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GEORGE POLYA December 13, 1887-September 7, 1985 BY R. P. BOAS GEORGE (GYORGY) P6LYA macle many significant con- tributions to mathematics and at the same time rather unusually for a distinguished research mathematician was an effective advocate of improver! methods for teaching mathematics. His research publications extend from 1912 to 1976; his publications about teaching began in 1919 and con- tinuec! throughout his life. For several clecacles, he was stead- ily initiating new topics and making decisive contributions to more establisher! ones. Although his main mathematical in- terest was in analysis, at the peak of his career he was con- tributing not only to real and complex analysis, but also to probability, combinatorics, occasionally to algebra and num- ber theory, and to the theory of proportional representation and voting. His work typically combined great power and great lucidity of exposition. Although much of his work was so technical that it can be fully appreciated only by specialists, a substantial number of his theorems can be stated simply enough to be appreciated by anyone who has a moderate knowledge of mathematics. As a whole, Polya's work is notable for its fruitfulness. All his major contributions have been elaborated on by other mathematicians and have become the foundations of impor- tant branches of mathematics. In aciclition to his more sub- 339

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340 BIOGRAPHICAL MEMOIRS stantial contributions, Polya macle many brief communica- tions, ranging from the many problems that he proposed to brief remarks a considerable number of which became the germs of substantial theories in the hands of other mathe- maticians. A student who needs a topic for research could do worse than look through Polya's short papers. Polya's papers were published in four volumes: the first two devoted to complex analysis, the third to other branches of analysis including mathematical physics, the fourth to probability, combinatorics, and teaching and learning in mathematics.] ORIGINS AND CAREER Polya was born in Budapest on December 13, 1887, and died in Palo Alto, California, September 7, 1985. In 1918 he married StelIa Vera Weber, who survived him; they had no children. He received his doctorate in mathematics first having studied law, language, and literature from the Uni- versity of Budapest in 1912. After two years at Gottingen and a short period in Paris, he accepted a position as Privatdocent at the Eidgenossische Technische Hochschule (Swiss Federal Institute of Technology) in Zurich in 1914 and rose to full professor there in 1928. In 1924 he was the first Interna- tional Rockefeller Fellow and spent the year in England. In 1933 he was again a Rockefeller Fellow at Princeton. He em- igrated to the United States in 1940, held a position at Brown for two years, spent a short time at Smith College, and in 1952 became a professor at Stanford. He retired in 1954 but continued to teach until 1978. He was elected to the National Academy of Sciences in 1976. ~ George Polya, Collected Papers, 4 vols. (Cambridge, Massachusetts, and London, England: MIT Press, 1974-1984).

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GE O RGE P6 LYA PROBABILITY 341 Polya's first paper was in this field, and during his career he contributed perhaps thirty papers to various problems in probability theory. These papers contain many results that have now become textbook material, or even exercises, so that every student of probability encounters Polya's work. One of Polya's best known results is typical of his style, being unexpected but simple enough to prove once it was thought of. The Fourier transform roe f(t) = | emit (ax) J _oo of a one-dimensional probability measure is known as the characteristic function. Polya discovered (191S,3; 1923,2) that a sufficient condition for a real-valued function to be a characteristic function is that f(0) = I, f(oo) = 0, f(t) = f(-t), and f is convex, t > 0. This is the only useful general test for characteristic functions, even though the most famous char- acteristic function, exp(-t2), is not covered by it. In 192 I, Polya initiated the study of random walks (which he named), proving the striking and completely unintuitive theorem that a randomly moving point returns to its initial position with probability ~ in one or two dimensions, but not in three or more dimensions (192l,4). His other contribu- tions to the subject are less easily explained informally but, like those just mentioned, have served as starting points for extensive theories. These include limit laws (Polya also named the central limit theorem), the continuity theorem for moments, stable distributions, the theory of contagion and exchangeable sequences of random variables, and the roots of random polynomials.

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342 BIOGRAPHICAL MEMOIRS COMPLEX ANALYSIS Complex analysis is the study of analytic functions in two dimensionsthe field! to which P6lya made his most numer- ous contributions. As every student of the subject learns at an early stage, a function f that is analytic at a point, say O (for the sake of simplicity), is represented by a convergent power series 00 ~ anzn; n=0 and conversely such a series, if convergent, represents an analytic function. In principle the sequence {an) of coeffi- cients contains all the properties of the function. The prob- lem is to make the sequence surrender the desirer! informa- tion. The most attractive results are those that connect a simple property of the coefficients with a simple property of the function. P6lya made many contributions to this subject. He proved that the circle of convergence of a power series is "usually" a natural boundary for the functionthat is, a curve past which the sum of the series cannot be continuer! analytically (1916,1; 1929,1). It is, in fact, always possible to change the signs of the coefficients in such a way that the new series cannot be contained outside the original circle of conver- gence. According to Fabry's famous gap theorem, the circle of convergence of a power series is a natural boundary if the density of zero coefficients is I. P6lya proved that no weaker condition will suffice for the same conclusion. He also ex- tenclect this theorem in several ways and found analogs of Fabry's theorem for DirichIet series, which have a more com- plex theory.

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GEORGE P6LYA 343 In 1929, P6lya systematized his methods for dealing with problems about power series (1929,11. This very influential paper deals with densities of sequences of numbers, with con- vex sets anc} with entire functions of exponential type that is, with functions analytic in the whole complex plane whose absolute values grow no faster than a constant multiple of some exponential function eAlZl Functions of this kind have proved widely applicable in physics, communication theory, and in other branches of mathematics. The central theorem is P6lya's representation of a function f as a contour integral that resembles a Laplace transform, f(Z) = 2 if Few, a representation important in contexts far beyond! those that P6lya originally envisioned. Another topic that interested P6lya was how the general character of a function is revealed by the behavior of the function on a set of isolated points. The whole subject orig- inatect with P6lya's discovery (1915, 2) that 2Z is the "smallest" entire function, not a polynomial, that has integral values at the positive integers. There are many generalizations, on which research continued at least into the 1970s, and the theory is still far from complete. P6lya also contributed to many other topics in complex analysis, including the theory of conformal mapping and its extensions to three dimen- s~ons. One of P6lya's favorite topics was the connections between properties of an entire function ant! the set of zeros of poly- nomials that approximate that function. He and I. Schur introduced two classes (now known as P6lya-Schur or I~aguerre-P6lya functions) that are limits of polynomials that have either only real zeros or only real positive zeros. There are now many more applications, both in pure and applied

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344 BIOGRAPHICAL MEMOIRS mathematics, than Polya himself envisaged, including, for ex- ample, the inversion theory of convolution transforms ant! the theory of interpolation by spline functions. Another series of papers starting in 1927 was devotee! to ,. . . . zeros ot trigonometric ~ntegra s, Jf~tleiz~dt. Polya was much interested in the Riemann hypothesis about the zeros of the zeta function. His work on trigonometric integrals was inspires! by the fact that a sufficiently strong theorem about their zeros wouIc! establish the hypothesis. That this hope has, so far, prover! illusory, cloes not diminish the importance of Polya's results in both mathematics anc! physics. Maya devoted a great deal ot attention to the question ot how the behavior in the large of an analytic or meromorphic function affects the distribution of the zeros of the derivatives of the function. One of the simplest results (simplest to state, that is) is that when a function is meromorphic in the whole plane (has no singular points except for poles), the zeros of its successive derivatives become concentrated near the poly- gon whose points are equidistant from the two nearest poles. The situation for entire functions is much more complex, ant! Polya conjectured a number of theorems that are only now becoming possible to prove. in, . . . . . ~ REAL ANALYSIS, APPROXIMATION THEORY, NUMERICAL ANALYSIS Polya's most important contributions to this area are con- tainec! in the book on inequalities he wrote in collaboration with Hardy and LittIewooct (1934,21. This was the first sys- tematic study of the inequalities user! by all working analysts in their research and has never been fully superseded by any of the more recent books on the subject.

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GEORGE POLYA 345 Peano's space-fi~ling curve passes through every point of a plane area but passes through some points four times. In 1913, Polya produced a construction for a similar curve that has, at most, triple points, the smallest possible number. In keeping with P6lya's principle of (1rawing pictures whenever possible, the construction is quite geometrical ~9~3,~). Polya cievoted two papers more than fifty years apart to Graeffe's method] for numerical solution of polynomial equa- tions (1914,2; 196S,I). Although this method is useful for functions other than polynomials as well, it was not highly regarclec! originally because of the large amount of compu- tation it requires. With the availability of modern high-speed computers, however, the method is becoming more useful. His pioneering investigation of the theory of numerical in- tegration (1933,1) is still important today in numerical anal- ys~s. COMBINATORICS Combinatorics aciciresses questions about the number of ways there are to clo something that is too complicated to be analyzed intuitively. Polya's chief discovery was the enumer- ation of the isomers of a chemical compound, that is, the chemical compounds with different properties but the same numbers of each of their constituent elements. The problem had baffled chemists. P6lya treated it abstractly as a problem in group theory and was able to obtain formulas that macle the solution of specific problems relatively routine. With the abstract theory in hand, Polya couIct solve many concrete problems in chemistry, logic, ant! graph theory. His ideas and methods have been still further developed by his successors. A related problem is the study of the symmetry of geo- metric figures, for example, tilings of the plane by tiles of particular shapes. Polya's paper (1937,2) came to the atten- tion of the artist M. C. Escher, who used it in constructing his famous pictures of interIockec} figures.

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346 BIOGRAPHICAL MEMOIRS The theory of symmetries also plays an important role in Polya's work in mathematical physics. MATHEMATICAL PHYSICS Physical problems in two or three dimensions usually cle- penc! in essential ways on the shape of the domain in which the problems are considered. For example, the shape of a drumbeat affects the sound of the drum; the electrostatic capacitance of an object depends on its shape. Except for very simple shapes, such as circles or spheres, the mathematical equations that describe the properties are too difficult to solve exactly; the solutions must be approximated in some way. Polya's contributions to mathematical physics consisted of de- veloping methods for such approximations. These metho(ls, like his work in other fields, were subsequently cievelopec! further by others. P6lya was interested in estimating quantities of physical interest connected with particular domains, as, for example, electrostatic capacitance, torsional rigidity, and the lowest vi- bration frequency. Usually one wants an estimate for some property of a domain in terms of another. The simplest prob- lem of this kind (and the oldest it goes back to antiquity) is the isoperimetric problem, in which the area inside a curve is compared with the perimeter, or the volume of a solid is compared with its surface area. Problems of this kind, con- sequently, go by the generic name of isoperimetric problems. One method to which Polya devoted a great deal of work, including the production of a widely read book (195l,1), is to replace a given domain by a more symmetric one with one property (say, the area inside a curve) the same, and for which the other property is more easily discussed. If we know that symmetrization increases or decreases the quantity in which we are interested, the result is an inequality for the other property.

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GEORGE P6LYA 347 One of the earliest successes of this technique was a simple proof of Rayleigh's conjecture that a circular membrane has the lowest vibration frequency (that is, the smallest eigen- value of the corresponding differential equation) among all membranes of a specified area. For different problems, dif- ferent kincis of symmetrization are needed. Many physical quantities that are determined as the so- lutions of extremal problems can be estimated by making appropriate changes of variable, a technique known as trans- plantation. Polya exploited this technique in a long paper in collaboration with M. Schiffer (1954,11. He also contributed several refinements to the standard! technique of approxi- mating solutions of partial differential equations by solving ifference equations ~ ~ 952, I; ~ 954,2~. TEACHING AND LEARNING MATHEMATICS Polya believed that one should learn mathematics by solv- ing problems. This led him to write, with G. Szego, Problems and Theorems in Analysis (1925,1 t2 vole., in German]; 1972,} tvol. I], and 1976,} tvol. 2] revised and enlarged English translation) in which topics are developer! through series of problems. Besicles their use for systematic instruction, these volumes are a convenient reference for special topics and methods. Polya thought a great deal about how people solve problems and how they can learn to do so more effectively. His first book on this subject (1945,2) was very popular and has been translates! into many languages. He wrote two acI- clitional books (1954,3; 1962,1) and many articles on the same general theme. Polya also stressed the importance of heuristics (essen- tially, intelligent guessing) in teaching mathematics and in mathematical research. In the preface to Problems and Theo- rems, for example, he and Szego remarked that since a straight line is determined by a point and a parallel geom-

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348 BIOGRAPHICAL MEMOIRS etry suggests, by analogy, ways of approaching problems that have nothing to do with geometry. One can hope both to generate new problems and to guess methods for solving them lay generalizing a well-understooc! problem, by inter- polating between two problems or by thinking of a parallel situation. One can see these principles at work in some of Polya's research, and many other mathematicians have found them helpful. Whether heuristics can really be successful on a large scale as a teaching technique has not yet been established. Some researchers in artificial intelligence have not found it elective for teaching mathematics. It is not clear, however, whether these results reflect more unfavorably on P61ya or artificial intelligence. It does seem clear that putting Polya's Pleas into practice on a large scale would entail major changes both in the mathematics curriculum and in the training of teachers of mathematics. P61ya also stressed geometric visualization of mathematics wherever possible, and "Draw a figure!" was one of his fa- vorite adages. ~ N P R E P A R ~ N G T H ~ S M E M O ~ R I have drawn on the introductions and notes in the Collected Papers and, to a large extent, on the more detailed memoir prepared by G. L. Alexanderson and L. H. Lange for the Bulletin of the London Mathematical Society, which I had the opportunity of seeing in manuscript and to which I also contrib- uted.

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GEORGE P6LYA SELECTED BIBLIOGRAPHY 349 1913 Uber eine Peanosche Kurve. Bull. Acad. Sci. Cracovie, A, 305-13. Sur un algorithme toujours convergent pour obtenir les polynomes de meilleure approximation de Tchebychef pour une fonction continue quelconque. C. R. Acad. Sci. (Paris), 1957:840-43. Uber Annaherung durch Polynome mit lauter reelen Wurzeln. Rend. Circ. Mat. Palermo, 36:279-95. Uber Annaherung durch Polynome deren samtliche Wurzeln in einen Winkelraum fallen. Nachr. Ges. Wiss. Gottingen, 1913:326-30. 1914 With G. Lindwart. Uber einen Zusammenhang zwischen der Kon- vergenz von Polynomfolgen und der Verteilung ihrer Wurzeln. Rend. Circ. Mat. Palermo, 37:297-304. Uber das Graeffesche Verfahren. Z. Mat. Phys., 63:275-90. With I. Schur. Uber zwei Arten von Faktorenfolgen in der Theo- rie der algebraischen Gleichungen. J. Reine Angew. Math., 144:89-113. Sur une question concernant les fonctions entieres. C. R. Acad. Sci. (Paris), 158:330-33. 1915 Algebraische Untersuchungen uber ganze Functionen vom Ge- schlechte Null und Eins. }. Reine Angew. Math., 145:224-49. .. Uber ganzwertige ganze Funktionen. Rend. Circ. Mat. Palermo, 40: 1-16. 1916 With A. Hurwitz. Zwei Beweise eines von Herrn Fatou vermuteten Satzes. Acta Math., 40:179-83. .. Uber den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grossten Gliede der zugehori- gen Taylorschen Reihe. Acta Math., 40:311- 19. Uber Potenzreihen mit ganzzahligen Koeffizienten. Math. Ann., 77:497-513.

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350 BIOGRAPHICAL MEMOIRS 1917 .. Uber geometrische Wahrscheinlichkeiten. S.-B. Akad. Wiss. Denksch. Philos. Hist. K1., 126:319-28. .. Uber die Potenzreihen, deren Konvergenzkreis naturliche Grenze ist. Acta Math., 41:99-118. 1918 Uber Potenzreihen mit endlich vielen verscheidenen Koeffizienten. Math. Ann., 78:286-93. Uber die Verteilung der quadratischen Reste und Nichtreste. Nachr. Ges. Wiss. Gottingen, 1918:21-29. Uber die Nullstellen gewisser ganzer Funktionen. Math. Z.,2:352- 83. 1919 .. Uber das Gauss'sche Fehlergesetz. Astronom. Nachr., 208: 186-91; 209:111. Proportionalwahl und Wahrscheinlichkeitsrechnung. Z. Gesamte Staatswiss., 74:297-322. 1920 Arithmetische Eigenschaften der Reihenentwicklungen rationaler Funktionen. J. Reine Angew. Math., 151: 1-31. .. Uber den zentralen Grenzwertsatz der Wahrscheinlichkeits- rechnung und das Momentenproblem. Math. Z., 8: 171-81. Uber ganze ganzwertige Funktionen. Nachr. Ges. Wiss. Gottingen, 1920: 1-10. 1921 Bestimmung einer ganzen Funktion endlichen Geschlechts durch viererlei Stellen. Mat. Tidsskr. B.: 16-21. Ein Mittelwertsatz fur Funktionen mehrerer Veranderlichen. Tohoku Math. }., 19: 1-3. Uber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz. Math. Ann., 84: 149-60. 1922 Uber die Nullstellen sukzessiver Derivierten. Math. Z., 12:36-60.

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GEORGE P6LYA 1923 351 Sur les series entieres a coefficients entiers. Proc. London Math. Soc., 21:22-38. Herleitung des Gauss'schen Fehlergesetzes aus einer Funktional- gleichung. Math. Z., 18:96-108. Bemerkungen uber unendliche Folgen und ganze Funktionen. Math. Ann., 88: 169-83. Uber die Existenz unendlich vieler singularer Punkte auf der Kon- vergenzgeraden gewisser Dirichletscher Reihen. S.-B. Preuss. Akad. Wiss. Gottingen Math. Phys. K1. Abh. Folge 3., 1923:45- 50. With F. Eggenberger. Uber die Statistik verketteter Vorgange. Z. Angew. Math. Mech., 3:279-89. On the zeros of an integral function represented by Fourier's in- tegral. Mess. Math., 52: 185 -88. 1924 IJber die Analogie der Krystallsymmetrie in der Ebene. Z. Kristall., 60:278-82. On the mean-value theorem corresponding to a given linear ho- mogeneous differential equation. Trans. Am. Math. Soc., 24:312-24. 1925 With G. Szego. Aufgaben und Lehrsatze aus derAnalys?s. 2 vols. Berlin: Springer-Verlag. 1926 On an integral function of an integral function. J. London Math. Soc., 1: 12-15. On the minimum modulus of integral functions of order less than unity. }. London Math. Soc., 1:78-86. 1927 With G. H. Hardy and A. E. Ingham. Theorems concerning mean values of analytic functions. Proc. R. Soc. A., 113:542-69. Uber trigonometrische Integrale mit nur reelen Nullstellen. J. Reine Angew. Math., 158:6-18. Uber die algebraisch-funktionentheoretischen Untersuchungen

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352 BIOGRAPHICAL MEMOIRS von J. L. W. V. Jensen. Kgl. Danske Vidensk. Selsk. Math.-Fys. Medd., 7~17~. Eine Verallgemeinerung des Fabryschen Luckensatzes. Nachr. Ges. Wiss. Gottingen, 1927:187-95. 1928 Uber gewisse notwendige Determinantenkriterien fur die Fortsetz- barkeit einer Potenzreihe. Math. Ann., 99:687-706. Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehr- fach zusammenhangende Gebiete. S.-B. Preuss. Akad. Wiss. Gottingen Math. Phys. K1. Abh. Folge 3., 1928:228-32, 280- 82; 1929:55-62. 1929 Untersuchungen uber Lucken und Singularitaten von Potenz- reihen. Math. Z., 29:549-640. 1931 With G. Szego. Uber den transfiniten Durchmesser (Kapazitats- konstante) von ebenen und raumlichen Punktmengen. J. Reine Angew. Math., 165 :4-49. 1932 With A. Bloch. On the roots of certain algebraic equations. Proc. London Math. Soc., 33:102-14. 1933 Uber die Konvergenz von Quadraturverfahren. Math. Z., 37:264- 86. Qualitatives uber Warmeausgleich. Z. Angew. Math. Mech., 13: 125-28. Untersuchungen uber Lucken und Singularitaten von Potenz- reihen. II. Ann. of Math. (2), 34:731-77. 1934 .. Uber die Potenzreihenentwicklung gewisser mehrdeutiger Funk- tionen. Comment. Math. Helv., 7:201-21. With G. H. Hardy and J. E. Littlewood. Inequalities. Cambridge: Cambridge University Press.

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GEORGE P6LYA 1936 353 Algebraische Berechnung der Anzahl der Isomeren einiger organ- ischer Verbindungen. Z. Kristall. (A), 93:315-43. 1937 With M. Plancherel. Fonctions entieres et integrates de Fourier multiples. Comment. Math. Helv., 9:224-48; 10:110-63. Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen und chemische Verbindungen. Acta Math., 68: 145-254. Uber die Realitat der Nullstellen fast alter Ableitungen gewisser ganzer Funktionen. Math. Ann., 114:622-34. 1938 Sur la promenade au hasard dans un reseau de rues. Actual. Sci. Ind., 734:25-44. 1942 On converse gap theorems. Trans. Am. Math. Soc., 52:65-71. With R. P. Boas. Influence of the signs of the derivatives of a func- tion on its analytic character. Duke Math. }., 9:406-24. 1943 On the zeros of the derivatives of a function and its analytic char- acter. Bull. Am. Math. Soc., 49:178-91. 1945 With G. Szego. Inequalities for the capacity of a condenser. Am. }. Math., 67:1-32. How To Solve It: A New Aspect of Mathematical Method. Princeton: Princeton University Press. 1948 Torsional rigidity, principal frequency, electrostatic capacity and symmetrization. Q. Appl. Math., 6:276-77. 1949 With H. Davenport. On the product of two power series. Can Math., 1: 1-5. ,, ].

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354 BIOGRAPHICAL MEMOIRS 1950 With A. Weinstein. On the torsional rigidity of multiply connected cross sections. Ann. Math., 52: 154-63. 1951 With G. Szego. Isoper?metric Inequalities in Mathematical Physics. Princeton: Princeton University Press. 1952 Sur une interpretation de la methode des differences finies qui pent fournir des bornes superieures ou inferieures. C. R. Acad. Sci. (Paris), 235: 1079-81. 1954 With M. Schiffer. Convexity of functionals by transplantation. l. Analyse Math., 3:245-345. Estimates for eigenvalues. In: Studies in Mathematics and Mechanics Presented to Richard von M?ses, New York: Academic Press, pp. 200-7. Mathematics and Plausible Reasoning. Vol. 1, Induction and Analogy in Mathematics. Vol. 2, Patterns of Plausible Inference. Princeton: Princeton University Press. 1956 With L. E. Payne and H. F. Weinberger. On the ratio of consecutive eigenvalues. I. Math. Phys., 35:289-98. 1958 With I. J. Schoenberg. Remarks on de la Vallee-Poussin means and convex conformal maps of the circle. Pacific I. Math., 8:295- 334. 1959 With M. Schiffer. Sur la representation conforme de l'exterieur d'une courbe fermee convene. C.R. Acad. Sci. (Paris), 248:2837-39.

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GEORGE P6LYA 1961 355 On the eigenvalues of vibrating membranes, In memoriam Her- mann Weyl. Proc. London Math. Soc., 11:419-33. 1962 Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving. 2 vols. New York: John Wiley & Sons. 1968 Graeffe's method for eigenvalues. Numer. Math., 11:315-19. 1972 With G. Szego. Problems and Theorems in Analysis, Vol. 1. New York, Heidelberg, Berlin: Springer-Verlag. Revised and enlarged En- glish language version of 1925,1. (See 1976,1, for Vol. 2~. 1974 Collected Papers. Vol. 1, Singularities of Analytic Functions. Vol. 2, Location of Zeros, ed. R. P. Boas. Cambridge: MIT Press. 1976 With G. Szego. Problems and Theorems in Analysis, vol. 2. New York, Heidelberg, Berlin: Springer-Verlag. Revised and enlarged En- glish language version of 1925,1. (See 197Y,1, for Vol. 1~. 1984 Collected Papers, Vol. 3, Analysis, eds. J. Hersch and G. C. Rota. Vol. 4, Probability, Combinatorics, Teaching and Learning Mathematics, ed. G. C. Rota. Cambridge: MIT Press.