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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
dimensions—the 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 function—that 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
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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
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1914
With G. Lindwart. Uber einen Zusammenhang zwischen der Kon-
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Sur une question concernant les fonctions entieres. C. R. Acad. Sci.
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1915
Algebraische Untersuchungen uber ganze Functionen vom Ge-
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..
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1916
With A. Hurwitz. Zwei Beweise eines von Herrn Fatou vermuteten
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..
Uber den Zusammenhang zwischen dem Maximalbetrage einer
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350
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1917
..
Uber geometrische Wahrscheinlichkeiten. S.-B. Akad. Wiss.
Denksch. Philos. Hist. K1., 126:319-28.
..
Uber die Potenzreihen, deren Konvergenzkreis naturliche Grenze
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1918
Uber Potenzreihen mit endlich vielen verscheidenen Koeffizienten.
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Uber die Verteilung der quadratischen Reste und Nichtreste.
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..
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Proportionalwahl und Wahrscheinlichkeitsrechnung. Z. Gesamte
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1920
Arithmetische Eigenschaften der Reihenentwicklungen rationaler
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Uber den zentralen Grenzwertsatz der Wahrscheinlichkeits-
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1921
Bestimmung einer ganzen Funktion endlichen Geschlechts durch
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Ein Mittelwertsatz fur Funktionen mehrerer Veranderlichen.
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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.
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Herleitung des Gauss'schen Fehlergesetzes aus einer Funktional-
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Math. Ann., 88: 169-83.
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1924
IJber die Analogie der Krystallsymmetrie in der Ebene. Z. Kristall.,
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On the mean-value theorem corresponding to a given linear ho-
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1925
With G. Szego. Aufgaben und Lehrsatze aus derAnalys?s. 2 vols. Berlin:
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On an integral function of an integral function. J. London Math.
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On the minimum modulus of integral functions of order less than
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1927
With G. H. Hardy and A. E. Ingham. Theorems concerning mean
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von J. L. W. V. Jensen. Kgl. Danske Vidensk. Selsk. Math.-Fys.
Medd., 7~17~.
Eine Verallgemeinerung des Fabryschen Luckensatzes. Nachr. Ges.
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1928
Uber gewisse notwendige Determinantenkriterien fur die Fortsetz-
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Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehr-
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With G. Szego. Uber den transfiniten Durchmesser (Kapazitats-
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With A. Bloch. On the roots of certain algebraic equations. Proc.
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..
Uber die Potenzreihenentwicklung gewisser mehrdeutiger Funk-
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GEORGE P6LYA
1936
353
Algebraische Berechnung der Anzahl der Isomeren einiger organ-
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1937
With M. Plancherel. Fonctions entieres et integrates de Fourier
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Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen
und chemische Verbindungen. Acta Math., 68: 145-254.
Uber die Realitat der Nullstellen fast alter Ableitungen gewisser
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1938
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1942
On converse gap theorems. Trans. Am. Math. Soc., 52:65-71.
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1943
On the zeros of the derivatives of a function and its analytic char-
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1945
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How To Solve It: A New Aspect of Mathematical Method. Princeton:
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1948
Torsional rigidity, principal frequency, electrostatic capacity and
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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.
Representative terms from entire chapter:
biographical memoirs
