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  
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This free software is compliments of the authors.
John H. Mathews,   mathews@fullerton.edu 
Russell W. Howell,  howell@westmont.edu

CHAPTER 3  ANALYTIC and HARMONIC FUNCTIONS

u = v = w = x = y = z = Z = 
                    0; 
Clear[a, b, difQuotient, dfdt, 
        dfdx, Δf, ΔfΔx, ΔfΔy, ΔfΔz, Δw, Δx, Δy, Δz, E1, 
        E2, E3, E4, E5, f, F, g, g1, g2, g3, g4, g5, lap, lim, L1, L2, sub, 
        u, Ux, Uy, Uθ, v, v1, v2, v3, Vx, Vy, Vθ, w, x, x0, y, 
        y0, z, z0, Z];

Section 3.1	Differentiable Functions

Section 3.2	The Cauchy-Riemann Equations

Section 3.3  Analytic Functions and Harmonic Functions

Let ϕ (x, y) be a real-valued function of the two real variables x and y. The partial differential equation  

			(Subscript[ϕ, xx]) (x, y) + (Subscript[ϕ, yy]) (x, y) = 0  

is known as Laplace's Equation equation and is sometimes referred to as the potential equation. If ϕ,(Subscript[ϕ, x]),(Subscript[ϕ, y]),(Subscript[ϕ, xx]),(Subscript[ϕ, xy]),(Subscript[ϕ, yx])and (Subscript[ϕ, yy]) are all continuous and if ϕ (x, y) satisfies Laplace's equation, then ϕ (x, y) is called a Harmonic Function. Harmonic functions are important in the areas of applied mathematics, engineering, and mathematical physics. They are used to solve problems involving steady state temperatures, two-dimensional electrostatics, and ideal fluid flow. In Chapter 10 we will see how complex analysis techniques can be used to solve some problems involving harmonic functions. We begin with an important theorem relating analytic and harmonic functions.

Theorem 3.8,  Page 114.  Let f (z) = f (x + iy) =  
          u (x, y) + i v (x, y)  be an analytic function in the domain  D.  
Then both  u(x,y)  and  v(x,y)  are harmonic function in  D. In other words, the real and imaginary parts of an analytic function are harmonic.

Proof of Theorem 3.8,  see text Page 114.

If we are given a function u(x,y) that is harmonic in the domain D and if we can find another harmonic function v(x,y), such that the partial derivatives for u and v satisfy the Cauchy-Riemann equations throughout D, then we say that v(x,y) is a harmonic conjugate of u(x,y). It then follows that the function f(z) =u(x,y)+iv(x,y) is analytic in D.

Example 3.11,  Page 115.  If u (x, y) = (x^2) - (y^2), then (Subscript[u, xx]) (x, y) + (Subscript[u, yy]) (x, y) = 2 - 2 = 0;  hence u is a harmonic function. We find that v (x, y) = 2 x y is also a harmonic function and that 

			(Subscript[u, x]) = (Subscript[v, y]) = 2  x and (Subscript[u, y]) = -(Subscript[v, x]) = -2 y.

Therefore v is a harmonic conjugate of u, and the function f given by  

			f (z) = (x^2) - (y^2) + i 2 x y = (z^2).

Solution 3.11.

Enter the functions u[x,y] and v[x,y] and show that the Cauchy-Riemann equations hold.

Clear[u, Ux, Uy, v, Vx, Vy, w, x, y, z, 
        Z]; 
TagSet[x, Im[x], 0]; 
TagSet[y, Im[y], 0]; 
TagSet[x, Re[x], x]; 
TagSet[y, Re[y], y]; 
u[x_, y_]  =  (x^2) - (y^2); 
v[x_, y_]  =  2 x y; 
Ux  =  (∂_x) u[x, y]; 
Uy  =  (∂_y) u[x, y]; 
Vx  =  (∂_x) v[x, y]; 
Vy  =  (∂_y) v[x, y]; 
Print[" u[x,y] = ", u[x, y]]; 
Print[" v[x,y] = ", v[x, y]]; 
Print[""]; 
\(\(Print[\*""\!\(u\_x\)[x,y] = "", Ux];\)\ \)
\(\(Print[\*""\!\(u\_y\)[x,y] = "", Uy];\)\ \)
\(\(Print[\*""\!\(v\_x\)[x,y] = "", Vx];\)\ \)
\(\(Print[\*""\!\(v\_y\)[x,y] = "", 
        Vy];\)\ \)
Print[""]; 
\(\(Print[\*""Is \!\(u\_x\)[x,y] =  \!\(v\_y\)[x,y] ?      "", 
        Ux == Vy];\)\ \)
\(\(Print[\*""Is \!\(u\_y\)[x,y] = -\!\(v\_x\)[x,y] ?      "", 
        Uy == \(-Vx\)];\)\ \)

The Cauchy-Riemann equations hold, therefore  f (x + i y) = 
        u (x, y) + i v (x, y)  is analytic.  It follows that both  u (x, y) = Re[f (x + i y)] =  
          (x^2) - (y^2)  and   v (x, y) = Im[f (x + i y)] =  2 x y are harmonic functions.

Aside.  The underlying complex function is  f[z] = (z^2).

f[z_] = (z^2); 
Z  =  x  +  I y; 
Print["  w = f[z] = ", f[z]]; 
Print["  w = f[z] = ", f[Z]]; 
Print["  w = f[z] = ", 
        ComplexExpand[f[Z]]]; 
Print[""]; 
Print["  f[", 
        Z, "] = u[x,y] + \[ImaginaryI] v[x,y]"]; 
Print["  f[", Z, "] = (", 
        u[x, y], ") + \[ImaginaryI](",  
        v[x, y], ")"]; 
Print["  f[", Z, "] = ", 
        u[x, y] + I v[x, y]];

Theorem 3.8 makes the construction of harmonic functions from known analytic functions an easy task.

Example 3.12,  Page 115. The function f (z) = (z^3) = 
          (x^3) - 3 x (y^2) + i (3 (x^2) y - (y^3)) is analytic for all values of z, hence it follows that  

			u (x, y) = Re[f (z)] = (x^3) - 3 x (y^2)  
is harmonic, and
			v (x, y) = Im[f (z)] = 3 (x^2) y - (y^3)  
		
is a harmonic conjugate of u(x,y).

Solution 3.12.

Enter the functions u[x,y] and v[x,y] and show that the Cauchy-Riemann equations hold.

Clear[u, Ux, Uy, v, Vx, Vy, w, x, y, z, 
        Z]; 
TagSet[x, Im[x], 0]; 
TagSet[y, Im[y], 0]; 
TagSet[x, Re[x], x]; 
TagSet[y, Re[y], y]; 
u[x_, y_]  =  (x^3) - 3 x (y^2); 
v[x_, y_]  =  3 (x^2) y - (y^3); 
Ux  =  (∂_x) u[x, y]; 
Uy  =  (∂_y) u[x, y]; 
Vx  =  (∂_x) v[x, y]; 
Vy  =  (∂_y) v[x, y]; 
Print[" u[x,y] = ", u[x, y]]; 
Print[" v[x,y] = ", v[x, y]]; 
Print[""]; 
\(\(Print[\*""\!\(u\_x\)[x,y] = "", Ux];\)\ \)
\(\(Print[\*""\!\(u\_y\)[x,y] = "", Uy];\)\ \)
\(\(Print[\*""\!\(v\_x\)[x,y] = "", Vx];\)\ \)
\(\(Print[\*""\!\(v\_y\)[x,y] = "", 
        Vy];\)\ \)
Print[""]; 
\(\(Print[\*""Is \!\(u\_x\)[x,y] =  \!\(v\_y\)[x,y] ?      "", 
        Ux == Vy];\)\ \)
\(\(Print[\*""Is \!\(u\_y\)[x,y] = -\!\(v\_x\)[x,y] ?      "", 
        Uy == \(-Vx\)];\)\ \)

The Cauchy-Riemann equations hold, therefore  f (x + i y) = u (x, 
                y) + i v (x, y) =  
          (x^3) - 3 x (y^2) + i (3 (x^2) y - (y^3))  is analytic.  It follows that both  u (x, y) = Re[f (x + i y)] =  
          (x^3) - 3 x (y^2)  and   v (x, y) = Im[f (x + i y)] =  
          3 (x^2) y - (y^3) are harmonic functions.

Aside.  The underlying complex function is  f[z] = (z^3).

f[z_] = (z^3); 
Z  =  x  +  I y; 
Print["  w = f[z] = ", f[z]]; 
Print["  w = f[z] = ", f[Z]]; 
Print["  w = f[z] = ", 
        ComplexExpand[f[Z]]]; 
Print[""]; 
Print["f[", 
        Z, "] = u[x,y] + \[ImaginaryI] v[x,y]"]; 
Print["f[", Z, "] = (", u[x, y], ") + \[ImaginaryI](",  
        v[x, y], ")"]; 
Print["f[", Z, "] = ", 
        u[x, y] + I v[x, y]];

Complex analysis can be used to show easily that certain combinations of harmonic functions are harmonic. For example, we can show that if v is a harmonic conjugate of u, then their product ϕ (x, y)  = u (x, y) v (x, y) is a harmonic function. This can be verified directly by computing the partial derivatives and showing that Laplace's equation holds true, but the details are tedious. If we use complex variable techniques instead, we can start with the fact that f (z) = u (x, y) + i v (x, y) is an analytic function. Then we observe that the square of f is also an analytic function, which is

		\(\([f \((z)\)]\)\^2 = \([u \((x, y)\)]\)\^2 - \([v \((x, y)\)]\)\^2 + i\ 2\ u \((x, y)\) v \((x, y)\)\).  

	We then know immediately that the imaginary part, 2u(x,y)v(x,y), is a harmonic function by theorem 3.8. Since a constant multiple of a harmonic function is harmonic, it follows that ϕ (x, y)  = u (x, y) v (x, y) is harmonic. It is left as an exercise to show that if (Subscript[u, 1]) (x, y) and (Subscript[u, 2]) (x, y) are two harmonic functions that are not related in the preceding fashion, then their product need NOT be harmonic.

Theorem 3.9 (Construction of a Conjugate),  Page 116.  Let  u(x,y)  be harmonic in an ϵ-neighborhood of the point ((Subscript[x, 0]), (Subscript[y, 0])).  Then there exists a conjugate harmonic function  v(x,y)  defined in this neighborhood, and  f (z) = u (x, y) + i v (x, y) is an analytic function in this neighborhood.

Proof of Theorem 3.9,  see text Page 116.

Method. Construction of the Harmonic Conjugate of u(x,y).

Execute this cell to activate the construction of the harmonic conjugate subroutine.

u = v = x = y = 
              0; 
Clear[con, lap, u, v, v1, v2, v3, x, y]; 
\(\(con[u0_]\  := \ 
        Module[{}, 
lap\  = \ 
            Simplify[∂\_\(x, x\)u[x, 
                    y]\  + \ ∂\_\(y, y\)u[x, 
                    y]]; \ 
If[lap =!= 0, 
            Print["u[x,y] was NOT harmonic"]; \ 
Print["u(x,y) = ", 
              u[x, y]]; \ 
Print[\*""(Subscript[u, xx]) + (Subscript[u, yy]) = ""]; \ \ 
Return[
              Print[Expand[
                  lap]]]]; \ 
v1\  = ∫∂\_x\ u[x, 
                    y] \[DifferentialD]y; \ 
v2\  = \ \(-\ ∂\_y\ u[x, y]\)\  - \ ∂\_x\ 
                v1; \ 
v3\  = ∫v2 \[DifferentialD]x; \ 
Return[
            v1\  + \ v3];\ \ ];\)\ \)

Example 3.13,  Page 117.  Show that  u (x, y) = x (y^3) - (x^3) y  is a harmonic function and find the harmonic conjugate  v(x,y).

Solution 3.13.

Enter the function u[x,y], and determine if it is a harmonic function.  If so, proceed with the construction of the harmonic conjugate v[x,y].

Clear[f, F, u, v, x, 
        y]; 
u[x_, y_]  =  x (y^3)  -  (x^3) y; 
v[x_, y_]  =  con[u[x, y]]; 
Print["u(x,y) = ", u[x, y]]; 
Print[""]; 
Print["Constructing the harmonic conjugate we get"]; 
Print[""]; 
Print["v(x,y) = ", v[x, y]];

Form the analytic function  f[z] = u[x,y]+ i v[x,y]  and verify that the Cauchy-Riemann equations hold.

Ux  =  (∂_x) u[x, y]; 
Uy  =  (∂_y) u[x, y]; 
Vx  =  (∂_x) v[x, y]; 
Vy  =  (∂_y) v[x, y]; 
f[x_, y_] = u[x, y]  +  I v[x, y]; 
Print["   f[z] = ", f[x, y]]; 
Print["   f[z] = u[x,y] + \[ImaginaryI] v[x,y]"]; 
Print[""]; 
Print[" u[x,y] = ", u[x, y]]; 
Print[" v[x,y] = ", v[x, y]]; 
Print[""]; 
\(\(Print[\*""\!\(u\_x\)[x,y] = "", Ux];\)\ \)
\(\(Print[\*""\!\(u\_y\)[x,y] = "", Uy];\)\ \)
\(\(Print[\*""\!\(v\_x\)[x,y] = "", Vx];\)\ \)
\(\(Print[\*""\!\(v\_y\)[x,y] = "", 
        Vy];\)\ \)
Print[""]; 
\(\(Print[\*""Is \!\(u\_x\)[x,y] =  \!\(v\_y\)[x,y] ?      "", 
        Ux == Vy];\)\ \)
\(\(Print[\*""Is \!\(u\_y\)[x,y] = -\!\(v\_x\)[x,y] ?      "", 
        Uy == \(-Vx\)];\)\ \)

Therefore  v[x, y] = ((x^4)/4) - ((3 (x^2) (y^2))/2) + ((y^4)/4) is the harmonic conjugate of  u[x, y] = -(x^3) y + x (y^3).  

Aside.  We can use Mathematica to get the standard form of an analytic function.

F[z_] = 
        ReplaceAll[
          f[x, y], {x -> ((z + Conjugate[z])/2), 
            y -> ((z - Conjugate[z])/(2  I))}]; 
F[z_] = ComplexExpand[F[z]]; 
\(\(W = 
        ReplaceAll[
          f[x, y], {x -> \(z + \(\(z\)\(\ \)\)\&_\)\/2, 
            y -> \(z - \(\(z\)\(\ \)\)\&_\)\/\(2  I\)}];\)\ \)
Print["f[z] = u[x,y] + \[ImaginaryI] v[x,y]"]; 
Print["f[z] = ", f[x, y]]; 
Print[""]; 
Print["f[z] = ", W]; 
Print[""]; 
Print["f[z] = ", F[z]];

Harmonic functions are solutions to many physical problems. Applications include two-dimensional models of heat flow, electrostatics, and fluid flow. We now give an example of the latter.
	
	We assume that an incompressible and frictionless fluid flows over the complex plane and that all cross sections in planes parallel to the complex plane are the same. Situations such as this occur when fluid is flowing in a deep channel. The velocity factor at the point (x,y) is 

		V(x, y)=p (x, y) + i q (x, y).  

	The assumption that the flow is irrotational and has no sources or sinks implies that both the curl and divergence vanish, that is, (Subscript[q, x]) - (Subscript[p, y]) = 0 and (Subscript[p, x]) + (Subscript[q, y]) = 0. Hence p and q obey the partial differential equations 

	(Subscript[p, x]) (x, y) = -(Subscript[q, y]) (x, y) and (Subscript[p, y]) (x, y) = (Subscript[q, x]) (x, y).

These equations are similar to the Cauchy-Riemann equations and permit us to define a special complex function:

	f (z) = u (x, y) + i v (x, y) =p (x, y) - i q (x, 
              y).  

Here we have (Subscript[u, x]) = (Subscript[p, x]),(Subscript[u, y]) = (Subscript[p, y]),(Subscript[v, x])=-(Subscript[q, x]) and (Subscript[v, y])=-(Subscript[q, y]). Using the above equations we can verify that the Cauchy-Riemann equations are satisfied for f:  

		(Subscript[u, x]) (x, 
            y) = (Subscript[p, x]) (x, 
              y) = -(Subscript[q, y]) (x, y) = (Subscript[v, y]) (x, y)  
	and
		(Subscript[u, y]) (x, 
            y) = (Subscript[p, y]) (x, 
              y) = (Subscript[q, x]) (x, y) = -(Subscript[v, x]) (x, y).  

Assuming the functions p and q have continuous partials, Theorem 3.5 guarantees that function f is analytic, and the fluid flow is the conjugate of an analytic function, that is,

		V(x, y)=\(\(f \((z)\)\)\&_\).

In Chapter 6 we will prove that every analytic function f has an analytic antiderivative F;  assuming this to be the case, we can write  

			F (z) = ϕ (x, y) + i ψ (x, y),  

where F' (z) = f (z).  

	Theorem 3.8 tells us that ϕ (x, y) is a harmonic function. If we use the vector interpretation of a complex number we see that the gradient oft ϕ can be written as follows:

			grad ϕ (x, 
            y) =  (Subscript[ϕ, x]) (x, 
              y) + i (Subscript[ϕ, y]) (x, y).

The Cauchy-Riemann equations applied to F(z) give us (Subscript[ϕ, y]) (x, y) = - (Subscript[ψ, x]) (x, y);  making this substitution in the last equation yields  

			grad ϕ (x, 
            y) =  (Subscript[ϕ, x]) (x, 
              y) - i (Subscript[ψ, x]) (x, y)\(\(\(\ \)\(\(=\)\(\(\(ϕ\_x\) \((x, 
                  y)\) + i\ \(ψ\_x\) \((x, 
                  y)\)\)\&_\)\)\)\).  

Theorem (3-17) says that (Subscript[ϕ, x]) (x, y) + i (Subscript[ψ, x]) (x, y) = 
        F' (z), which by the last equation and equations imply that  

			\(grad\ ϕ \((x, 
            y)\) = \(\(F' \((z)\)\)\&_ = \(f \((z)\)\)\&_\)\).  

From the above work we finally see that ϕ (x, y) is the scalar potential function for the a fluid flow, that is,  

			V(x, y)=grad ϕ (x, y).

The curves given by {(x, y) :   ϕ (x, y) = constant} are called equipotentials. The curves {(x, y) :  ψ (x, y) = constant} are called streamlines and describe the path of fluid flow. In Chapter 10 we will see that the family of equipotentials is orthogonal to the family of streamlines.

Example 3.14,  Page 119.  Show that ϕ (x, y) =  (x^2) - (y^2)  is the scalar potential function for the fluid flow  V(x, y) =2 x - 2 i y.

Solution 3.14.

Enter the complex potential  F[z] = (z^2).

Clear[F, ϕ, φ, z]; 
F[z_]  =  (z^2); 
ϕ[x_, y_]  =  
        ComplexExpand[Re[F[x + I y]]]; 
φ[x_, y_]  =  
        ComplexExpand[
          Im[F[x + I y]]]; 
Print["The complex potential is"]; 
Print["  F(z) = ", F[z]]; 
Print[" F'(z) = ", 
        Expand[F'[x + I y]]]; 
Print[""]; 
Print["The vector field is"]; 
\(\(Print[\*""V(x,y) = \!\(\(F' \((z)\)\)\&_\) = "", 
\ \ \ \ \ \ \ ComplexExpand[
          Conjugate[\(F'\)[
              x + I\ y]]]];\)\ \)
Print[""]; 
Print["The equipotential curves are"]; 
Print["ϕ(x,y) = ", ϕ[x, 
          y]]; 
Print[""]; 
Print["The streamline curves are"]; 
Print["φ(x,y) = ", φ[x, 
          y]];

Use Mathematica to make a contour plot of the velocity potential and streamline curves

g1 = 
        ContourPlot[ϕ[x, y], {x, -8, 8}, {y, 0, 
            8}, 
Contours  ->  10, 
          PlotPoints -> 25, 
          AspectRatio -> (1/2), 
ContourShading -> 
            False, AxesLabel -> {"x", "y"}]; 
g2 = 
        ContourPlot[φ[x, y], {x, -8, 8}, {y, 0, 
            8}, 
Contours  ->  10, 
          PlotPoints -> 25, 
          AspectRatio -> (1/2), 
ContourShading -> 
            False, AxesLabel -> {"x", "y"}]; 
Show[g1, g2]; 
Print["The  equipotential  curves  ϕ(x,y) = ", ϕ[x, 
          y]]; 
\(\(Print["and the streamline curves   φ(x,y) = ", φ[x, y]];\)\ \)

ContourPlot[ϕ[x, y], {x, -8, 8}, {y, 0, 
          8}, 
Contours  ->  10, 
        PlotPoints -> 25, 
        AspectRatio -> 
          (1/2), 
AxesLabel -> {"x", "y"}, 
        ColorFunction -> Hue]; 
Print["The equipotential curves are"]; 
Print["ϕ(x,y) = ", ϕ[x, 
          y]];

ContourPlot[φ[x, y], {x, -8, 8}, {y, 
          0, 8}, 
Contours  ->  10, 
        PlotPoints -> 25, 
        AspectRatio -> 
          (1/2), 
AxesLabel -> {"x", "y"}, 
        ColorFunction -> Hue]; 
Print["The streamline curves are"]; 
Print["φ(x,y) = ", φ[x, 
          y]];

Library Research Experience for Undergraduates

Project I. Write a report on harmonic functions.

1. Axler, Sheldon, ''Harmonic Functions from a Complex Analysis Viewpoint,'' Am. Math. M., Vol. 93, No. 4. (Apr., 1986), pp. 246-258, Jstor.   

2. Bivens, Irl C., (1992), ''When Do Orthogonal Families of Curves Possess a Complex Potential ?,'' Math. Mag., V. 65, No. 4, pp. 226-235.

3. Curtiss, J. H., ''On Extending the Definition of a Harmonic Function (in Questions, Discussions, and Notes),'' Am. Math. M., Vol. 47, No. 4. (Apr., 1940), pp. 225-228, Jstor.   

4. Edgar, Gerald A. and Lee A. Rubel, (1990), ''The harmonic conjugate of an algebraic function Am. Math. M. V. 97 pp. 165-166.

5. Fisher, Stephen D., (1990), Complex Variables, 2nd ed., Monterey, Calif., Wadsworth & Brooks/Cole.

6. Flanigan, Francis J., (1972), Complex Variables: Harmonic and Analytic Functions, Boston, Allyn and Bacon.

7. Fuchs, Watson, (1993), Complex Variables and Introduction, New York, Marcel Dekker, Inc.

8. Jeffrey, Alan, (1992), Complex Analysis and Applications, Boca Raton, FL., CRC Press.

9. Johnson, L. H., ''The Cauchy Problem for Laplace's Equation in Three Dimensions,'' Am. Math. M., Vol. 42, No. 2. (Feb., 1935), pp. 65-74, Jstor.   

10. Kasner, Edward and John De Cicco, ''Note on Conjugate Harmonic Functions (in Math. Notes),'' Am. Math. M., Vol. 54, No. 7, Part 1. (Aug. - Sep., 1947), pp. 405-406, Jstor.   

11. Laitone, E. V., (1977), ''Relation of the Conjugate Harmonic Functions to f(z) (in Math. Notes),'' Am. Math. M., Vol. 84, No. 4. (Apr., 1977), pp. 281-283., Jstor.   

12. Mann, H. B., ''On a Property of Harmonic Functions (in Classroom Notes),'' Am. Math. M., Vol. 66, No. 5. (May, 1959), p. 414, Jstor.   

13. Miles, E. P., ''Three Dimensional Harmonic Functions Generated by Analytic Functions of a Hypervariable (in Math. Notes),'' Am. Math. M., Vol. 61, No. 10. (Dec., 1954), pp. 694-697, Jstor.   

14. Needham, Tristan, ''The Geometry of Harmonic Functions,'' Math. Mag., Vol. 67, No. 2, pp. 92-108, April 1994.

15. Nilson, R. H. and Y. G. Tsuei, (1974), ''Inverted Cauchy Problem for the Laplace equation in Engineering Design,'' J. of Engineering Math., Vol 8, No. 4, pp. 329-337.

16. Redheffer, Raymond M., ''On Pairs of Harmonic Functions,'' Proceedings of the American Mathematical Society, Vol. 8, No. 3. (Jun., 1957), pp. 450-457, Jstor.   

17. Sacksteder, Richard C., (1978), ''On Oscillatory Flows,'' Math. Intell., V. 1, No. 1, pp. 45- 51.

18. Srinivasan, V. K., (1977), ''A Note on Harmonic Functions and Harmonic Conjugates,'' Int. J. of Math. Ed. in Sci. and Tech., V. 8, No. 3, pp. 323-328.

Project II. Write a report about computer graphics for harmonic functions and complex functions.

1. Braden, Bart, (1991), ''The Vector Field Approach in Complex Analysis,'' Visualization in Teaching and Learning Math., Providence, R.I., Math. Assoc. of Amer., pp. 191-196.

2. Brandt, Siegmund and Hermann Schneider, (1976), ''Computer-Drawn Field Lines and Potential Surfaces for a Wide Range of Field Configurations,'' Am. J. Phy., V. 44, No. 12, pp. 1160-1171.

3. Bruch, John C., (1975), ''The Use of Interactive Computer Graphics in the Conformal Mapping Area,'' Computers and Graphics, V. 1, pp. 361-374.

4. Kranc, S. C., (1986), ''Plotting Streamlines and Pathlines on a Microcomputer,'' Comp. in Ed. Div. of ASEE, V. VI, No. 3, pp. 20-21.

5. Mathews, John H., (1988), ''Using a Symbol Manipulation Program to Construct a Harmonic Conjugate Function,'' Computers in Education Division of ASEE, Vol. VIII, No. 3, pp. 67-71.

6. Piele, Donald T., Morris W. Firebaugh and Robert Manulik, ''Applications of Conformal Mapping to Potential Theory Through Computer Graphics,'' Am. Math. M., Vol. 84, No. 9. (Nov., 1977), pp. 677-692., Jstor.   

7. Shilgalis, Thomas W., (1982), ''Geometric Transformations on a Microcomputer,'' The Math. Teach., V. 75, No. 1, pp. 16-19.

Project III. Write a report on fluid flow and how it is related to harmonic and analytic functions.
		
1. Burington, R. S., ''On the Use of Conformal Mapping in Shaping Wing Profiles,'' Am. Math. M., Vol. 47, No. 6. (Jun. - Jul., 1940), pp. 362-373, Jstor.   

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

3. Harlow, F. H., ''Numerical Fluid Dynamics,'' Am. Math. M., Vol. 72, No. 2, Part 2: Computers and Computing. (Feb., 1965), pp. 84-91, Jstor.   

4. Jeffrey, Alan, (1992), Complex Analysis and Applications, Boca Raton, FL., CRC Press.

5. Mancill, J. D. and Betty Thomas, ''On the Equation of Joukowski's Aerofoils (in Discussions and Notes),'' Am. Math. M., Vol. 53, No. 3. (Mar., 1946), pp. 147-149, Jstor.   

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

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

8. Sacksteder, Richard C., (1978), ''On Oscillatory Flows,'' Math. Intell., V. 1, No. 1, pp. 45- 51.

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

Project IV. Write a report on the Polya vector field.

1. Boas, R. P., ''Selected Topics from Polya's Work in Complex Analysis,'' Math. Mag., Vol. 60, No. 5, pp. 271-274, December 1987.

2. Braden, Bart, (1985), ''Picturing Functions of a Complex Variable,'' Coll. Math. J., Vol. 16, No. 1, pp. 63-72.

3. Braden, Bart, (1987), ''Polya's Geometric Picture of Complex Contour Integrals, Math. Mag., V. 60, No. 5, pp. 321-327.

4. Braden, Bart, (1991), ''The Vector Field Approach in Complex Analysis,'' Visualization in Teaching and Learning Math., Providence, R.I., Math. Assoc. of Amer., pp. 191-196.

5. Gluchoff, Alan D., (1993), ''Complex Power Series -- a Vector Field Visualization,'' Math. Mag., V. 66, No. 3, pp. 189-191.

Section 3.3  Exercises for Analytic Functions and Harmonic Functions
See textbook page 121.

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