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

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 some surprises, and in this chapter we will encounter on 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 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  (Subscript[D, r]) (α),  its Taylor series about  α  will converge to the function at every point in this disk.  Thus, analytic functions are locally nothing more than "glorified polynomials."
	
		We shall also see that complex functions are the key to unlocking many of the mysteries encountered when power series are first introduced in a calculus course.  We begin by discussing an important property associated with power series - uniform convergence.

Recall that if we have a function  f(z)  defined on a set  T,  the sequence of functions  {(Subscript[S, n]) (z)}  converges to the function  f  at the point  (Subscript[z, 0]) ϵ T  provided that   (lim_(n  ->  ∞)) S ((Subscript[z, 0]))  =  
        f ((Subscript[z, 0])).  Thus, for the particular point  (Subscript[z, 0]),  this means that for every  ϵ > 0,  there exists a positive integer  (Subscript[N, ∈,(Subscript[z, 0])])  (which depends on both   ϵ  and  (Subscript[z, 0]))  such that

		if  n ≥ (Subscript[N, ∈,(Subscript[z, 0])]),  then  |S ((Subscript[z, 0]))  -  
          f ((Subscript[z, 0]))|< ϵ.  
	
If  (Subscript[S, n]) (z)  is the  (n^th)  partial sum of the infinite series (∑_(k = o)^∞) (Subscript[c, k]) (((z - α))^k), this statement becomes

		if  n ≥ (Subscript[N, ∈,(Subscript[z, 0])]),  then  \(\(\(|\)\(∑\+\(k = o\)\%\(n - 1\)\(c\_k\) \((z\_0 - α)\)\^k\  - \ f \((z\_0)\)\)\(|\)\(\(<\)\(\ \)\(ϵ\)\)\)\).  

	It is important to stress that for a given value of  ϵ,  the integer  (Subscript[N, ∈,(Subscript[z, 0])])  we need to satisfy the inequality will often depend on our choice of point  (Subscript[z, 0]).  This is not the case if the sequence   {(Subscript[S, n]) (z)}  converges uniformly.  For a uniformly convergent sequence, it is possible to fine an integer  N  (which depends only on  ϵ)  that guarantees that  |S ((Subscript[z, 0]))  -  
          f ((Subscript[z, 0]))|< ϵ  no matter which value  (Subscript[z, 0]) ϵ T  we pick.  In other words, if  n  is large enough, the function  (Subscript[S, n]) (z)  is uniformly close to  f(z).  Formally, we have the following definition.

Definition 7.1 (Uniform Convergence), Page 262. The sequence {(Subscript[S, n]) (z)} converges uniformly to f(z) on the set T if for every  ϵ > 0,  there exists a positive integer  (Subscript[N, ϵ])  (which depends only on  ϵ)  such that 

		if  n ≥ N,  then |(Subscript[S, n]) (z) - 
          f (z)|<∈  for all  z ϵ T.

If in the preceding, (Subscript[S, n]) (z)  is the (n^th)  partial sum of the infinite series  (∑_(k = o)^∞) (Subscript[c, k]) (((z - α))^k),  we say that the series  (∑_(k = o)^∞) (Subscript[c, k]) (((z - α))^k)  converges uniformly to  f(z)  on the set  T.

Example 7.1,  Page 262.  The sequence {(Subscript[S, n]) (z)} = {(e^z) + (1/n)} converges uniformly to  (e^z) on the entire complex plane because for any ϵ > 0, statement (7-2) is satisfied for all z for n ≥ (Subscript[N, ϵ]), where (Subscript[N, ϵ]) is any integer greater than (1/ϵ). We leave the details of showing this as an exercise.

A good example of a sequence of functions that does not converge uniformly is the sequence of partial sums comprising the geometric series. Recall that the geometric series has (Subscript[S, n]) (z) (∑_(k = o)^n) (z^k) converging to f (z) = (1/(1  -  z)) for all z ∈ (Subscript[D, 1]) (0).  Because the real numbers are a subset of the complex numbers, we can show statement (7-2) is not satisfied by demonstrating it does not hold when we restrict our attention to the real numbers. In that context, (Subscript[D, 1]) (0) becomes the open interval (-1,1), and the inequality, |(Subscript[S, n]) (z) - 
          f (z)|<∈, becomes |(Subscript[S, n]) (x) - 
          f (x)|<∈, which for real variables is equivalent to \(\(\(f \((x)\)\)\(-\)\) ∈ \(\(<\)\(\(S\_n\) \((x)\)\)\(<\)\(\(f \((x)\)\)\(+\)\)\) ∈ \).  If Statement (7-2) were to be satisfied, then given ϵ > 0, (Subscript[S, n]) (x)would be within an ϵ-bandwidth of f(x) for all x in the interval (-1,1) provided n were large enough. However, there is an ϵ such that no matter how large n is, we can find (Subscript[x, 0]) ∈ (-1, 1) such that (Subscript[S, n]) ((Subscript[x, 0])) lies outside this bandwidth. In other words, this illustrates the negation of Statement (7-2), which in technical terms is the following.

	There exists ϵ > 0, such that for all positive integers N, 
	there is some n ≥ N and some (Subscript[z, 0]) ∈ T 
	such that |(Subscript[S, n]) ((Subscript[z, 0])) - 
          f ((Subscript[z, 0]))|≥∈.  

In the exercises, we ask you to use this latter statement to show that the partial sums of the geometric series do not converge uniformly to f (z) = (1/(1  -  z)) for z ∈ (Subscript[D, 1]) (0).

A useful procedure known as the Weierstrass M-test, which can help determine whether an infinite series is uniformly convergent.

Theorem 7.1,  Page 263.  (Weierstrass M-Test)  Suppose the infinite series  (∑_(k = 0)^∞) (Subscript[u, k]) (z)  has the property that for each  k,  \(\(\(|\)\(\(u\_k\) \((z)\)\)\(|\)\(\(≤\)\(\ \)\(M\_k\)\)\)\) for all  z ϵ T.  If  (∑_(k = 0)^∞) (Subscript[M, k])  converges,  then  (∑_(k = 0)^∞) (Subscript[u, k]) (z)  converges uniformly on  T.

Proof of Theorem 7.1,  see text Page 263.

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

Theorem 7.2,  Page 264.  Suppose the power series  (∑_(k = o)^∞) (Subscript[c, k]) (((z - α))^k)  has radius of convergence  ρ > 0.  Then for each  r,  (where  0 < r < ρ),  the series converges uniformly on the closed disk  \(\(\(D\_r\)\&_\) \((α)\) = {z : \(\(|\)\(z - α\)\(|\)\(\(≤\)\(r\)\)\)}\).

Proof of Theorem 7.2, see text Page 264.

An immediate consequence of this theorem is the following result.

Corollary 7.1,  Page 265.  For each  r,  (where  0 < r < 1),  the geometric series converges uniformly on the closed disk  \(\(\(D\_r\)\&_\) \((0)\) = {z : \(\(|\)\(z\)\(|\)\(\(≤\)\(r\)\)\)}\).

Theorem 7.3 gives important properties of uniformly convergent sequences.

Theorem 7.3,  Page 265.  Suppose {(Subscript[S, n]) (z)} is a sequence of continuous functions defined on a set  T  containing the contour  C.   
If  {(Subscript[S, n]) (z)}  converges uniformly to  f(z)  on the set  T,  then  

	(i)  f(z)   is continuous on  T,  and  
	
	(ii)  \(\(\(lim\+\(k\  -> \ ∞\)\ \((∫\_C\ \(S\_n\) \((z)\) dz)\)\)\(\ \)\(=\)\(\ \)\)\)\(∫\_C\((lim\+\(k\  -> \ ∞\)\ \(S\_n\) \((z)\))\) dz\  = \ ∫\_C 
            f \((z)\) dz\).

Proof of Theorem 7.3, see text Page 265.

Corollary 7.2,  Page 266.  If the series  (∑_(k = o)^∞) (Subscript[c, k]) (((z - α))^k)  converges uniformly to  f(z)  on the set  T,  and  C  is a contour contained in  T,  then  

		\(\(\(∑\+\(k = o\)\%∞\((∫\_C\( 
                    c\_k\) \(\((z - α)\)\^k\) dz)\)\)\(=\)\(\ \)\)\)\(∫\_C\((lim\+\(k\  -> \ ∞\)\ \(S\_n\) \((z)\))\) dz\  = \ ∫\_C 
            f \((z)\) dz\).  
	or
		\(\(\(∑\+\(k = o\)\%∞\((∫\_C\( 
                    c\_k\) \(\((z - α)\)\^k\) dz)\)\)\(=\)\(\ \)\)\)\(∫\_C\((∑\+\(k = o\)\%∞\( 
                      c\_k\) \((z - α)\)\^k)\) dz\  = \ ∫\_C f \((z)\) dz\).

Example 7.2,  Page 266.  Show that Log (1 - 
              z) = -(∑_(n = 1)^∞) ((z^n)/n) for all |z|< 1.

Solution 7.2.

Remove[z]; 
Clear[f, n, S17, S, S∞]; 
f[z_]  =  Log[1 - z]; 
S[z_]  =  Normal[Series[f[z], {z, 0, 17}]]; 
Print[" f[z] = ", f[z]]; 
Print[""]; 
\(\(Print[\*""\!\(S\_17\)[z] = "", 
        S[z]];\)\ \)

Sum up infinitely many terms to verify that we have things right.

S17[
          z_]    = -(∑_(n = 1)^17) ((z^n)/n); 
S∞[
          z_]  = -(∑_(n = 1)^∞) 
            ((z^n)/n); 
\(\(Print[\*""\!\(S\_17\)[z] = "", 
        S17[z]];\)\ \)
Print[""]; 
\(\(Print[\*""(Subscript[S, ∞])[z] = -(∑_(n = 1)^∞) ((z^n)/n) = "", 
        S∞[z]];\)\ \)

We can use Mathematica to investigate how well the series is "converging" for real numbers.

Plot[{f[x], S17[x]}, {x, -0.99999, 
          0.99999}, 
PlotRange -> {{-1, 
              1}, {-6, 
              1}}, 
Ticks -> {Range[-1, 1, 0.5], 
            Range[-6, 1, 1]}, PlotStyle -> {Red, Green}, 
        AxesLabel -> {"x", "s"}]; 
\(\(Print[\*""s = \!\(S\_17\)[z]""];\)\ \)
\(\(Print[\*""s = \!\(S\_∞\)[z] = "", 
        S∞[z]];\)\ \)

We can use Mathematica to investigate how well the complex series is "converging" in the complex plane.

\(
\(PolarMap[
        S17\ , {0, 0.9, 
          0.15}, {\(-π\), \ π, \ π\/12}, 
PlotRange -> {{\(-2.5\), 0.8}, {\(-1.2\), 1.2}}, 
        AspectRatio -> 
          2.4\/3.3, 
Ticks -> {Range[\(-4\), 4, 1], 
            Range[\(-4\), 4, 1]}, AxesLabel -> {"u", "v"}, 
        PlotStyle -> {DarkGreen, Green}];\)\ \)
\(Print["w = ", S17[z], ",  where |z| < 0.9"]; \ 
    PolarMap[f, {0, 0.9, 
        0.15}, {\(-π\), \ π, \ π\/12}, 
PlotRange -> {{\(-2.5\), 0.8}, {\(-1.2\), 1.2}}, 
      AspectRatio -> 
        2.4\/3.3, 
Ticks -> {Range[\(-4\), 4, 1], 
          Range[\(-4\), 4, 1]}, AxesLabel -> {"u", "v"}, 
      PlotStyle -> {Magenta, Pink}];\ \)
Print["w = ", 
        f[z], ",  where |z| < 0.9"];

\(
\(PolarMap[
        S17\ , {0, 0.99, 
          0.165}, {\(-π\), \ π, \ π\/12}, 
PlotRange -> {{\(-4.8\), 0.8}, {\(-1.7\), 1.7}}, 
        AspectRatio -> 
          3.4\/5.6, 
Ticks -> {Range[\(-4\), 4, 1], 
            Range[\(-4\), 4, 1]}, AxesLabel -> {"u", "v"}, 
        PlotStyle -> {DarkGreen, Green}];\)\ \)
\(Print["w = ", S17[z], ",  where |z| < 0.99"]; \ 
    PolarMap[f, {0, 0.99, 
        0.165}, {\(-π\), \ π, \ π\/12}, 
PlotRange -> {{\(-4.8\), 0.8}, {\(-1.7\), 1.7}}, 
      AspectRatio -> 
        3.4\/5.6, 
Ticks -> {Range[\(-4\), 4, 1], 
          Range[\(-4\), 4, 1]}, AxesLabel -> {"u", "v"}, 
      PlotStyle -> {Magenta, Pink}];\ \)
Print["w = ", 
        f[z], ",  where |z| < 0.99"];

\(
\(PolarMap[
        S17\ , {0, 0.999, 
          0.1665}, {\(-π\), \ π, \ π\/12}, 
PlotRange -> {{\(-4.8\), 0.8}, {\(-1.8\), 1.8}}, 
        AspectRatio -> 
          3.6\/5.6, 
Ticks -> {Range[\(-4\), 4, 1], 
            Range[\(-4\), 4, 1]}, AxesLabel -> {"u", "v"}, 
        PlotStyle -> {DarkGreen, Green}];\)\ \)
\(Print["w = ", S17[z], ",  where |z| < 0.999"]; \ 
    PolarMap[f, {0, 0.999, 
        0.1665}, {\(-π\), \ π, \ π\/12}, 
PlotRange -> {{\(-4.8\), 0.8}, {\(-1.8\), 1.8}}, 
      AspectRatio -> 
        3.6\/5.6, 
Ticks -> {Range[\(-4\), 4, 1], 
          Range[\(-4\), 4, 1]}, AxesLabel -> {"u", "v"}, 
      PlotStyle -> {Magenta, Pink}];\ \)
Print["w = ", 
        f[z], ",  where |z| < 0.999"];

Library Research Experience for Undergraduates

Project I. Write a report on Taylor series.

1. Arcache, Alexander, ''Expansion of Analytic Functions in Infinite Series and Infinite Products with Application to Multiple Valued Functions'', Am. Math. M., Vol. 72, No. 8. (Oct., 1965), pp. 861-864, Jstor.   

2. Boas, R. P., ''Expansions of Analytic Functions,'' Trans. of the Am. Math. Soc., Vol. 48, No. 3. (Nov., 1940), pp. 467-487, 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. .Small, Robert D., ''Modified Convergence of Taylor Series (in Classroom Notes),'' Am. Math. M., Vol. 88, No. 6. (Jun. - Jul., 1981), pp. 439-441, Jstor.

Project II. Write a report on the topic of Analytic Continnuation.

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. Cunningham, John, (1965), Complex Variable Methods in Science and Technology, London, N.Y., Van Nostrand.

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

5. Derrick, William R., (1984), Complex Analysis and Applications, 2nd Ed., Belmont, CA, Wadsworth Pub. Co.

6. Henrici, Peter, (1974), Applied and Computational Complex Analysis, Vol. I, N.Y., Wiley.

7. Kodaira, Kunihiko, (1984), Introduction to Complex Analysis, N.Y., Cambridge University Press.

8. Markushevich, Aleksei Ivanovich, (1967), Theory of Functions of a Complex Variable, Vol. III, Englewood Cliffs, N.J., Prentice-Hall.

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

10. Paliouras, John D., (1990), Complex Variables for Scientists and Engineers, 2nd ed., N.Y., Macmillan.

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

12. Stewart, Ian and David Tall, (1983), Complex Analysis (The Hitchhiker's Guide to the Plane), N.Y.: Cambridge University Press.

Section 7.1  Exercises for Uniform Convergence
See textbook page 267.

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

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