COMPLEX ANALYSIS: Mathematica 4.1 Notebooks
(c) John H. Mathews, and
Russell W. Howell, 2002
Complimentary software to accompany our textbook:

COMPLEX ANALYSIS: for Mathematics and Engineering, 
4th Edition,  2001,  ISBN: 0-7637-1425-9
Jones & Bartlett Publishers, Inc.  
40 Tall Pine Drive, Sudbury, MA 01776  
Tele.  (800) 832-0034, FAX:  (508)  443-8000  
E-mail:  mkt@jbpub.com  
Internet:  http://www.jbpub.com/  
This free software is compliments of the authors.
John H. Mathews,   mathews@fullerton.edu 
Russell W. Howell,  howell@westmont.edu

CHAPTER 7  TAYLOR AND LAURENT SERIES

a = b = c = x = y = z = Z = 
                    0; 
Clear[a, A, b, c, f, f1, f2, g, h, i, lim1, lim2, L, L1, L2, L3, L4, 
        m, n, p, solset, S, SI, Sinf, S0, S7, S9, S11, S12, S13, S17, S21, 
        S22, S25, x, y, z, Z, Z1, Z2, Z3, Z4]; 
Needs["Graphics`ComplexMap`"]; 
Needs["Graphics`Colors`"];

Section 7.1	Uniform Convergence

Section 7.2	Taylor Series Representations

Section 7.3	Laurent Series Representations

Section 7.4	Singularities, Zeros and Poles

Section 7.5  Applications of Taylor and Laurent Series

In this section we show how you can use Taylor and Laurent series 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  α  of a set  T  is called isolated if there exists a disk  (Subscript[D, r]) (α)  about  α  that does not contain any other points of  T.

Theorem 7.13,  Page 298.  Suppose f(z) is analytic in a domain D containing the point z = α  and that  f (α) = 0.   If f(z) is not identically zero, then there exists a punctured disk  \(\(\(\ \)\(\(D\_r\^*\) \((α)\)\)\)\)  in which f(z) has no zeros.

Proof of Theorem 7.13, see text Page 298.

The following corollaries are given as exercises.

Corollary 7.8,  Page 299.  Suppose that  f(z)  is analytic in the domain  D,  and that  α ∈ D.  If there exists a sequence of points  {(Subscript[z, n])}  in  D  such that  (Subscript[z, n])  ->  α,  and  f ((Subscript[z, n])) = 0,  then  f (z) ≡ 0  for all  α ∈ D.

Corollary 7.9,  Page 299.  Suppose that  f(z)  and  g(z)  are analytic in the domain  D,  where  α ∈ D.  If there exists a sequence  {(Subscript[z, n])}  in  D  such that  (Subscript[z, n])  ->  α,  and  f ((Subscript[z, n])) = g ((Subscript[z, n]))  for all  n,  then  f (z) ≡ g (z)  for all  α ∈ D.

Theorem 7.14 allows us to give a simple proof for one version of L'Hôpital's rule.

Corollary 7.10,  Page 299.  (L'Hospital's Rule)  Suppose f(z) and g(z) are analytic at  z = α.  If  f (α) = 0  and  g (α) = 0  but g' (α) ≠ 0,  then  

	\(lim\+\(z\  -> \ α\)\ \(f \((z)\)\)\/\(g \((z)\)\) = \(lim\+\(z\  -> \ α\)\ \(f' \((z)\)\)\/\(g' \((z)\)\) = \(f' \((α)\)\)\/\(g' \((α)\)\)\)\).

Proof of Corollary 7.9, see text Page 299.

We can use Theorem 7.14 to get Taylor series for quotients of analytic functions. Its proof involves ideas from Section 7.2, and we leave it as an exercise.

Theorem 7.15  (Division of Power Series),  Page 300.  Suppose f and g are analytic at z = α with power series representations  

\(f \((z)\) = ∑\+\(n = 0\)\%∞\( 
                  a\_n\) \((z - α)\)\^n\ \ \ for\ \ z ∈ \(D\_r\) \((α)\)\)  and  \(g \((z)\) = ∑\+\(n = 0\)\%∞\( 
                  b\_n\) \((z - α)\)\^n\ \ \ for\ \ z ∈ \(D\_r\) \((α)\)\).  

If  g (α) ≠ 0, then the quotient  ((f (z))/(g (z))) has the power series representation  

\(\(f \((z)\)\)\/\(g \((z)\)\) = ∑\+\(n = 0\)\%∞\( 
                  c\_n\) \((z - α)\)\^n\ \ \ for\ \ z ∈ \(D\_r\) \((α)\)\),  

where the coefficients satisfy the equations  (Subscript[a, n]) = (Subscript[b, 0]) (Subscript[c, n]) + (Subscript[b, 1]) 
            (Subscript[c, n - 1]) + ⋯ + (Subscript[b, n - 1]) 
            (Subscript[c, 1]) + (Subscript[b, n]) (Subscript[c, 0]).  

In other words, the series for the quotient ((f (z))/(g (z))) can be obtained by the familiar process of dividing the series for f(z) by the series for g(z) using the standard long division algorithm.

Proof of Theorem 7.15, see text Page 300.

Example 7.15,  Page 300.  Find the first few terms of the Maclaurin series for h (z) = sec (z) and use it to compute Derivative[4][h](0).

Solution 7.15.

Method (i).  Use Mathematica to expand sec(z) in a Taylor series about z = 0.

Remove[c, z]; 
Clear[f, g, h, k, n, S]; 
h[z_]  =  Sec[z]; 
S[z_]  =  Normal[Series[Sec[z],  {z, 0, 12}]]; 
Print["h[z] = ", h[z]]; 
Print[""]; 
Print["S[z] = ", 
        S[z], "+ ... "];

Find the fourth derivative of the series and evaluate it to compute Derivative[4][h](0).

Print["h''''[z] = ", 
        Expand[h''''[z]]]; 
Print["S''''[z] = ", S''''[
          z], "+ ... "]; 
Print[""]; 
Print["h''''[0] = ", h''''[0]]; 
Print["S''''[0] = ", S''''[
          0]];

Method (ii).  Use division of power series.  For this problem we have  1 = cos(z) S(z)  where S(z) is the desired Maclaurin series.  Mathematica is used to expand cos(z) up to degree n, and solve the resulting equation for coefficients of S(z) up to n.

n = 9; 
f[z_] = 1;  
g[z_] = Cos[z];  
S1[z_] = Series[f[z], {z, 0, n}];  
S2[z_] = Series[g[z], {z, 0, n}];  
S3[z_] = (∑_(k = 0)^n) (Subscript[c, k]) (z^k);  
coeff = Table[(Subscript[c, k]), {k, 0, n}];  
eqn1 = S1[z] == S2[z] S3[z];  
eqn2 = LogicalExpand[eqn1];  
eqn3 = ReplaceAll[eqn2 , And -> List];  
solset = Solve[eqn3, coeff];  
S4[z_] = 
      ReplaceAll[(∑_(k = 0)^n) (Subscript[c, k]) (z^k), 
        (Subscript[solset, [[1]]])];  
Print["f[z] = ", f[z]];  
Print["g[z] = ", g[z]];  
\(Print[\*""\!\(f[z]\/g[z]\) = "", f[z]\/g[z]]; \ \)
Print[""];  
Print["f[z] = ", S1[z]];  
Print["g[z] = ", S2[z]];  
Print[""];  
Print[eqn1];  
Print[""];  
Print[eqn3];  
Print[""];  
Print[solset];  
Print[""];  
\(\(Print[h[z], \*"" = \!\(f[z]\/g[z]\)""];\)\ \)
\(\(Print[
        h[z], \*"" = (∑_(k = 0)^∞)(Subscript[c, k])(z^k)""];\)\ \ \)
Print[h[z], " = ", 
        S4[z], "+ ..."];

Remark. This is the same as the first few terms in the Maclaurin series that was computed using methods (i).

\(
\(Print[
        h[z], \*"" = \!\(f[z]\/g[z]\)""];\)\ \)
Print[h[z], " = ", 
        S[z], "+ ..."];

We close this section with some results concerning the behavior of complex functions at points near the different types of isolated singularities. Our first theorem is due to the German mathematician G. F. Bernhard Riemann (1826--1866).

Theorem 7.15  (Riemann's Theorem),  Page 300.  Suppose that f(z) is analytic in \(\(D\_r\^*\) \((α)\)\).  If f(z) is bounded in  \(\(\(\ \)\(\(D\_r\^*\) \((α)\)\)\)\),  then either f(z) is analytic at z = α or f(z) has a removable singularity at z = α.

Proof of Theorem 7.15, see text Page 300.

Corollary 7.12,  Page 301.  Suppose that  f(z)  is analytic in  \(\(D\_r\%*\) \((α)\)\).  Then  f  can be defined to be analytic at  α  if and only if  (lim_(z  ->  α)) f (z) exists (and is finite).

Theorem 7.16,  Page 301.  Suppose that  f(z)  is analytic in  \(\(D\_r\%*\) \((α)\)\).  The function  f  has a pole of order  k  at  z = α  if and only if  \(lim\+\(z\  -> \ α\)\ \(\(|\)\(f \((z)\)\)\(|\)\) = ∞\).

Proof of Theorem 7.16, see text Page 301.

Theorem 7.17,  Page 302.  The function f(z) has an essential singularity at z = α if and only if  (lim_(z  ->  α)) |f (z)|  does not exist.

Proof of Theorem 7.17, see text Page 302.

Example 7.16,  Page 302.  Show that the function  g (z) = (e^(((-1)/(z^2))))  is NOT continuous at  z = 0.

Solution 7.16.

Enter the function  g[z] =  (e^(((-1)/(z^2))))  and find the limits (lim_(x -> 0)) 
        g[x + 0  i]  and  (lim_(y -> 0)) 
            g[0 + i y].

Remove[x, y, z]
Clear[g, lim1, lim2]; 
g[
          z_]  =  (E^(((-1)/(z^2)))); 
G[x_, y_] = 
        MapAll[Together, 
          ComplexExpand[g[x + I y], 
            TargetFunctions -> {Re, Im}]]; 
lim1  =  
        Limit[g[x  +  0 I],  x -> 0]; 
lim2  =  
        Limit[g[0  +  I y],  y -> 0]; 
Print["      g[z] = ", g[z]]; 
Print["   g[x+\[ImaginaryI]y] = ", 
        g[x + I y]]; 
Print["   g[x+\[ImaginaryI]y] = ", 
        G[x, y]]; 
Print[""]; 
Print["   g[x+0\[ImaginaryI]] = ", 
        g[x + 0 I]]; 
\(\(Print[\*""\!\(lim\+\(x -> 0\)\) g[x+0I] = "", 
        lim1];\)\ \)
Print[""]; 
Print["   g[0+\[ImaginaryI]y] = ", 
        g[0 + I y]]; 
\(\(Print[\*""\!\(lim\+\(y -> 0\)\) g[0+Iy] = "", 
        lim2];\)\ \)

Since the limits are NOT the same,  g (z) = (e^(((-1)/(z^2))))  is NOT continuous at   z = 0.

Show that the function g[z]  =  (e^(((-1)/(z^2)))) has an essential singularity at the origin.

\(
\(S[z_]\  = \ 
        Normal[Series[
            g[z], {z, ∞, 9}]];\)\ \)
lim0 = 
        Limit[g[z], z -> 0, 
          Direction -> -1]; 
lim1 = 
        Limit[g[z], z -> 0, 
          Direction -> -I]; 
lim2 = 
        Limit[g[z], z -> 0, 
          Direction -> 1]; 
lim3 = 
        Limit[g[z], z -> 0, 
          Direction -> I]; 
Print["g[z] = ", g[z]]; 
Print[""]; 
Print["S[z] = ... + ", S[z] - 1, " + ", 
        1]; 
Print[""]; 
\(\(Print[\*""\!\(lim\+\(z -> \(0\^+\)\)\) "", 
        g[z], " = ", lim0];\)\ \)
Print[""]; 
\(\(Print[\*""\!\(lim\+\(z -> \(I0\^+\)\)\) "", 
        g[z], " = ", lim1];\)\ \)
Print[""]; 
\(\(Print[\*""\!\(lim\+\(z -> \(0\^-\)\)\) "", 
        g[z], " = ", lim2];\)\ \)
Print[""]; 
\(\(Print[\*""\!\(lim\+\(z -> \(I0\^-\)\)\) "", 
        g[z], " = ", lim3];\)\ \)

The Laurent series for  g(z) has "infinitely many" negative powers of z.  Hence, g[z]  =  (e^(((-1)/(z^2)))) has an essential singularity at the origin.

Library Research Experience for Undergraduates

Project I. Write a report on series representation of analytic functions.

1. Boas, R. P., (1989), ''When is a (C^∞) Function Analytic?'', Math. Intell., V. 11, No. 4, pp. 34-37.

2. Cater, F. S., (1984), ''Differentiable, Nowhere Analytic Functions,'' Am. Math. M., Vol. 91, No. 10. (Dec., 1984), pp. 618-624, Jstor.   

3. Eidswick, Jack A., ''Alternatives to Taylor's Theorem in Proving Analyticity (in Classroom Notes),'' Am. Math. M., Vol. 82, No. 9. (Nov., 1975), pp. 929-931., Jstor.   

4. Gray, J. D. and S. A. Morris, ''When is a Function that Satisfies the Cauchy-Riemann Equations Analytic?,'' Am. Math. M., Vol. 85, No. 4. (Apr., 1978), pp. 246-256., Jstor.   

5. Snider, Arthur David, (1978), ''On the Definition of Analyticity,'' Int. J. of Math. Ed. in Sci. and Tech., V. 9, No. 3, pp. 373-374.

Project II. Write a report on infinite products of complex numbers and/or functions.

1. Bak, Joseph, and Donald J. Newman, (1982), Complex Analysis, N.Y., Springer-Verlag.

2. Boas, R. P., (1987), Invitation to Complex Analysis, N.Y., Random House.

3. DePree, John D. and Charles C. Oehring, (1969), Elements of Complex Analysis, Reading, Mass., Addison-Wesley Pub. Co.

4. Marsden, Jerrold E. and Michael J. Hoffman, (1987), Basic Complex Analysis, 2nd Ed., San Francisco, W. H. Freeman

5. Pennisi, Louis Legendre, (1976), Elements of Complex Variables, 2nd Ed., N.Y., Holt, Rinehart and Winston.

6. Wermuth, Edgar M.E., ''Some Elementary Properties of Infinite Products,'' Am. Math. M., Vol. 99, No. 6. (Jun. - Jul., 1992), pp. 530-537., Jstor.

Project III. Write a report on the Bieberbach conjecture.

1. DeBranges, Louis, (1985), ''A Proof of the Bieberbach Conjecture,'' Acta Mathematica, V. 154, pp. 137-152.

2. Fomenko, O. M. and G. V. Kuz'mina, (1986), ''The Last 100 Days of the Bieberbach Conjecture,'' Math. Intell., V. 8, No. 1, pp. 40-47.

3. Korevaar, J., ''Ludwig Bieberbach's Conjecture and Its Proof by Louis de Branges,'' Am. Math. M., Vol. 93, No. 7. (Aug. - Sep., 1986), pp. 505-514, Jstor.   

4. Pommerenke, Ch., (1985), ''The Bieberbach Conjecture,'' Math. Intell., V. 7, No. 2, pp. 23-32.

5. Zorn, Paul, (1986), ''The Bieberbach Conjecture: A Famous unsolved problem and the story of de Branges' surprising proof.,'' Math. Mag., V. 59, No. 3, pp. 131-148.

Section 7.5  Exercises for Applications of Taylor and Laurent Series
See textbook page 303.

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