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 10  APPLICATIONS of HARMONIC FUNCTIONS

a = A = b = B = c = k = R = T = u = U = v = V = w = W = x = X = y = Y = z = Z = 
                                        0; 
Clear[a, a0, a1, a2, a3, A, b, B, 
        CR, Δx, Δy, eq1, eq2, eqns, f, f1, F, F1, 
        F2, F3, F4, g, G, lines, n, P, Ψ, ϕ, Φ, 
        r, r0, solset, θ, t, T, T1, T2, Q, u, u0, U, U1, v, v0, V, V1, 
        w, w0, W, W1, x, x1, x2, x3, y, z, z1, z2, z3, 
        z4]; 
Needs["Graphics`ComplexMap`"]; 
Needs["Graphics`ParametricPlot3D`"]
Needs["Graphics`Colors`"];

Section 10.1	Preliminaries

Section 10.2	Invariance of Laplace's Equation and the Dirichlet Problem

Section 10.3	Poisson's Integral Formula for the Upper Half Plane

Section 10.4  Two-Dimensional Mathematical Models

We now turn our attention to problems involving steady state heat flow, electrostatics, and ideal fluid flow that can be solved by conformal mapping techniques. The method uses conformal mapping to transform a region in which the problem is posed to one in which the solution is easy to obtain.  Since our solutions will involve only two independent variables,  x  and  y,  we first mention a basic assumption needed for the validity of the model.

	The physical problems we just mentioned are real-word applications and involve solutions in three-dimensional Cartesian space.  Such problems generally would involve the Laplacian in three variables and the divergence and curl of thee dimensional vectors.  Since complex analysis involves only  x  and  y,  we consider the special case in which the solution does not vary with the coordinate along the axis perpendicular to the xy-plane.  For steady state heat flow and electrostatics this assumption will mean that the temperature  T,  or potential  V,  varies only with  x  and  y.  For the flow of  ideal fluids this means that the fluid motion is the same in any plane that is parallel to the z-plane.  Curved drawn in the z plane are to be interpreted as cross sections that correspond to infinite cylinders perpendicular to the z-plane.  Since an infinite cylinder is the limiting case of a "long" physical cylinder, the mathematical model we present is valid provided that the three dimensional problem involves a physical cylinder long enough that the effects at the ends can be reasonably neglected.
	
	In Section 10.1 and Section 10.2 we learned how to obtain solutions  ϕ (x, y)  for harmonic functions.  For applications it is important to consider the family of level curves  {ϕ (x, y) = (Subscript[K, 1]):  (Subscript[K, 1])  is a real constant}  and the conjugate harmonic function  φ (x, y)  and its family of level curves  {φ (x, y) = (Subscript[K, 1]):  (Subscript[K, 1])  is a real constant}.  It is convenient to introduce the terminology complex potential for the analytic function   F (z)  =  ϕ (x, 
              y)  +  i φ (x, y).  The following result regarding the orthogonality of the above mentioned families of level curves will be use in developing the ideas concerning the physical applications.

Theorem 10.4 (Orthogonal Families of Level Curves),  Page 407. Let ϕ (x, y) be harmonic in a domain D. Let φ (x, y) be the harmonic conjugate and let   F (z)  =  ϕ (x, 
              y)  +  i φ (x, y)  be the complex potential.  Then the two families of level curves  

	{ϕ (x, y)=(Subscript[K, 1]):  (Subscript[K, 1])  is a real constant}  and  {φ (x, y)=(Subscript[K, 2]):  (Subscript[K, 2])  is a real constant},  

respectively, are orthogonal in the sense that if  (a,b)  is a point in common to the two curves  ϕ (x, y)=(Subscript[K, 1])  and  φ (x, y)=(Subscript[K, 2]),  and  if   F' (a + ib)≠0  ,  then these two curves intersect orthogonally.

Proof of Theorem 10.4, see text Page 407.

Solution for Theorem 10.4.

Use the Cauchy Riemann equations  (∂_x) u (x, y) = (∂_y) v (x, y)  and (∂_y) u (x, y) = -(∂_x) v (x, y),  which are implemented by using the Mathematica  substitutions:

Remove[x, y, u, v]; 
Clear[CR]; 
CR  =  {(∂_y) v[x, y] -> (∂_x) 
              u[x, y], (∂_y) 
              u[x, y] ->  -(∂_x) 
                v[x, y]};

The normal vectors to the curves  u (x, y) = (Subscript[K, 1])  and  v (x, y) = (Subscript[K, 2])  are

(Subscript[N, 1])  =  {(∂_x) 
            u[x, y], (∂_y) u[x, y]}; 
(Subscript[N, 2])  =  {(∂_x) v[x, y], (∂_y) 
            v[x, y]}; 
\(\(Print[\*""\!\(N\_1\) = "", N\_1];\)\ \)
\(\(Print[\*""\!\(N\_2\) = "", 
        N\_2];\)\ \)

We will show that  (Subscript[N, 1])  is orthogonal to  (Subscript[N, 2])  by forming the dot product:

\(
\(Print[\*""\!\(N\_1\).\!\(N\_2\) = "", 
        N\_1, ".", N\_2];\)\ \)
\(\(Print[\*""\!\(N\_1\).\!\(N\_2\) = "", 
        N\_1 . N\_2];\)\ \)

Using the Cauchy Riemann equations, this is simplified to get:

\(\(\(
\)\(\(Print[\*""(Subscript[N, 1]).(Subscript[N, 2]) = "", ReplaceAll[N\_1 . N\_2, CR]];\)\(\ \)\(
\)
    \)\)\)

Therefore  (Subscript[N, 1])  is orthogonal to  (Subscript[N, 2]) .

Library Research Experience for Undergraduates

Project I. Write a report on mathematical modeling with complex variables.

1. Borgiotti, Giorgio V.;  Sarkissian, Angie;  Williams, Earl G., ''Conformal generalized near-field acoustic holography for axisymmetric geometries,'' The J. of the Acoustical Soc. of Am., V. 88, pp. 199-209.

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. Bryngdahl, Olof, (1974), ''Geometrical Transformations in Optics,'' J. of the Optical Soc. of Am., V. 64, No. 8, pp. 1092-1099.

4. Friedan, Daniel and Stephen Shenker, (1987), ''Conformal invariance and critical exponents in two dimensions,'' Physics Today, V. 40, pp. S19-S20.

5. Celik, I. and H. M Cekirg and I. I. Esen,\ (1977), ''Pollution Control in Bays with Currents,'' Letters in Applied Engineering Science, Vol. 5, No. 3, pp. 241-252.

6. Collett, D. E. and R. C. Hewson-Browne and D. W. Windle, (1970), ''A Complex Variable Approach to Electrochemical Machining Problems,'' J. of Engr. Math., Vol. 4, No. 1, pp. 29-37.

7. Geyi,Wen and Liu Xueguan and Wu Wanchun, (1989), ''Solution of the characteristic impedance of an arbitrarily shaped TEM transmission line using complex variable boundary element method (CVBEM),'' IEE Proc., Part H, Microwaves, Antennas and Propagation V. 136, pp. 73-75.

8. Gilicz, Andras, (1987), ''Alternative, useful conformal mapping method can be used for odd-shaped fields,'' Oil & Gas J., V. 85, pp. 57-58.

9. Goldstein, Marvin and Robert Siegel, (1970),''Conformal Mapping for Heat Conduction in a Region with an Unknown Boundary,'' International J. of Heat Mass Transfer, Vol. 13, pp. 1632-1636.

10. Harryman, R. R., and T. V. Hromadka, J. L. Vaughn and D. P. Watson, ''The CVBEM for Multiply Connected Domains Using a Linear Trial Function,'' Appl. Math. Modeling, 1990, Vol. 14, February, pp. 104-110.

11. Harryman, R. R., and J. L. Vaughn and D. P. Watson, ''The Complex Variable Boundary Element Method Using a Cubic Hermite Trial Function,'' Software for Engineering Workstations, 1988, Vol. 4, October, pp. 193-205.

12. Hayes, John K., David K. Kahaner and Richard G. Kellner, ''A Numerical Comparison of Integral Equations of the First and Second Kind for Conformal Mapping,'' Math. of Computation, Vol. 29, No. 130. (Apr., 1975), pp. 512-521., Jstor.   

13. Heyser, Richard C., (1984), ''Speaker impedance;  more complex than one number,'' Audio, V. 68, pp. 42-46.

14. Hurst, William, (1985), ''Conformal mapping solves intricate problems,'' Oil & Gas J., V. 83, p. 138.

15. Laura, Patricio A., (1975), ''A Survey of Modern Applications of the Method of Conformal Mapping,'' Revista de la Union Matematica Argentina, Vol. 27, pp. 167-179.

16. Miyamoto, S and A. W. Warrick and H. L. Bohn, (1974), ''Land Disposal of Waste Gases: I. Flow Analysis of Gas Injection Systems,'' J. of Environmental Quality, Vol. 3, No. 1, pp. 49-60.

17. Moorer, James A., (1983), ''The manifold joys of conformal mapping: applications to digital filtering in the studio,'' J. of the Audio Engr. Soc., V. 31, pp. 826-841.

18. Nitsche, Joachim A., ''Finite-Element Methods for Conformal Mappings,'' SIAM J. on Numerical Analysis, Vol. 26, No. 6. (Dec., 1989), pp. 1525-1533, Jstor.   

19. Ochs, Robert L., ''The Limited Aperture Problem of Inverse Acoustic Scattering: Dirichlet Boundary Conditions,'' SIAM J. on Applied Math., Vol. 47, No. 6. (Dec., 1987), pp. 1320-1341, Jstor.   

20. 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.   

21. Pocinki, Leon S., (1956), ''The Application of Conformal Transformations to Ocean Wave Refraction Problems,'' Trans. American Geophysical Union, Vol. 31, No. 6, pp. 856-866.

22. Rajaiah, K. and R. R. Kumar, (1984), ''Optimum opening in circular cylindrical shells under uniaxial tension and torsion by conformal transformation,'' J. of Applied Mechanics, V. 51, pp. 693-694.

23. Reichel, Lothar, ''On Polynomial Approximation in the Complex Plane with Application to Conformal Mapping,'' Math. of Computation, Vol. 44, No. 170. (Apr., 1985), pp. 425-433, Jstor.   

24. Reichel, Lothar, ''Polynomials by Conformal Mapping for the Richardson Iteration Method for Complex Linear Systems,'' SIAM J. on Numerical Analysis, Vol. 25, No. 6. (Dec., 1988), pp. 1359-1368, Jstor.   

25. Russell, D. L., (1986), ''The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region,'' SIAM J. on Control and Optimization, V. 24, pp.199-229.

26. Shiue, Sham-Tsong and Sanboh Lee, (1990), ''Conformal Mapping Functions Between Different Slits,'' Int. J. of Math. Ed. in Sci. and Tech., V. 21, No. 4, pp. 595-598.

27. Szego, Gabor, ''Conformal Mapping of the Interior of an Ellipse onto a Circle (in Math. Notes),'' Am. Math. M., Vol. 57, No. 7. (Aug. - Sep., 1950), pp. 474-478, Jstor.   

28. Truesdell, C., ''Notes on the History of the General Equations of Hydrodynamics,'' Am. Math. M., Vol. 60, No. 7. (Aug. - Sep., 1953), pp. 445-458, Jstor.   

29. Vincenty, T., (1987), ''Conformal transformations between dissimilar plane coordinate systems,'' Surveying and Mapping, V. 47, pp. 271-274.

30. Weaver, Warren, (1932), ''Conformal Representation, with Applications to Problems in Applied Math.,'' Am. Math. M., V. XXXIX, pp. 448-473.

Section 10.5	Steady State Temperatures

Section 10.6	Two-Dimensional Electrostatics

Section 10.7	Two-Dimensional Fluid Flow

Section 10.8	The Joukowski Airfoil

Section 10.9	The Schwarz-Christoffel Transformation

Section 10.10	Image of a Fluid Flow

Section 10.11	Sources and Sinks

GO TO  THE  Table of Contents