COMPLEX ANALYSIS: Maple Worksheets, 2001
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

Complimentary software to accompany the textbook:

COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
Jones and Bartlett Publishers, Inc., 40 Tall Pine Drive, Sudbury, MA 01776
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CHAPTER 4 SEQUENCES, JULIA and MANDELBROT SETS, and Power Series

Section 4.4 Power Series Functions

The function is called a power series .

Theorem 4.14 Suppose that .

Then the set of points for which the series converges is one of the following:

i. The single point .

ii. The disk = ,

along with part (either none, some, or all) of the circle = .

iii. The entire complex plane.

Another way to phrase case (ii) is to say that the power series converges if , and diverges if .

We call the number the r adius of convergence of the power series. If we are in case (i), we say that the radius of convergence is zero, and that the radius of convergence is infinity if we are in case (iii).

Theorem 4.14 (Radius of convergence)

Given the power series function , we can find , its radius of convergence, by any of the following methods:

i. Cauchy's root test: . (Provided the limit exists.)

ii. d'Alembert's ratio test: . (Provided the limit exists.)

Example 4.21, Page 153. Find the radius of convergence of the series .

> c:='c': C:='C': n:='n': P:='P':
C := n -> ((n+2)/(3*n+1))^n:
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
`The general term is `, c[n]= C(n); ` `;
`The n-th root is:`;
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
P := C(n)^(1/n):
abs(c[n])^(1/n) = P;
P := simplify(P, assume=positive):
abs(c[n])^(1/n) = P;

Use the root test and find the limit and then the radius of convergence .

> L:='L': R:='R':
L0 := limit(P, n=infinity):
R := 1/L0:
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
c[n] = C(n);
abs(c[n])^(1/n) = P;
`L = `, limit(abs(c[n])^(1/n), n=infinity) = L0;
`The radius of convergence is R = `, 1/L = R;

Example 4.23, Page 153. Find the radius of convergence of the series .

> c:='c': C:='C': n:='n': Q:='Q':
C := n ->1/n!:
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
`The general term is `, c[n] = C(n); ` `;
`The ratio of consecutive terms is:`;
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
Q := C(n+1)/C(n):
c[n+1]/c[n] = Q;
Q := simplify(Q):
c[n+1]/c[n] = Q;

Use the ratio test and find the limit and then the radius of convergence .

> L:='L': R:='R':
L0 := limit(Q, n=infinity):
R := limit(1/Q, n=infinity):
c[n+1]/c[n] = Q;
`L = `, limit(c[n+1]/c[n], n=infinity) = L0;
`The radius of convergence is R = `, 1/L = R;

The sum of the infinite series can be computed.

> S:='S':
`The general term is `, c[n] = C(n)*z^n;
`A partial sum is:`;
S[5](z) = sum(C(n)*z^n, n=0..5); ` `;
`The sum of the infinite series is:`;
`f(z) = `, Sum(C(n)*z^n, n=0..infinity) = sum(C(n)*z^n, n=0..infinity);

Example 4.24, Page 156. Show that .
Use the fact that and perform termwise differentiation.

> s:='s': S:='S': S1:='S1': z:='z': Z:='Z':
s := sum(Z^n, n=0..infinity):
S := z -> subs(Z=z, s):
S1 := z -> subs(Z=z, diff(S(Z),Z)):
`S(z) ` = S(z);
`S'(z) = f(z) ` = S1(z);

Or sum the series directly.

> c:='c': C:='C': n:='n': S:='S': z:='z':
C := n ->(n+1):
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
`The general term is `, c[n] = C(n)*z^n;
`A partial sum is:`;
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
S[5](z) = sum(C(n)*z^n, n=0..5); ` `;
`The sum of the infinite series is:`;
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
`f(z) = `, Sum(C(n)*z^n, n=0..infinity) = sum(C(n)*z^n, n=0..infinity);

Example 4.25, Page 156. Termwise differentiate the Bessel function and get .

> J:='J': s:='s': S:='S': t:='t': T:='T': z:='z':
S := series(BesselJ(0,z), z=0, 14):
T := series(BesselJ(1,z), z=0, 13):
s := convert(S, polynom):
t := convert(T, polynom):
s1 := diff(s, z):
`S(z) ` = s, `...`;
`T(z) ` = t, `...`;;
`S '(z) ` = s1, `...`;;
`Does T(z) = - S '(z) ?`;
evalb(t = - s1);

Or use Maple's built in Bessel functions and use differentiation.

> J[0](z) = BesselJ(0,z);
diff(J[0](z),z) = diff(BesselJ(0,z) , z);

End of Section 4.4.