations
for and for .
If , then the quotient has the power series representation
, where the coefficients satisfy the equations
.
In other words, the series for the quotient can be obtained by the familiar process of
dividing the series for by the series for using the standard long division algorithm.

Example 7.15, Page 300. Find the firt few terms of the
Maclaurin series for and use it to compute .

> f:='f': p:='p': P:='P': S:='S': z:='z': Z:='Z':
f := z -> sec(z):
S := series(f(Z), Z=0, 14):
p := z -> subs(Z=z,convert(S, polynom)):
`f(z) ` = f(z);
`f(z) ` = subs(Z=z,S);
P[12](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 -> sec(z):
S := taylor(f(z), z=0, 14):
p:=unapply(convert(taylor(f(z),z=0,14),polynom),z):
`f(z) ` = f(z);
`f(z) ` = S;
P[12](z) = p(z);

> D4 := simplify(diff(f(Z), Z$4)):
f4 := z -> subs(Z=z, D4):
S4 := simplify(diff(p(Z), Z$4)):
p4 := z -> subs(Z=z, S4):
`f'(z) ` = simplify(diff(f(z), z));
`p'(z) ` = simplify(diff(p(z), z)); ` `;
`f''(z) ` = simplify(diff(f(z), z$2));
`p''(z) ` = simplify(diff(p(z), z$2)); ` `;
`f'''(z) ` = simplify(diff(f(z), z$3));
`p'''(z) ` = simplify(diff(p(z), z$3)); ` `;
`f''''(z) ` = f4(z);
`p''''(z) ` = p4(z); ` `;
`f''''(0) ` = eval(f4(0));
`p''''(0) ` = p4(0);

Theorem 7.15 (Riemann) Suppose that is analytic in .
If is bounded in , then either is analytic at or has a removable
singularity at .

Theorem 7.16 The function has a pole of order k at
if and only if .

Theorem 7.18 The function has an essential singularity at
if and only if does not exist.

Example 7.16, Page 302. Show that the function
is NOT continuous at .

> g:='g': G:='G': x:='x': y:='y': z:='z':
G := z -> exp(-1/z^2):
`g(z) ` = G(z); ` `;
`g(x + I*0) ` = evalc(G(x + I*0)),` and `,
`g(0 + I*y) ` = evalc(G(0 + I*y));
Limit(g(x), x=0, right),` = `,
Limit(G(x), x=0, right) = limit(G(x), x=0, right);
Limit(g(0 + I*y), y=0, right),` = `,
Limit(G(I*y), y=0, right) = limit(G(I*y), y=0, right);

Since the limits are NOT the same, is NOT continuous at .
REMARK. Although is known to be infinitely differentiable,
it does not have a Maclaurin series expansion. However, it does have a Laurent series.

> f:='f': p:='p': S:='S': z:='z':
f := z -> exp(-1/z^2):
s := z -> series(f(z), z=infinity, 14):
`f(z) ` = f(z);
`f(z) ` = s(z);

End of Section 7.5.