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 4  SEQUENCES, JULIA and MANDELBROT SETS, and POWER SERIES

c = q = S = t = x = y = z = 
                    0; 
Clear[a, absZn, c, cx, cy, dot, i, juliaC, k, lim, limX, limY, limZ, 
        list, mandelbrotC, n, newton, Pn, q, Qn, R, s, S, Sn, set, sum, sumN, 
        sumX, sumY, sumZ, t, table, x, Xn, y, Yn, z, Z, 
        Zn]; 
Needs["Graphics`ComplexMap`"]; 
Needs["Graphics`Colors`"];

Section 4.1	Sequences and Series

Section 4.2  Julia and Mandelbrot Sets

An impetus for studying complex analysis is the comparison of properties of real numbers and functions with their complex counterparts.  In this section we take a look at Newton's method for finding solutions to the equation  f (x) = 0.  We then examine the more general topic of iteration.

	Recall from calculus that Newtons Method method proceeds by starting with a function  f(x)  and an initial "guess"  (Subscript[x, 0])  as a solution to  f (x) = 0.  We then generate a new guess  (Subscript[x, 1])  by the computation  (Subscript[x, 1]) = (Subscript[x, 0]) - ((f ((Subscript[x, 0])))/(f' ((Subscript[x, 0])))).  Using  (Subscript[x, 1]) in place of  (Subscript[x, 0]),  this process is repeated, giving us  (Subscript[x, 2]) = (Subscript[x, 1]) - ((f ((Subscript[x, 1])))/(f' ((Subscript[x, 1])))).  Thus we obtain a sequence of points  {(Subscript[x, k])}, where  (Subscript[x, k + 1]) = (Subscript[x, k]) - ((f ((Subscript[x, k])))/(f' ((Subscript[x, k])))).  The points  {(Subscript[x, k])}_(k = 0)^∞  are called the iterates of  (Subscript[x, 0]).  For functions defined on the real numbers, this method gives remarkably good results so that the sequence  {(Subscript[x, k])}  often converges to a solution of  f (x) = 0  rather quickly.  In the late 1800's the british mathematician Arthur Cayley investigated the question as to whether Newton's method can be applied to complex functions.  He wrote a paper giving an analysis for how this method works for quadratic polynomials and indicated his intention to publish a subsequent paper for cubic polynomials.  Unfortunately, Cayley died before producing this paper.  As you will see, The extension of Newton's method to the complex domain and the more general question of iteration are quite complicated.

Example 4.7,  Page 134.  Trace out the next five iterates of Newton's method given an initial guess of  (Subscript[z, 0]) = (1/4) + (i/4)  as a solution to the equation  f[z] = 0,  where  f[z] = (z^2) + 1.

Solution 4.7.

Enter the function  f[z] = (z^2) + 1  and find the next five iterates.

Remove[z, Z]; 
Clear[f, i, k, K, list, set]; 
f[z_]  =  (z^2)  +  1 ; 
K = Table[i, {i, 0, 5}]; 
Z = Table[0, {6}]; 
(Subscript[Z, [[1]]]) = 
        N[(1/4) + (I/4)]; 
g[z_] = z - f[z]/(f')[z]; 
Print["f[z] = ", f[z]]; 
Print[""]; 
Print["The formula for iteration is"]; 
\(\(Print[\*""(Subscript[z, k + 1]) = (Subscript[z, k]) - f[(Subscript[z, k])]/(f')[(Subscript[z, k])] = "", 
        g[\*""(Subscript[z, k])""]];\)\ \)

Now compute the next five iterates  (Subscript[z, k])  and the corresponding values of  f[(Subscript[z, k])].

For[k = 0, 
        k ≤ 
          4, k++, 
(Subscript[Z, [[k + 2]]]) = 
            g[(Subscript[Z, [[k + 1]]])];  ]; 
\(\(Print[
        TableForm[
          Transpose[{K, Z, 
              f[Z]}], 
TableHeadings -> {None, {"k", \*""(Subscript[z, k])"", \*""f[(Subscript[z, k])]""}}]];\)\ \)

Plot the iterates  (Subscript[z, k])  and observe that they are converging to  z = i.

set  =   ReplaceAll[Z, 
          xx_Real -> {xx, 0}]; 
list  =   ReplaceAll[set, 
          Complex[xx_, yy_] -> {xx, yy}]; 
\(\(ListPlot[list, PlotRange -> {{\(-0.9\), 0.5}, {0.0, 1.2}}, 
        Prolog -> {PointSize[0.02], 
            Magenta}, 
AspectRatio -> 1.2\/1.4, 
        AxesLabel -> {"x", "y"}, 
Ticks -> {Range[\(-1\), 1, 0.2], 
            Range[0, 1.2, 0.2]}];\)\ \)
\(\(Print[
        TableForm[
          Transpose[{K, 
              Z}], 
TableHeadings -> {None, {"k", \*""(Subscript[z, k])""}}]];\)\ \)

Notice that the points  (Subscript[z, 4])  and  (Subscript[z, 5])  are co close together that they appear to coincide and that the value for   (Subscript[z, 5])  agrees to six decimal places with the actual solution  z = i.

The complex version of Newton's method appears also to work quite well.  You may recall, however, that with functions defined on the reals, it is not the case that every initial guess produces a sequence that converges to the solution.  Example 4.21 shows that the same is true in the complex case.

Example 4.8,  Page135.  Show that Newton's method fails for the function  f[z] = (z^2) + 1  if our initial guess is a real number.

Solution 4.8.

From Example 4.20 we know that given any guess  (Subscript[z, 0])  as a solution of  (z^2) + 1 = 0, the next guess at a solution is (Subscript[z, 1]) = (Subscript[z, 0]) - 
            f[(Subscript[z, 0])]/(f')[(Subscript[z, 0])] = (((Subscript[z, 0]^2)  -  1)/(2  (Subscript[z, 0]))).  Let  {(Subscript[z, k])}  be the sequence of iterations produced by the initial seed  (Subscript[z, 0]).  If for any k,  (Subscript[z, k]) = 0, the iteration terminates with an undefined result.  If all of the terms of the infinite sequence  {(Subscript[z, k])}  are defined, an easy induction argument shows that all the terms in the sequence will be real valued.  Since the solutions of  (z^2) + 1 = 0  are  z = ±i, the sequence  {(Subscript[z, k])}  cannot possibly converge to either solution.  In the exercises we ask you to explore in detail what happens when  (Subscript[z, 0])  is in the upper or lower half plane.

Extra Example for Page135.  Investigate Newton's method for finding the roots of  f[z] = (z^3) + 1.
Given the initial seed  (Subscript[z, 0])  determine if the sequence  {(Subscript[z, k])}  converges to one of the roots  z = -1, (1/2) + ((i (Sqrt[3]))/2), 
      (1/2) - ((i (Sqrt[3]))/2).

Extra Solution for Page 135.

Enter the function  f[z] = (z^3) + 1  and find its roots.

Remove[z, Z]; 
Clear[f]; 
f[z_] = (z^3) + 1; 
solset = Solve[f[z] == 0, z]; 
(Subscript[Z, 1]) = 
        ComplexExpand[
          (Subscript[solset, [[1, 1, 2]]]), 
          TargetFunctions -> {Im, Re}]; 
(Subscript[Z, 2]) = 
        ComplexExpand[
          (Subscript[solset, [[2, 1, 2]]]), 
          TargetFunctions -> {Im, Re}]; 
(Subscript[Z, 3]) = 
        ComplexExpand[
          (Subscript[solset, [[3, 1, 2]]]), 
          TargetFunctions -> {Im, Re}]; 
Print["f[z] = ", f[z]]; 
Print[""]; 
Print["The roots are"]; 
\(\(Print[\*""\!\(z\_1\) = "", 
        Z\_1];\)\ \)
\(\(Print[\*""\!\(z\_2\) = "", 
        Z\_2];\)\ \)
\(\(Print[\*""\!\(z\_3\) = "", 
        Z\_3];\)\ \)

We color each point (Subscript[z, 0])  in the rectangle  R = {z = 
            x + iy : -2 ≤ x ≤ 
                  2, -2 ≤ x ≤ 2} in the following fashion.  Color the point yellow, blue, or red, if the sequence of iterations produced by the seed  (Subscript[z, 0])  converges to -1, (1/2) + ((i (Sqrt[3]))/2), 
      (1/2) - ((i (Sqrt[3]))/2), respectively.  The iteration formula is

g[z_] = 
        Together[z - f[z]/(f')[z]]; 
Print["g[z] = ", g[z]];

The formulas for  f[z] and  g[z]  and approximations to the roots z = -1, (1/2) + ((i (Sqrt[3]))/2), 
      and (1/2) - ((i (Sqrt[3]))/2) are used in the function  newton[x,y]  to make the density plot.

\(\(\(
\)\(\(newton = 

        Compile[{x, y}, 
Module[{z, ct = 0}, 
            z = x + I\ y; \ 
While[
              Abs[Evaluate[z\^3 + 1]] > 0.001\  && \ 
                ct ≤ 
                  50, 
z = \(\(-1\) + 2\ z\^3\)\/\(3\ z\^2\); \ \(++ct\)]; \ 
Which[
Abs[
                  z - \((\(-1\))\)] < 0.1, 
              0.15, 
Abs[
                  z - \((0.5 + 0.866\ I)\)] < 
                0.1, 0.5, 
Abs[
                  z - \((0.5 - 0.866\ I)\)] < 
                0.1, 0.9]]\ ];\)\(\ \)\(
\)
    \)\)\)

Make a plot showing the initial seeds  (Subscript[z, 0]),  and the nature of the sequence {(Subscript[z, k])}  that is produced by it.

DensityPlot[
        newton[x, y], 
{x, -2, 2. },  {y, -2, 
          2}, 
PlotPoints -> 100, Mesh -> False, 
        AspectRatio -> 1, 
ColorFunction -> 
          Hue]; 
Print["Yellow points converge to  -1."]; 
\(\(Print[\*""Blue points converge to  (1/2)+((I (Sqrt[ 3]))/2).""];\)\ \)
\(\(Print[\*""Red  points converge to  (1/2)-((I (Sqrt[ 3]))/2).""];\)\ \)

In 1918 the French mathematicians Gaston Julia and Pierre Fatou noticed the fractal phenomenon when exploring iterations of functions not necessarily connected with Newton's method.  Beginning with a function  f(z)  and a point  (Subscript[z, 0]),  they computed the iterates  (Subscript[z, 1]) = f ((Subscript[z, 0])), (Subscript[z, 2]) = f ((Subscript[z, 1])), ... , (Subscript[z, k + 1]) = f ((Subscript[z, k])), and investigated properties of the sequences  {(Subscript[z, k])}.  Their findings did not receive a great deal of attention, largely because computer graphics were not available at this time.  With the recent proliferation of computers, it is not surprising that these investigations were revived in the 1980's.  Detailed studies of Newton's method and the more general topic of iteration were undertaken by a host of mathematicians including Curry, Douady, Garnett, Hubbard, Mandelbrot, Milnor and Sullivan.  We now turn our attention to some of their results by focusing on the iterates produced by quadratics of the form   (Subscript[f, c]) (z) = (z^2) + c.

Example 4.9,  Page 137.  Given  (Subscript[f, c]) (z) = (z^2) + c,  analyze all possible iterations for the function  (Subscript[f, 0]) (z) = (z^2) + 0.

Solution 4.9.

In the exercises we will ask you to verify that if  |(Subscript[z, 0])|< 1  the sequence will converge to  0,  if  |(Subscript[z, 0])|> 1  the sequence will be unbounded, and if  |(Subscript[z, 0])| =  1  the sequence will either oscillate chaotically around the unit circle or converge to  1.

Given the function  (Subscript[f, c]) (z) = (z^2) + c,  and an initial seed  (Subscript[z, 0]),  the set of iterates given by  (Subscript[z, 1]) = (Subscript[f, c]) ((Subscript[z, 0])), (Subscript[z, 2]) = (Subscript[f, c]) ((Subscript[z, 1])), etc., are also called the orbits of  (Subscript[z, 0])  generated by  (Subscript[f, c]) (z).  Let  (Subscript[K, c])  denote the set of points with bounded orbits for  (Subscript[f, c]) (z).  Example 4.9 shows that  (Subscript[K, 0])  is the closed unit disk \(\(\(D\_1\)\&_\) \((0)\)\).  The boundary of  (Subscript[K, c])  is known as the Julia Set for the function  (Subscript[f, c]) (z).  The Julia set for  (Subscript[f, 0]) (z)  is the unit circle  (Subscript[C, 1]) (0).  It turns out that  (Subscript[K, c])  is a nice simple set only when  c = 0  or  c = -2.  Otherwise,   (Subscript[K, c])  is fractal.  

	Julia and Fatou independently discovered a simple criterion that can be used to tell when the Julia set for  (Subscript[f, c]) (z)  is connected or disconnected.  We state their result, but omit the proof, as it is beyond the scope of this text.

Theorem 4.8, Page 138.  The boundary of  (Subscript[K, c])  is connected if and only if  0  ϵ (Subscript[K, c]).  In other words, the Julia set for  (Subscript[f, c]) (z)  is connected if and only if the orbits of  0  are bounded.

Example 4.10,  Page 138.  Show that the Julia set for  (Subscript[f, i]) (z) = (z^2) + i  is connected.

Solution 4.10.

We apply Theorem 4.8 and compute the orbits of  0  for  (Subscript[f, i]) (z) = (z^2) + i.

Remove[z, Z]; 
Clear[f, i]; 
f[z_] = (z^2) + I; 
Z = NestList[f, 0, 7]; 
Print["f[z] = ", f[z]]; 
Print[""]; 
\(\(For[i = 0, 
        i ≤ 
          6, \(i++\), 
\(Print["f[", 
            Z\_\(\([[\)\(i + 1\)\(]]\)\)\ , "] = ", 
            Z\_\(\([[\)\(i + 2\)\(]]\)\)];\)\ \ ];\)\ \)

Thus, the orbits of  0  is the sequence  \({0, i, \(-1\) + i, \(-i\), \(-1\) + i, \(-i\), \(-1\) + i, \(-i\),  ... }\)  which is clearly a bounded sequence.  Thus, by Theorem 4.14, the Julia set for  (Subscript[f, i]) (z)  is connected.

Extra Example for  Page 138. Plot the Julia set.

Extra Solution for Page 138.

Here is the subroutine from the Mathematica  Journal.

\(\(\(
\)\(\(juliaC\  = 

        Compile[{x, y, lim, cx, cy}, 
Module[{z, ct = 0}, 
            z = x + I\ y; \ 
While[
              Abs[z] < 2.0\  && \ ct ≤ lim, 
z = 
                z\^2 + \((cx + I\ cy)\); \ \(++ct\)]; \ 
            ct]\ ];\)\(\ \)\(
\)
    \)\)\)

Here is the Julia set generated by the value  c = -1.25.  This is a fuzzy picture of Color Plate 4.  Increase the number of PlotPoints to get a sharper picture.  Also, it will increase the computing time.  It takes quite a bit of time with  PlotPoints -> 200.  Things start to look better with  PlotPoints -> 500.

DensityPlot[-juliaC[x, y, 50, -1.25, 
            0], 
{x, -1.5, 1.5},  {y, -1.0, 
          1.0}, 
PlotPoints -> 200, 
        Mesh -> False, 
        AspectRatio -> (2/3), 
ColorFunction -> 
          Hue]; 
Print["The Julia set generated by the value c = -1.25"];

In 1980, the Polish born mathematician Benoit Mandelbrot used computer graphics to study the following set:  

\(M = \({c : \ the\ Julia\ set\ for\ \(f\_c\) \((z)\)\ is\ connected}

        \ \ \(\(=\)\({c : \ 
              the\ orbits\ of\ \ 0\ \ determined\ by\ \(f\_c\) \((z)\)\ are\ bounded}\)\)\)\)  

The set  M  has come to be known as the Mandelbrot Set.  The remainder of this section looks at some of the properties of this amazing set.

Example 4.11,  Page 139.  Show that {c : |z|≤(1/4)} is a subset of the Mandelbrot set  M.

Solution 4.11.  See page 139 in the text.

There are other methods for determining which points belong to  M.  To see what they are, we need some additional vocabulary.

Definition 4.5 (Fixed Point), Page 139.  The point  (Subscript[z, 0])  is a fixed point for the function  f  if  f ((Subscript[z, 0])) = (Subscript[z, 0]).

Definition 4.6 (Attracting Point), Page 139.  The point  (Subscript[z, 0])  is an attracting point for the function  f  if  |f' ((Subscript[z, 0]))|< 1.

Theorem 4.9, Page 139.  Suppose (Subscript[z, 0])  is an attracting fixed point for the function  f.  Then there is a disk  (Subscript[D, r]) ((Subscript[z, 0]))  about  (Subscript[z, 0])  such that the iterates of all the points in (Subscript[D, r]) ((Subscript[z, 0]))  are dawn toward the point  (Subscript[z, 0])  in the sense that if  \(z ∈ \(D\_r\%*\) \((z\_0)\)\), then  |f (z) - 
          (Subscript[z, 0])|<|z - (Subscript[z, 0])|.

Proof of Theorem 4.15,  see text Page 139.

In 1905, Pierre Fatou showed that if the function  (Subscript[f, c]) (z)  has attracting fixed points, the orbits of  0  determined by  (Subscript[f, c]) (z)  must converge to one of them.  Since a convergent sequence is bounded, this implies that  c  must belong to  M.  In the exercises we ask you to show that the main cardioid-shaped body of  M is composed of those points  c  for which  (Subscript[f, c]) (z)  has attracting fixed points.  You will find that Theorem 4.16 is a useful characterization of them.

Theorem 4.10, Page 140.  The points  c  for which  (Subscript[f, c]) (z)  has attracting fixed points satisfy  \(\(\(|\)\(1 + \((1 - 4  c)\)\^\(1/2\)\)\(|\)\(\(<\)\(\ \)\(1\)\)\)\)  or  \(\(\(|\)\(1 - \((1 - 4  c)\)\^\(1/2\)\)\(|\)\(\(<\)\(\ \)\(1\)\)\)\),  where the square root is the principal square root function.

Proof of Theorem 4.16,  see text Page 140.

Definition 4.7 (n-Cycle), Page 140.  An n-cycle for a function  f  is a set  {(Subscript[z, 0]), (Subscript[z, 1]),  ... , (Subscript[z, n - 1])}  of  n  complex numbers such that  (Subscript[z, k]) = f ((Subscript[z, k - 1]))  for  k = 1, 2,  ... , n - 1,  and  f ((Subscript[z, n - 1])) = (Subscript[z, 0]).

Definition 4.8, Page 141.  An n-cycle  {(Subscript[z, 0]), (Subscript[z, 1]),  ... , (Subscript[z, n - 1])}  for a function  f   is said to be attracting if  \(\(\(|\)\(\(g\_n\%'\) \((z\_0)\)\)\(|\)\(\(<\)\(\ \)\(1\)\)\)\),  where  (Subscript[g, n])  is the composition of  f  with itself  n  times.  
For example, if n = 2, then (Subscript[g, 2]) (z)  =  f (f (z)).

Example 4.12,  Page 141.  Example 4.23 shows that  {-1 + i, -i}  is a 2-cycle for the function  (Subscript[f, i]) (z) = (z^2) + i.  Show that it is not an attracting 2-cycle.

Solution 4.12.

Enter the function  f[z] = (z^2) + i  and point  (Subscript[z, 0]) = -1 + i  and determine if  \(\(\(|\)\(\(g\_n\%'\) \((z\_0)\)\)\(|\)\(\(<\)\(\ \)\(1\)\)\)\).

Remove[z, Z]; 
Clear[f, g2]; 
f[z_] = (z^2) + I; 
g2[z_] = 
        Apart[ComplexExpand[f[f[z]]]]; 
(Subscript[Z, 0]) = -1 + I; 
\(\(Print[\*""\!\(f\_I\)[z] = "", 
        f[z]];\)\ \)
Print[""]; 
\(\(Print[\*""\!\(g\_2\)[z] = "", 
        g2[z]];\)\ \)
Print[""]; 
\(\(Print[\*""\!\(\(g\_2\%'\)\)[z] = "", \(g2'\)[
          z]];\)\ \)
Print[""]; 
\(\(Print[\*""\!\(\(g\_2\%'\)\)["", 
        Z\_0, "] = ", \(g2'\)[Z\_0]];\)\ \)
\(\(Print[\*""|\!\(\(g\_2\%'\)\)["", Z\_0, "]| = ", 
        Abs[\(g2'\)[Z\_0]]];\)\ \)
\(\(Print[\*""Is  |\!\(\(g\_2\%'\)\)["", Z\_0, "]| > 1 ?  ", 
        Abs[\(g2'\)[Z\_0]] > 1];\)\ \)

Therefore,  {-1 + i, -i}  is not an attracting 2-cycle.

Extra Example for  Page 141. Plot the Mandelbrot set.

Extra Solution for Page 141.

This modification of the  juliaC  code can be used to plot the Mandelbrot set.

Here is the subroutine from the Mathematica  Journal.

mandelbrotC = 
        Compile[{x, y, lim}, 
Module[{z, ct = 0}, 
            z = x + I y;  
While[
              Abs[z] < 2.0  &&  ct ≤ lim, 
z = 
                (z^2) + (x + I y);  ++ct];  
            ct] ];

This is a fuzzy picture of Color Plate 6. Increase the number of PlotPoints to get a sharper picture.  Also, it will increase the computing time.  It takes quite a bit of time with  PlotPoints -> 200.  Things start to look better with  PlotPoints -> 500.

DensityPlot[-mandelbrotC[x, y, 
            50], 
{x, -2.0, 0.55},  {y, -1.15, 
          1.15}, 
PlotPoints -> 200,  
        Mesh -> False, 
Frame -> False, 
        ColorFunction -> Hue]; 
Print["The Mandelbrot set."];

Library Research Experience for Undergraduates

Project I. Write a report on how complex analysis is used in the study of dynamical systems.

1. Devaney, Robert L., (1986), An Introduction To Chaotic Dynamical Systems, Menlo Park, CA., Benjamin/Cummings.

2. Devaney, Robert L., (1990), Chaos, Fractals and Dynamics: Computer Experiments in Math., N.Y., Addison-Wesley Pub. Co.

Project II. Write a report on how complex analysis is used in the study of fractals.

1. Bannon, Thomas J., (1991), ''Fractals and Transformations,'' Math. Teach., V. 81, No. 3, pp. 178-185.

2. Barcellos, Anthony, (1984), ''The Fractal Geometry of Mandelbrot,'' T. Y. C. Math. J., V. 15, No. 2, pp. 98-114.

3. Gilbert, William J., (1982), ''Fractal Geometry Derived from Complex Bases,'' Math. Intell., V. 4, pp. 78-86.

4. Goffinet, Daniel, (1991), ''Number Systems With a Complex Base: a Fractal Tool for Teaching Topology,'', Am. Math. M., V. 98, No. 3, pp. 249-255.

5. Mandelbrot, Benoit B., (1983), ''Self-Inverse Fractals Osculated by Sigma-Discs and the Limit Sets of Inversion Groups,'' Math. Intell., V. 5, No. 2, pp. 9-17.

6. Nievergelt, Yves, (1991), ''Fractals Illustrate the Mathematical Way of Thinking,'' Coll. Math. J., V. 22, No. 1, pp. 60-64.

7. Straffin, Philip D., (1991), ''Newton's Method and Fractal Patterns,'' UMAP J., V. 12, No. 2, pp. 147-164, and UMAP, Module 716, pp. 1-18, COMAP, Inc. Lexington, MA 02420 USA. COMAP   

8. Zobitz, Jennifer, (1987), ''Fractals: Math. Monsters,'' Pi Mu Epsilon J., V. 8, No. 7, pp. 425-440.

Project III. Write a report on how complex analysis is used the study Julia sets.

1. Peitgen, H. O. and D. Saupe and F. V. Haeseler, (1984), ''Cayley's Problem and Julia Sets,'' Math. Intell., V. 6, No. 2, pp. 11-20.

2. Vrscay, Edward R., ''Julia Sets and Mandelbrot-Like Sets Associated With Higher Order Schroder Rational Iteration Functions: A Computer Assisted Study,'' Math. of Computation, Vol. 46, No. 173. (Jan., 1986), pp. 151-169., Jstor.

Project IV. Write a report on how complex analysis is used to study the Mandelbrot set.

1. Bridger, Mark, (1988), ''Looking at the Mandelbrot Set,'' Coll. Math. J., V. 19, No. 4, pp. 353-363.

2. Crowe, W. D. and R. Hasson and P. J. Rippon and P. E. D. Strain-Clark, (1989), ''On the Structure of the Mandelbar Set,'' Nonlinearity, V. 2, pp. 541-553

3. Devaney, Robert L., (1991), ''The Orbit Diagram and the Mandelbrot Set,'' Coll. Math. J., V. 22, No. 1, pp. 23-38.

4. Ewing, John, (1995), ''Can We See the Mandelbrot Set?,'' College Math J.: Volume 26, Number 2, Pages: 90-99.

5. Frantz, Marny and Sylvia Lazarnick, (1991), ''The Mandelbrot Set in the Classroom,'' The Math. Teach., V. 84, No. 3, pp. 173-177.

Project V. Write a report on how complex functions are used in the study of chaos.

1. Barton, Ray, (1990), ''Chaos and Fractals,'' Math. Teach., V. 83, No. 7, pp. 524-529.

2. Devaney, Robert L., (1987), ''Chaotic Bursts in Nonlinear Dynamical Systems, Science, V. 235, pp. 342-345.

3. Devaney, Robert L. and Linda Keen, (Editors), (1989), Chaos and Fractals: The Math. Behind the Computer Graphics, Providence R.I., Amer. Math. Soc.

4. Devaney, Robert L. and Marilyn B. Durkin, ''The Exploding Exponential and Other Chaotic Bursts in Complex Dynamics,'' Am. Math. M., Vol. 98, No. 3. (Mar., 1991), pp. 217-233, Jstor.   .

5. Parris, Richard, (1991), ''The Root-Finding Route to Chaos,'' Coll. Math. J., V. 22, No. 1, pp. 48-55.

6. Strang, Gilbert, (1991), ''The Chaotic Search for i,'' Coll. Math. J., V. 22, No. 1, pp. 3-12.

Section 4.3  Exercises for Julia and Mandelbrot Sets
See textbook page 141.

Section 4.3	Geometric Series and Convergence Theorems

Section 4.4	Power Series Functions

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