The Schwarz Function and Its Applications

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VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS 17 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 17

H. A. Schwarz showed us how to extend the notion of re ection in The Schwarz Function and Its Applications Davis straight lines and circles to re ection in an arbitrary analytic arc. Notable applications were made to the symmetry principle and The Schwarz to problems of analytic continuation. Re ection, in the hands of Schwarz, is an antianalytic mapping. By taking its complex conjugate, we arrive at an analytic function that we have called here the Schwarz Function of the analytic arc. This function is worthy of study Function and Its in its own right and this essay presents such a study. In dealing with certain familiar topics, the use of the Schwarz Function lends a point of view, a clarity and elegance, and a degree of generality which might otherwise be missing. It opens up a line of inquiry which has yielded Applications numerous interesting things in complex variables; it illuminates some functional equations and a variety of iterations which interest the numerical analyst. Philip J. Davis The perceptive reader will certainly nd here some old wine in rela- belled bottles. But one of the principles of mathematical growth is that the relabelling process often suggests a new generation of problems. Means become ends; the medium rapidly becomes the message.

This book is not wholly self-contained. Readers will nd that they should be familiar with the elementary portions of linear algebra and of the theory of functions of a complex variable. M A PRESS / MAA AMS

4-color Process 241 pages spine: 1/2" finish size: 5.5" X 8.5" 50 lb stock 10.1090/car/017

THE SCHWARZ FUNCTION AND ITS APPLICATIONS

By PHILIP J. DAVIS

THE

CARUS MATHEMATICAL MONOGRAPHS

Published by THE MATHEMATICAL ASSOCIATION OF AMERICA

Committee on Publications E. F. BECKENBACH, Chairman Subcommittee on Cams Monographs D. T. FINKBEINER II, Chairman R. P. BOAS D. E. CHRISTIE HE CARUS MATHEMATICAL MONOGRAPHS are an Texpression of the desire of Mrs. Mary Hegeler Carus, and of her son, Dr. Edward H. Carus, to contribute to the dissemination of mathematical knowledge by making accessible at nominal cost a series of expository presentations of the best thoughts and keenest researches in pure and applied mathematics. The publication of the first four of these monographs was made possible by a notable gift to the Mathematical Association of America by Mrs. Carus as sole trustee of the Edward C. Hegeler Trust Fund. The sales from these have resulted in the Carus Monograph Fund, and the Mathematical Association has used this as a revolving book fund to publish the succeeding monographs. The expositions of mathematical subjects which the monographs contain are set forth in a manner comprehensible not only to teachers and students specializing in mathematics, but also to scientific workers in other fields, and especially to the wide circle of thoughtful people who, having a moderate acquaintance with elementary mathematics, wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises. The scope of this series includes also historical and biographical monographs.

The following monographs have been published: No. 1. Calculus of Variations, by G. A. BLISS No. 2. Analytic Functions of a Complex Variable, by D. R. CURTISS No. 3. Mathematical Statistics, by H. L. RIETZ No. 4. Projective Geometry, by J. W. YOUNG No. 5. A History of Mathematics in America before 1900, by D. E. SMITH and JEKUTHIEL GINSBURG (out of print) No. 6. Fourier Series and Orthogonal Polynomials, by DUNHAM JACKSON No. 7. Vectors and Matrices, by C. C. MACDUFFEE No. 8. Rings and Ideals, by Ν. H. McCOY No. 9. The Theory of Algebraic Numbers, by HARRY POLLARD No. 10. The Arithmetic Theory of Quadratic Forms, by B. W. JONES No. 11. Irrational Numbera, by IVAN NIVEN No. 12. Statistical Independence in Probability, Analysis and Num- ber Theory, by MARK KAC No. 13. A Primer of Real Functions, by RALPH P. BOAS, JR. No. 14. Combinatorial Mathematics, by HERBERT JOHN RYSER No. 15. Noncommutative Rings, by I. N. HERSTEIN No. 16. Dedekind Sums, by HANS RADEMACHER and EMIL GROSSWALD No. 17. The Schwarz Function and its Applications, by PHILIP J. DAVIS The Carus Mathematical Monographs

NUMBER SEVENTEEN

THE SCHWARZ FUNCTION

AND ITS APPLICATIONS

By PHILIP J. DAVIS Professor of Applied Mathematics Brown University

Library of Congress Catalog Card Number 74-77258

eISBN 978-1-61444-017-8 Paperback ISBN 978-0-88385-046-6 Hardcover (out of print) ISBN 978-0-88385-017-6

Printed in the United States of America

Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 To Stefan and Edy Bergman In Friendship

CONTENTS

CHAPTER PAGE 1. Prologue 1 2. Conjugate Coordinates in the Plane 5 3. Elementary Geometrical Facts 7 4. The Nine-Point Circle 13 5. The Schwarz Function for an Analytic Arc 19 β. Geometrical Interpretation of the Schwarz Function; Schwarzian Reflection 29 7. The Schwarz Function and Differential Geometry. . 41 8. Conformal Maps, Reflections, and their Algebra ... 49 9. What Figure is the Power of a Circle? .... 89 10. Properties in the Large of the Schwarz Function . . 103 11. Derivatives and Integrals 109 12. Application to Elementary Fluid Mechanics .... 135 13. The Schwarz Function and the Dirichlet Problem . . . 149 14. Schwarz Functions of Specified Type 153 15. Schwarz Functions and Iterations 173 16. Dictionary of Functional Relationships 207 17. Bibliographical and Supplementary Notes 209 18. Bibliography 219

INDEX 225 ix

ACKNOWLEDGEMENTS

I should like to acknowledge my pleasure at the en- couragement provided me by the Committee on Carus Monographs. I am indebted to the following individuals: Eleanor M. Addison, Katrina Avery, Maylun Buck-Lew, Ezoura Fonseca, Frances Gajdowski, Carol Kemmler, Carol Salvatore, Dr. Henry O. Pollak, Professor J. H. Ahlberg, Professor Martin Braun, Professor Daniel Finkbeiner, Professor Peter Henrici, Professor Herbert Kolsky, Pro- fessor Philip Rabinowitz and Professor Frank Stenger. The motif on the title page is the da Vinci rendition of the human figure according to the proportions of Vitruvius. It has been subjected to a Mobius transformation leaving the bounding circle invariant. This was kindly provided to me by Professors R. Vitale of Brown and K. Long of the Rhode Island School of Design. The end piece was created by Jonathan Sachs. The manuscript was prepared partially under the sup- port of the National Science Foundation under Grant GP-35398 with Brown University.

PHILIP J. DAVIS Brown University xi

INDEX

analytic arc, 30 circle matrices, 211 analytic continuation, 56, 115, circle theorem, 144 124 circulation, 138 anti-analytic function, 112 clinant, 9, 43, 209 anti-analytic transformation, 59 complex dilation, 113 attractive point, 174 complex plane, 5 autonomous differential equa- dual, 209 tion, 95 complex velocity, 135 computation, 77, 217 Babbage equation, 87, 214 confocal ellipses, 70, 215 Baker, I. N., 212 conformal invariance, 81, 212 Bergman, S., 214, 215 conformal invariants, 82, 84 Bergman's kernel function, 156 conformal mapping, principle of, bicircular quartic, 26, 130 142 biharmonic function, 115 conformal symmetry, 52, 211 bilinear transformation, 36 conic, 19, 25 Birkhoff, Garrett, 212 Birkhoff, George D., 212 conjugate analytic function, 112 bisection problem, 81, 85, 204 conjugate coordinates, 1, 5 Blasius' formula, 147 conjugate function, 34 Bochner, S., 214 conjugate stress deviator, 118 conjugates (Hermitian), 55 canonical form, 186 conjugation, 35 Cartan matrix, 212 contraction mapping theorem, Cartwright, Mary, 1 174, 216 Carver, W. B., 209 counteridentity, 38 Cauchy problem, elliptic equa- curvature, 45 tions, 120, 214 curvilinear angle, 82 circle, 11, 24 Darboux type formula, 130 circle of curvature, 47 Davis, P. J., 210, 211, 213-215 226 INDEX de Cicco, J., 211 Green's theorem, 125, 214 derivatives (complex), 109 Gross, F., 213 diagonalization of function, 192 Grunsky, H., 214 differential equation, Beltrami Haber, S., 170, 216 type, 113 harmonic function, 115 differential geometry, 41-48, harmonic function, continuation 211, 212 of, 115, 214 directional derivative, 110 Henrici, P., 212, 214, 216 Dirichlet problem, 149 Henrici fixed-point theorem, 176, distance, 7 divergence, 138 216 doublet, 138 Hilbert space L»(B), 155 Douglas, Jesse, 210 horn angle, 82 elliptic differential operator, 114 identity function, 34 images, method of, 150, 215 epi trochoid, 160 inscribed circle, 17 escribed circle, 17 integrals, 124 Euler line, 15 Eves, Η. B., 209 invariant arc, 57, 65, 93, 185, 213, 217 exterior product, 15, 112 functional equation, 185 Feuerbach's theorem, 17 invariant point set, 173 fixed point, 174 inversion, 29 function, class CtC, 153 inversive geometry, 3, 211 class 7(9), 123 involutory function, 33 function-theoretic center involutory, real, 61 problem, 196 isochromatic lines, 117 functional, class £), 153 isoclinic lines, 117 functional composition, 35, 72, isotropic coordinates, 5 212 iterates, fractional, 193 functional equations, Abel type, , continuous, 193, 204, 59 217 Babbage type, 87, 214 iteration, 173-204, 217 conjugate Abel type, 137 groups, 193 conjugate Schroeder type, 137 invariant curve, 185 Jabotinsky, E., 212 dictionary of, 207 Jacobian, 112 automorphic type, 59 Jacobsthal, E., 211 functional iteration, 71, 187 functional square root, 86, 204 Kasner, E., 210-212 Keckic, J. D., 214 Garabedian, P., 214 Kuczma, M., 212, 214, 217 Grave, D. Α., 215 Kutta-Joukowski theorem, 146 INDEX 227

Lagrange-Burmann theorem, 79, packing, complete, 169 212 Pado approximants, 48 Lami curve, 25 permutable functions, 58, 66, 67 Laplacian, 114 Pfeiffer, G., 211, 212 Laurent series, 72 photoelasticity, 116 Lehto, 0., 212, 214 Plemelj-Stieltjes formula, 167, lemniscate, generalized, 27 215 Lewis, J. Α., 214 Poincare's problem, 82 Lewy, H., 212 point slope formula, 9, 42 limacon, 53, 159 Pollak, H. O., 210, 211, 213, 214 linear fractional transformation, Polya, G., 213 36 Polya's theorem, 103, 213 loxodrome, 94 power series, formal, 73, 87 mapping function, Riemann, quadrature, complete, 170 interior, 52 quasi-conformal mapping, 114, matrix, Schur, 74 212, 214 for bilinear transformation, 36 Melzak, Ζ. Α., 215 Rail, L. Β., 1, 209 Menger, K., 211 reflection Mergelyan, S., 215 analytic arc, 30, 59 minimal coordinates, 5 circle, 29 Mobius transformation, 10, 36, non-analytic arc, 49 58, 213 straight line, 29, 57 Mobius transformation repeated, 71 invariant curves, 93, 213 reproducing kernel function, 156 iteration, 217 residue, 73 Morley, F., and Morley, F. V., rose, 25 209 Schoenberg, I. J., 214 Moser, J., 217 Schroeder function, 193 Motzkin, T., 214 table of, 194 Motzkin-Schoenberg-Grunsky for bilinear function, 195 theorem, 128 bisection problem, 204 Muskhelishvili, Ν. I., 215 Schroeder-Koenigs equation, 193 Nemenyi, P. F., 215 Schur, I., 212 Newton's method, 179 Schwarz, Η. Α., 4, 210 nine-point circle, 1, 13 Schwarz function, 21, 24, 210 normal derivative, 110 algebraic arc, 20, 215 analytic arc, 19-24 orbit, 188 and analytic continuation, 56 orthogonality, 11, 69 bicircular quartic, 26, 130 ovals of Cassini, 27 circle, 24 228 INDEX

Schwarz function (Continued) single-circle method, 119 and complex dilation, 114 singularity of differential sys- as conformal map, 51 tem, 98 conic section, 20 Sloss, J. M., 212 and directional derivatives, source, 140 111 spiral of Archimedes, 81 and Dirichlet problem, 149 spiral of Bernoulli, 91, 141, 186, ellipse, 25, 132 213 and fluid mechanics, 135 stable transformation, 174, 201 functional equation, 33, 49, 53, Stein, P. R., 217 57, 61, 185, 189, 190 Steiner's porism, 63 hyperbola, 25 Stenger, F., 166 and iteration, 173, 178, 182 straight line, 7, 24 Lame curve, 25 stream lines, 137 loxodrome, 95 symmetry and mapping function, 52 with respect to arcs, 66 meromorphicity of, 130, 150, with respect to circles, 67 154-159 with respect to lines, 67 and orbits, 188 symplectic geometry, 211 and ordinary differential equa- tangential derivative, 110 tions, 95 tangential triangle, 16 properties in the large, 103 transformation, plane, 184 rationality of, 104 trioperational algebra, 35 rose, 25 two-circle method, 120 singularities of, 126, 164, 167, 215 Ulam, S., 217 spiral of Archimedes, 81 uniform flow, 138 spiral of Bernoulli, 91 straight line, 24 Vekua, I. N., 214 Schwarzian derivative, 46, 211 Vekua's theorem, 123 Schwarzian reflection, 4, 29, 31, velocity equipotentials, 137 46, 211 Virtanen, Κ. I., 212, 214 Schwerdtfeger, H., 209, 213, 214, vortex, line, 140, 141 217 Walsh, J. L., 215 self-conjugate, 21 Walsh's theorem, 153, 215 separable function, 96 Wesler, O., 215 shear difference method, 118 Wilson, L. T., 212 Siegel, C. L., 211, 217 Siegel's theorem, 197 Yaglom, I. B., 209 VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS 17 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 17

The Schwarz Function and Its Applications

4-color Process 241 pages spine: 1/2" finish size: 5.5" X 8.5" 50 lb stock