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 6  COMPLEX INTEGRATION

a = b = c = d = R = t = u = U = v = V = w = W = x = y = z = Z = 
                                      0; 
Clear[a, b, c, center, centers, d, dot, dots, f, F, i, int, int1, 
        int2, Int, ImInt, ImVal, label, path, r, R, ReInt, ReVal, soset, 
        solset, t, u, U, U1, U2, v, V, V1, V2, val, Val, w, W, W1, W2, Wint, 
        x, x0, x1, y, y0, y1, z, z1, z2, Z, Z1, Z2, 
        Z3]; 
Needs["Graphics`Arrow`"]; 
Needs["Graphics`Colors`"];

Section 6.1	Complex Integrals

Section 6.2	Contours and Contour Integrals

Section 6.3	The Cauchy-Goursat Theorem

Section 6.4	The Fundamental Theorem of Integration

Section 6.5	Integral Representations for Analytic Functions

Section 6.6  The Theorems of Morera and Liouville and Some Applications

In this section we investigate some of the qualitative properties of analytic and harmonic functions.  Our first result shows that the existence of an antiderivative for a continuous function is equivalent to the statement that the integral of  f  is independent of the path of integration.  This result stated in a form that will serve as a converse to the Cauchy-Goursat theorem.

Theorem 6.13 (Morera's Theorem),  Page 251.  Let  f  be a continuous function in a simply connected domain D.  If  (∫_C) f (z) dz = 0  for every closed contour in D, then  f  is analytic in D.

Proof of Theorem 6.13, see text Page 251.

Cauchy's integral formula show how the value  f ((Subscript[z, 0]))  can be represented by a certain contour integral.  If we choose the contour of integration  C  to be a circle with center  (Subscript[z, 0]),  then we can show that the value  f ((Subscript[z, 0]))  is the integral average of the values of  f(z)  at points  z  on the circle  C.

Theorem 6.14 (Gauss's Mean Value Theorem),  Page 251.  If  f  is analytic in a simply connected domain  D  that contains the circle  C : |z - (Subscript[z, 0])| = R,  then  

			\(f \((z\_0)\) = \(1\/\(2  π\)\) \(∫\_0\%\(2\ π\)f \((z\_0 + 
                  R\ e\^\(i\ θ\))\) dθ\)\).

Proof of Theorem 6.14, see text Page 251.

Theorem 6.15 (Maximum Modulus Principle),  Page 252.  Let f be analytic and nonconstant in the bounded domain D. Then |f(
            z)| does not attain a maximum value at any point (Subscript[z, 0]) in D.

Proof of Theorem 6.15, see text Page 252.

We sometimes state the maximum modulus principle in the following form.

Theorem 6.16 (Maximum Modulus Principle),  Page 254.  Let f be analytic and nonconstant in the bounded domain D. If f is continuous on the closed region R that consists of D and all of its boundary points B, then |f(
            z)| assumes its maximum value, and does so only at point(s) (Subscript[z, 0]) on the boundary B.

Proof of Theorem 6.16, see text Page 254.

Example 6.26,  Page 254.  Let  f(z) = az + b,  where the domain is the disk  D : |z|< 1.  Then f(z) is continuous on the closed region  R : |z|≤ 1  and  \(max\+\(\(|\)\(z\)\(|\)\(\ \)\(\(≤\)\(\ \)\(1\)\)\)\(\(|\)\(f \((z)\)\)\(|\)\) = \(\(|\)\(a\)\(|\)\(+\(\(|\)\(b\)\(|\)\)\)\)\),  and this value is assumed by f(z) at a point  (Subscript[z, 0]) = (e^(i (Subscript[θ, 0])))  on the boundary of D.  
Solution.  The solution is given in the textbook.

Theorem 6.17 (Cauchy's Inequalities),  Page 255.  Let  f  be analytic and nonconstant in the domain D that contains the circle  (Subscript[C, R]) ((Subscript[z, 0])) = {z : |z - (Subscript[z, 0])| = R}.  If  |f (z)|≤ M  holds for all points z ∈ (Subscript[C, R]) ((Subscript[z, 0])),  then  

	\(\(\(|\)\(\(f\^\((n)\)\) \((z\_0)\)\)\(|\)\(\(≤\)\(\(\(n!\)\ M\)\/R\^n\)\)\)\)  for  n = 1, 2,  ... .

Proof of Theorem 6.17, see text Page 255.

Theorem 6.18 shows that a nonconstant entire function cannot be a bounded function.

Theorem 6.18 (Liouville's Theorem),  Page 255.  If  f  is an entire function and is bounded for all values of  z  in the complex plane,  then  f  is constant.

Proof of Theorem 6.18, see text Page 255.

Example 6.27,  Page 256.  Show that the function   sin(z)   is not a bounded function.

Solution 6.27.

Clear[f, F, y, z]
f[z_]  =  Sin[z]; 
F1[y_]  =  Sin[(π/2)  +  I y]; 
F2[y_]  =  TrigToExp[F1[y]]; 
lim1  =  Limit[ F1[y],  y -> ∞]; 
lim2  =  Limit[ F2[y],  y -> ∞]; 
Print["    f[z] = ", f[z]]; 
\(\(Print[\*""f[\!\(π\/2\)+Iy] = "", 
        f[π\/2\  + \ I\ y]];\)\ \)
Print[""]; 
\(\(Print[\*""    \!\(F\_1\)[y] = "", F1[y]];\)\ \)
\(\(Print[\*""    \!\(F\_2\)[y] = "", 
        F2[y]];\)\ \)
\(\(Print[\*""Is \!\(F\_1\)[y] = \!\(F\_2\)[y] = ? "", 
        F1[y] == F1[y]];\)\ \)
Print[""]; 
\(\(Print[\*""(lim_(y -> ∞)) (Subscript[F, 1])[y] = "", lim1];\)\ \)
\(\(Print[\*""(lim_(y -> ∞)) (Subscript[F, 2])[y] = "", lim2];\)\ \)

Therefore, we conclude that sin(z) is not a bounded function !

We can use Liouville's theorem to establish an important theorem of elementary algebra.

Theorem 6.16 (Fundamental Theorem of Algebra),  Page 256  If  P(z)  is a polynomial of degree n ≥ 1, then P has at least one zero.

Proof of Theorem 6.16, see text Page 256.

Corollary 6.4,  Page 257.  Let  P  be a polynomial of degree  n.  Then  P  can be expressed as the product of linear factors.  That is,  
			P (z) = A (z - (Subscript[z, 1])) (z - (Subscript[z, 2]))  ...  (z - 
              (Subscript[z, n]))  

where  (Subscript[z, 1]), (Subscript[z, 2]) ,  ... , (Subscript[z, n])  are the zeros of P counted according to multiplicity an A is a constant.

Proof of Corollary 6.4, see text Page 257.

Extra Example for Page 257.  Find the n zeros of the equation (z^n) - 1 = 0.

Solution for Extra Example, Page 257.

We use Mathematica to illustrate the cases n=1,2,...,8.

Clear[n, P, Q]; 
n = 2; 
P[z_] = (z^n) - 1; 
Q[z_] = Factor[P[z] ]; 
solset = Solve[P[z] == 0, z] ; 
Print[(Subscript[P, n]), "[z] = ", P[z] ]; 
Print[(Subscript[P, n]), "P[z] = ", Q[z] ]; 
Print[""]; 
Print["Solutions to ", 
        P[z], " = 0"];  
Print[solset];

n = 3; 
P[z_] = (z^n) - 1; 
Q[z_] = Factor[P[z] ]; 
solset = Solve[P[z] == 0, z] ; 
Print[(Subscript[P, n]), "[z] = ", P[z] ]; 
Print[(Subscript[P, n]), "P[z] = ", Q[z] ]; 
Print[""]; 
Print["Solutions to ", 
        P[z], " = 0"]; 
Print[solset]; 
Print[MapAll[ComplexExpand, solset]];

n = 4; 
P[z_] = (z^n) - 1; 
Q[z_] = Factor[P[z] ]; 
solset = Solve[P[z] == 0, z] ; 
Print[(Subscript[P, n]), "[z] = ", P[z] ]; 
Print[(Subscript[P, n]), "P[z] = ", Q[z] ]; 
Print[""]; 
Print["Solutions to ", 
        P[z], " = 0"]; 
Print[solset];

n = 5; 
P[z_] = (z^n) - 1; 
Q[z_] = Factor[P[z] ]; 
solset = Solve[P[z] == 0, z] ; 
Print[(Subscript[P, n]), "[z] = ", P[z] ]; 
Print[(Subscript[P, n]), "P[z] = ", Q[z] ]; 
Print[""]; 
Print["Solutions to ", 
        P[z], " = 0"]; 
Print[solset]; 
Print[MapAll[ComplexExpand, solset]];

n = 6; 
P[z_] = (z^n) - 1; 
Q[z_] = Factor[P[z] ]; 
solset = Solve[P[z] == 0, z] ; 
Print[(Subscript[P, n]), "[z] = ", P[z] ]; 
Print[(Subscript[P, n]), "P[z] = ", Q[z] ]; 
Print[""]; 
Print["Solutions to ", 
        P[z], " = 0"]; 
Print[solset]; 
Print[MapAll[ComplexExpand, solset]];

n = 7; 
P[z_] = (z^n) - 1; 
Q[z_] = Factor[P[z] ]; 
solset = Solve[P[z] == 0, z] ; 
Print[(Subscript[P, n]), "[z] = ", P[z] ]; 
Print[(Subscript[P, n]), "P[z] = ", Q[z] ]; 
Print[""]; 
Print["Solutions to ", 
        P[z], " = 0"]; 
Print[solset]; 
Print[MapAll[ComplexExpand, solset]];

n = 8; 
P[z_] = (z^n) - 1; 
Q[z_] = Factor[P[z] ]; 
solset = Solve[P[z] == 0, z] ; 
Print[(Subscript[P, n]), "[z] = ", P[z] ]; 
Print[(Subscript[P, n]), "P[z] = ", Q[z] ]; 
Print[""]; 
Print["Solutions to ", 
        P[z], " = 0"]; 
Print[solset]; 
Print[MapAll[ComplexExpand, solset]];

Library Research Experience for Undergraduates

Project I. Write a report on the Fundamental Theorem of Algebra.

1. Boas, R. P., ''A Proof of the Fundamental Theorem of Algebra (in Questions, Discussions, and Notes),'' Am. Math. M., Vol. 42, No. 8. (Oct., 1935), pp. 501-502, Jstor.   

2. Boas, R. P., ''Yet Another Proof of the Fundamental Theorem of Algebra (in Math. Notes),''Am. Math. M., Vol. 71, No. 2. (Feb., 1964), p. 180, Jstor.   

3. Brenner, J. L. and R. C. Lyndon,''Proof of the Fundamental Theorem of Algebra,'' Am. Math. M., Vol. 88, No. 4. (Apr., 1981), pp. 253-256., Jstor.   

4. Campbell, Paul J., (1967), ''The Fundamental Theorem of Algebra,'' Pi Mu Epsilon J., V. 4, No. 6, pp. 243-247.

5. Eaton, J. E., ''The Fundamental Theorem of Algebra (in Classroom Notes),'' Am. Math. M., Vol. 67, No. 6. (Jun. - Jul., 1960), pp. 578-579, Jstor.   

6. Fefferman, Charles, ''An Easy Proof of the Fundamental Theorem of Algebra (in Classroom Notes),'' Am. Math. M., Vol. 74, No. 7. (Aug. - Sep., 1967), pp. 854-855, Jstor.   

7. Hirschhorn, Michael D., ''The Fundamental Theorem of Algebra,'' College Math J., Vol. 29, No. 4, September 1998, pp. 276-277.

8. Redheffer, Raymond M., ''The Fundamental Theorem of Algebra (in Math. Notes),'' Am. Math. M., Vol. 64, No. 8. (Oct., 1957), pp. 582-585, Jstor.   

9. Redheffer, Raymond M., ''What! Another Note Just on the Fundamental Theorem of Algebra? (in Classroom,'' Notes) Am. Math. M., Vol. 71, No. 2. (Feb., 1964), pp. 180-185, Jstor.   

10. Terkelsen, Frode, ''The Fundamental Theorem of Algebra (in Classroom Notes),'' Am. Math. M., Vol. 83, No. 8. (Oct., 1976), p. 647, Jstor.   

11. Velleman, Daniel J., ''Another Proof of the Fundamental Theorem of Algebra, Math. Mag., Vol. 70, No. 3, June 1997.

12. Wolfenstein, S., ''Proof of the Fundamental Theorem of Algebra (in Classroom Notes),'' Am. Math. M., Vol. 74, No. 7. (Aug. - Sep., 1967), pp. 853-854, Jstor.

Project II. Write a report on zeros of polynomials and/or complex functions.

1. Andrushkiw, Joseph W. (1959), ''The Probability that the Roots of a Real Quadratic Equation Lie Inside or on the Circumference of the Unit Circle in the Complex Plane,'' Math. Mag. , Vol.. 32, pp. 123-128.

2. Ballantine, J. P., ''Complex Roots of Real Polynomials (in Classroom Notes),'' Am. Math. M., Vol. 66, No. 5. (May, 1959), pp. 411-414, Jstor.   

3. Best, G. C. ''The Determination of the Complex Zeros of A Polynomial,''Am. Math. M., Vol. 54, No. 5. (May, 1947), pp. 269-273, Jstor.   

4. Denton, Brian, (1985), ''Roots of unity revisited!'' The Math. Gazette, V. 69, No. 447, pp. 17-20.

5. Farnsworth, David, (1983), ''Measuring Complex Roots,'' Math. and Comp. Ed., V. 17, No. 3, pp. 191-194.

6. Fettis, Henry E., ''Complex Roots of sin(z) = az, cos(z) = az and cosh(z) = az,'' Math. of Computation, Vol. 30, No. 135. (Jul., 1976), pp. 541-545+s1-s32, Jstor.   

7. Gehman, H. M. ''Complex Roots of a Polynomial Equation,'' Am. Math. M., Vol. 48, No. 4. (Apr., 1941), pp. 237-239, Jstor.   

8. Gomez-Calderon, Javier and David M. Wells, ''Why Polynomials Have Roots,'' The College Math. J., Vol. 27, No. 2, March 1996.

9. Henriquez, Garcia, ''The Graphical Interpretation of the Complex Roots of Cubic Equations (in Questions, Discussions, and Notes),'' Am. Math. M., Vol. 42, No. 6. (Jun. - Jul., 1935), pp. 383-384., Jstor.   

10. Hirst, Holly P., (1997), ''Bounding the Roots of Polynomials,'' College Math J., Vol. 28, No. 4, pp. 292-295.

11. Kim, David and Robert Travers, (1982), ''Those elusive imaginary zeros,'' The Math. Teach., V. 75, pp. 62-64.

12. Kowalski, Robert and Helen Skala, (1990), ''Determining Roots of Complex Functions with Computer Graphics,'' Coll. Micro., V. VIII, No.1, pp. 51-54.

13. Long, Cliff and Thomas Hern, (1989), ''Graphing the Complex Zeros of Polynomials Using Modulus Surfaces,'' Coll. Math. J., V. 20, No. 2, pp. 98-105.

14. Long, Cliff, (1971), ''A note on the Geometry of Zeros of Polynomials,'' Math. Mag., V. 44, pp. 157-159.

15. Long, Cliff, (1972), ''The Quadratic Polynomial and its Zeros,'' Two-Year Coll. Math. J., V. 3, No. 1, pp. 23-29.

16. Lyness, J. N. and L. M. Delves, ''A Numerical Method for Locating the Zeros of an Analytic Function,'' Math. of Computation, Vol. 21, No. 100. (Oct., 1967), pp. 543-560, Jstor.   

17. Norton, Alec and Benjamin Lotto, (1984), ''Complex Roots Made Visible,'' Coll. Math. J., V. 15, No. 3, pp. 248-249.

18. Renka, Robert J. and Floyd Vest, (1988), ''Graphical Representation of Complex Function,'' Math. and Comp. Ed., V. 22, No. 1, pp. 33-45.

19. Sorkin, Sylvia, (1984), ''Using an Interactive Computer Program in the Calculus Classroom to Find Complex Roots of Polynomial Equations,'' Math. and Comp. Ed., V. 18, No. 2, pp. 93-99.

20. Travers, Robert and David Kim, (1982), ''Those Elusive Imaginary Zeros,'' The Math. Teach., V. 75, No. 1, pp. 62-64.

21. Trifan, D. ''Complex Roots of an Integral Rational Equation (in Classroom Notes) Am. Math. M., Vol. 61, No. 9. (Nov., 1954), pp. 640-641., Jstor.   

22. Vest, Floyd, (1985), ''Graphing the Complex Roots of a Quadratic Equation,'' Coll. Math. J., V. 16, No. 4, pp. 257-261.

23. Wallen, Lawrence J., (1992), ''The p-th Root of an Analytic Function,'' Math. Mag., V. 65, No. 4, pp. 260-261.

24. Ward, James A., (1937), ''Graphical Representation of Complex Roots,'' Math. Mag., Vol. 11 , pp. 297-303.

Project III. Write a report on the theorems of Morera and Liouville.

1. Lenard, A., ''A Note on Liouville's Theorem (in Notes),'' Am. Math. M., Vol. 93, No. 3. (Mar., 1986), p. 200-201., Jstor.   

2. Springer, George, ''On Morera's Theorem,'' Am. Math. M., Vol. 64, No. 5. (May, 1957), pp. 323-331., Jstor.

Section 6.6  Exercises for The Theorems of Morera and Liouville and Some Applications
See textbook page 257.

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