Appel W. Mathematics for Physics and Physicists
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Further reading
General books
• Methods of theoretical physics
P. M. Morse and H. Feshbach (McGraw-Hill, 1953)
A great classic, very complete.
• Mathematical methods for physicists
G. B. Arfken and H. J. Weber (Academic Press, 4th edition, 1995)
Many sub j ects are covered, in particular, differential equations, Bessel and L eg-
endre functions, other special functions, spherical harmonics.
Integration
• Lebesgue measure and integration, an introduction
F. Burk (Wiley Interscience, 1998)
A very clear book , with many examples guiding the reader step by step. Also a
very good historical introduction.
Complex analysis
• Function theory of one complex variable
R. E. Greene and S. G. Krantz (Wiley Interscience, 1997)
Very ped agogical.
Distributions and Fourier a nalysis
• Fourier analysis: an introduction
E. M. Steinz and R. Shakarchi (Princeton University Press, 2003)
A very clear introduction.
• A wavelet tour of signal processing
S. Mallat (Academic Press, 1999)
A nice and clear treatment of a difficult subject, with many illustrations and
examples of practical applications of wavelets.
618 Bibliography
Integral transforms, operator theory, Hilbert spaces
• Theory of linear operators in Hilbert space
N. I. Akhiezer and I. M. Glazman (2 volumes, Dover, 1993)
A very complete book, rather di fficult to read.
• Applied functional analysis
E. Zeidler (Springer-Verlag, 1995, Applied Mathematical Sciences 108)
This book is oriented toward physical applications, but remains quite dif ficult.
• Quantum mechanics in Hilbert space
E. Prugove
ˇ
cki (Academic Press, 1971, Pure and Applied Ma thematics 41)
Another very complete treatement, mathematically rigorous, using powerful
tools which go far beyond what we have seen in the present text.
Tensors, geometry
• Riemannian geometry
S. Gallot, D. Hulin and J. Lafontaine (Springer-Verlag, 1990)
• Leçon sur la théorie des spineurs
É. Cartan (Hermann, 1966)
An English version is published by Dover wi th the title The theory of spinors, 1981.
• Introduction to differentiable manifolds
L. Auslander and R. E. MacKenzie (Dover, 1977)
A clear introduction to d i fferential geometry.
• Modern geometry—Methods and applications
B. Dubrovin, S. Novikov and A. Fomenko (S pringer, 1991)
Three volumes: I. Geometry of surfaces, of transformation groups and fields; II. Geometry
and topology of varieties; III. Methods of homology theory. The first two volumes
are most likely to be of interest to physicists. The geometrical intuition is
emphasized throughout. The English edition published by Springer-Verlag is
much more expensive than the original Russian or French editions.
• Geometry, topology and physics
M. Nakahara (Adam Hilger, 1990)
A unique book that makes the link between theoretical physics and high-level
mathematics.
Groups
• Symmetries in quantum mechanics: from angular momentum to super-
symmetry
M. Chaichian and R. Hagedorn (Institute of Physics, 1998)
A very interesting book, with increasing di fficulty, giving a vast overview of sym-
metries, and explaining clearly the relation between symmetries and dynamics.
Bibliography 619
• Lie groups and algebra, with applications to physics, geometry and me-
chanics
D. H. Sattinger and O. L. Weaver (Spr inger-Verlag, 1986, Applied Math-
ematical Sciences 61)
Probabilities, statistics
• An introduction to probability theory
W. Feller (Wiley, 1968 (vol. I) and 1971 (vol. II))
The first volume, with little formalism, is a remarkable book directed toward
concrete applications, analyzed in great detail and depth.
• Stochastik
A. Engel (Ernst Klett Verlag, 2000)
This book has a fascinating approach to probability theory, full of applications
and wit h little formalism. Moreover, this is one of the rare books dealing with
probabilities and statistics simultaneously. (A French translation of the original
German edi tion is available: Les certitudes du hasard, ALÉAS Éditeur, Lyon, 1990).
References
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[4] Naum Ilit ch Akhiezer and Israël Markovitch Glazman. Theory of linear
operators in Hilbert sp ace. Dover, 1993. (two volumes bound as one).
[5] André Angot. Compléments de mathématiques à l’usage des ingénieurs de
l’électrotechnique et des télécommunications. Masson, 1982.
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[15] Henri Broch. Au cœur de l’extraordinaire. Collection Zététique, 2002.
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[16] Frank Burk. Lebesgue measure and integration. Wiley-Interscience, 1998.
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forms, Dover, 2006).
[18] Masud Chaichian and Rolf Hagedorn. Symmetries in quantum mechan-
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[19] Éric Charpentier, Annick Lesne, and Nikolaï Nikolski. L’héritage de
Kolmogorov en mathématiques. Belin, 2004.
[20] Claude Cohen-Tannoudji , Bernard Diu, and Frank Laloë. Quantum
Mechanics. Wiley–Interscience, 2001.
[21] Claude Cohen-Tanno udji , Jacques Dupont-Roc, and Gilbert Gryn-
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[22] Jean-François Colombeau. Elementary introduction to new generalized func-
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distributions. North Holland Mathematical Studies 84, 1984.
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[30] Boris Dubrovin, Sergeï Novikov, and Anatoli Fomenko. Modern geom-
etry — Methods and applications, volume 1. Springer, 1982. Old-fashioned
Russian notation with indices everywhere, but the text is very clear; the
first two volumes are especially recommended for physicists.
[31] Phil P. G. Dyke. An introduction to Laplace transforms and Fourier series.
Springer, 2001.
[32] Freeman J. Dyson. Divergences of perturbation theory in quantum
electrodynamics. Phys. Rev., 85, 631–632, 1952.
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2 volumes.
[34] Richard P. Feynman and A. R. Hibbs. Quantum mechanics and integrals.
McGraw-Hill, 1965.
[35] Richard P. Feynman, Robert B. Leighton, and Matthew Sands. The
Feynman lectures on physics, volume 2. Addison Wesley, 1977.
[36] Richard P. Feynman, Robert B. Leighton, and Matthew Sands. The
Feynman lectures on physics, volume 3. Addison Wesley, 1977.
[37] Claude Gasquet and Patrick Witomski. Fourier analysis and applications:
filtering, numerical computation, wavelets. Springer, 1998.
[38] Israël M. Gelfan d and G. E. Shilov. Generalized functions. Academic
Press, 1964.
[39] Josiah W. Gibbs. L etters to the editor. Nature, 200–201, December 1898.
[40] Josiah W. Gibbs. L etters to the editor. Nature, 606, April 1899.
[41] François Gieres. Mathematical surprises and Dirac’s formalism in quan-
tum mechanics. Rep. Prog. Phys, 63, 1893–1931, 2000.
[42] I. S. Gradstein and I. M. Ryshik. Summen-, Produkt- und Integral-Tafeln.
Deutscher Verlag der Wissenschaften, 1963.
[43] Robert E. Greene and Steven G. Krantz. Function theory of one c omplex
variable. Wiley Interscience, 1997. Very pedagogical.
[44] Étienne Guyon, Jean-Pierre Hulin, and Luc Petit. Physical hydrodynam-
ics. Oxford University Press, 2001.
[45] Paul Halmos. Naive set theory. Springer-Verlag, 1998. A very clear intro-
duction.
[46] Godfrey H. Hardy. A mathematician’s apology. Cambridge University
Press, 1992. Mathematics as seen by a profound and very appealing
mathemat ician. W ith an account of Ra manujan.
624 References
[47] Harry Hochstadt. The functions of mathematical physics. Dover, 1986.
[48] Bruce J. Hunt. Rigorous discipline: Oliver Heaviside versus the mathe-
maticians. In Peter Dear, editor, The literary structure of scientific argument,
chapter 3, 72–95. University of Pennsylvania Press, 1991.
[49] Claude Itzykson and Jean-Bernard Zuber. Quantum field theory. Mc-
Graw Hill, 1980.
[50] John D. Jackson. Classical electrodynamics. John Wiley and Sons, 1962.
[51] Winfried Just and Martin Weese. Discovering modern set theory, volume I.
American Mathematical Society, 1996.
[52] Jean-Pierre Kahane. Nécessité et pièges des définitions mathématiques, chapter
174. In Qu’est-ce que l’Univers ? Odile Jacob, 2001. (Université de tous
les savoirs).
[53] Jean-Pierre Kahane, Pierre Cartier, and Vladimir I. Arnold. Leçons de
mathématiques d’aujourd’hui. Ca ssini, 2000. Some of the lectures (in partic-
ular, the one by P. Cartier) are particularly relevant to physicists and give
examples of contemporary researches in pure and applied mathematics.
[54] Jean-Pierre Kahane and Pierre Gilles Lemarié-Rieusset. Séries de Fourier
et ondelettes. Cassini, 1998. In the first part of the book , there is a re-
markable description of Fourier series and their history, from the origin
to the present days. The s econd part deals with wavelet analysis and is
written by a specialist in this topic.
[55] J. I. Kapusta. Finite-temperature field theory. Cambridge University Press,
1989.
[56] Michael K.-H. Kiessling. The “Jeans swindle”. A true story — mathe-
matically speaking. Adv. Appl. Math., 31, 132–149, 2003.
[57] Lev Landau and Evgueni Lifchitz. The classical theory of fields.
Butterworth-Heinemann, 1995.
[58] Lev Landau and Evgueni Lifchitz. Quantum mechanics. Butterworth-
Heinemann, 1996.
[59] Serge Lang. Complex analysis. Springer-Verlag, 1993.
[60] Michel Le Bellac. Quantum and statistical field theory. Oxford University
Press, 1992.
[61] Jean-Claude Le Guillou a nd Jean Zinn-Justin, editors. Large-order be-
haviour of perturbation theory. Current physics sources and comments 7.
North-Holland, 1990.
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[62] André Lichnerowicz. É léments de calcul tensoriel. Librairie Armand Colin,
1958. With applications to classical dynamics.
[63] Eliott H. Lieb. The stability of matter. Rev. Mod. Phys., 48(4), 553–569,
1976.
[64] Eliott H. Lieb and Joel L. Lebowitz. The constitution of matter: ex-
istence of thermodynamical systems composed of electrons and nuclei.
Adv. Math., 9, 316–398, 1972.
[65] Stéphane Mallat. A wavelet tour of signal processing. Academic Press, 1999.
[66] Jean Meeus. Calculs astronomiques à l’usage des amateurs. Société as-
tronomique de France, 1986.
[67] Charles Misner, Kip Thorne, and John Archibald Wheeler. Gravi-
tation. Freeman, 1973. Tensor calculus is explained in detail, patiently,
on affi ne spaces as well as in Riemannian geometry. The mathematical
formalism is very well presented.
[68] Mikio Nakahara. Geometry, topology and physics. Adam Hilger, 1990.
[69] Sir Isaac Newton. Philosophiæ Naturalis Principia Mathematica (1687).
Prometheus Books, 1995.
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(sur 3). Gauthier-Villars, 1892–99. (Distributed by the Librairie Blanchard
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[71] Eduard Prugove
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plied mathematics 41. Academic Press, 1971.
[72] Jean-Pierre Ramis. Séries divergentes et théories asymptotiques. Preprint
Strasbourg, I.R.M.A., 1992–1993.
[73] Tanguy Rivoal. Irrationalité d’au moins un des neuf nombres ζ (5),
ζ(7),. . . , ζ(21), 2002.
[74] R. M. Robinson. On the decomposition of spheres. Fundamenta Mathe-
matica, 34, 246–266, 1947.
[75] François Rouvière. Petit guide de calcul différentiel à l’usage de la licence et
de l’agrégation. Cassini, 1999.
[76] Walter Rudin. Real and complex analysis. McGraw Hill, 1986. Rather
mathemat ical.
[77] D avid Ruelle. Chance and chaos. Princeton University Press, 1991.
[78] Peter Sarnak. Arithmetic quantum chaos. The Sch ur lectures (1992). In
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[80] Laurent Schwartz. Théorie des distribution. Hermann, second ed ition,
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