Chapter 7 Complex Function Theory and Differential Equations

For a detailed historical account of this and many other subjects that we treat in this chapter, see the relevant chapters of Burkhardt (1908).

Many interesting questions in the purely real theory of the special functions, in particular many ingenious integrals, are therefore not described; for example, the paths introduced by Hilbert in his proof that π is transcendental in (Hilbert 1893). For that matter, many clever moves in the truly complex theory have also been left undiscussed once the first phase of the development is over, around 1900.

See Wagner (1894) for a history of Bessel Functions from Fourier to 1858. It stops short of discussion complex function theory. Schlömilch and Lipschitz proposed the name Bessel functions in the late 1850s; Heine countered with Fourier–Bessel functions, but this never caught on. The notes and corrections in the second edition (1878–1881) of Heine’s Handbuch der Kugelfunctionen (1861) are very informative historically and are probably the source of many later historical passages, such as those by Watson (1922).

See Leibniz (1849–1863, 3, 75).

See Cannon and Dostrovsky (1981, 7 and 69) who find amongst other riches in these papers of Euler’s an integral representation for J _n (x) usually attributed to Poisson and Lommel, see (Sect. 7.2.3) below. This had been already pointed out by Watson (1922, 24) who added that, compared to Euler’s, “Poisson’s forms” of the integrals “are more elegant, and his study of them is more systematic”.

See Lützen (1987).

The paper built on an earlier one, written in 1818 on Kepler’s problem where Bessel re-obtained Lagrange’s expansion of r ∕ a in series. We cannot find that it supports Kline’s (1972, 710) claim that in 1818 Bessel showed that J ₀(x) has infinitely many real zeroes.

In the special case h = 0 Bessel’s integral is commonly named after Parseval, for it had been obtained in 1806 by the latter in the memoir (Parseval 1806b) that we have considered in Chap.​ 2.

The corresponding, more familiar equation in orthogonal coordinates \(\frac{{\partial }^{2}V } {d{x}^{2}} + \frac{{\partial }^{2}V } {d{y}^{2}} + \frac{{\partial }^{2}V } {d{z}^{2}} = 0\) was published by Laplace in (Laplace 1789, 278).

The functions Q _n (z) that Heine called spherical functions of the second kind are commonly called Legendre functions of the second kind. Heine (1861, 41) pointed out that they were first introduced by Gauss in 1816 under the form of hypergeometric series, see Gauss, Werke 3, 188–189.

See the 2nd ed. of Heine’s Handbuch, vol. 1, 141–143. See also (F. Neumann 1878).

They are polynomials in negative powers of z. For an account of their properties, see Watson (1922, 271–280).

For a detailed, technical account, rich in historical references, see (Watson 1922, Chap. III).

An integral formula substantially equivalent to it was given in 1812 by Laplace (1812a, 135).

More information on Fuchs’s work will be found in Ince (1926), and Gray (1984a) and Gray (2000a).

Letter to Casorati, quoted in (Neuenschwander 1978b, 46).

Abel derived differential equations, notably Legendre’s, for functions defined by definite integrals in his (1827).

See Schwarz (1872b, 218).

Schwarz now assumed that the coefficients of the corresponding hypergeometric function F(α, β, γ) are real.

Schwarz also dealt with the special cases where λ, μ, ν may be integers.

An excerpt of the letter Hermite sent to Fuchs was added by Schlesinger to the reprint of (Fuchs 1877) in vol. 2 of his Ges. Math. Werke. See Fuchs (1877, 113).

See the lengthy extract published as (Fuchs 1877).

Hermite to Fuchs, November 27th 1876, added by Schlesinger to the reprint of (Fuchs 1877) in vol. 2 of his Ges. Math. Werke. See Fuchs (1877, 113–114).

This refers to (Koenigsberger 1874). The letter is quoted in (Koenigsberger 1919, 147).

Another linearly independent solution is \({x}^{1-\gamma }\phi (1 +\alpha -\gamma, 2-\gamma )\).

Beginning in 1877, Hermite published a series of notes in the Comptes rendus of the French Académie (Hermite 1877–1882) that he re-published separately in 1885 as a book (Hermite 1885).

See (Klein 1894a, 40–42) and (Bôcher 1894, 193–194).

See (Whittaker and Watson 1927, 337–354).

See the letter quoted in (Schlissel 1976–1977, 317n).

Stokes, in a footnote he added to the reprint of his (1864, 80) in his collected papers, observed that he had found the same formulae for the Bessel functions that Hankel gave, eleven years before Hankel, who then gave “the same explanation of the discontinuities of their constants”. He went on: “At that time the method of complex contour integration, developed by Riemann in his fundamental memoir on the hypergeometric series, a few months before the present paper, afforded the natural means of procedure”. He observed that his method worked even “when no expression of the form of a complex integral is available”.

See (Stieltjes 1886). For a more modern introduction to the subject, see (Erdélyi 1956).

Hille (1976, 284) incorrectly ascribed the discovery of this result to Hurwitz, but it was known earlier, to Poisson. He also neglected Klein’s contributions entirely and called the whole approach function-theoretic, thereby slighting the geometrical aspects.

Klein had made the subject part of his fifth Evanston Colloquium lecture, and mentioned some of van Vleck’s earlier work there, see Klein (1894b, 39).

All these results were also obtained in (Schafheitlin 1908).

For the correspondence between Pincherle and Poincaré see Dugac (1989, 210–217).

This letter is in the first of 55 manuscript volumes by Pincherle kept in the library of the Mathematical Department of the Bologna University. See Bottazzini and Francesconi (1989), and Dugac (1989, 215–217).

This section follows (Gray 1994).

We pass over here without discussion Poincaré’s introduction for the first time of the idea that the heat equation can be studied as an eigenvalue and eigenvector problem: the v _i are eigenvectors and Poincaré was the first to show that there are infinitely many eigenvalues and to study the spectrum.

In fact, (Zaremba 1899) shows that Poincaré’s “solution” to the problem was indeed a solution.

Volterra’s first contribution to Riemann’s approach to function theory was his (1882), a paper that he wrote when he was still a student. Schwarz gave this paper a very detailed review (JFM 15.0358.02) in which he pointed to some incorrect statements and proofs.

(Jacobi 1830) is worth mentioning in this connection. After an early reference to Cauchy’s theory of residues, there Jacobi introduced multidimensional residues (without naming them this way) as the coefficients of the powers of degree − 1 in the expansions of the roots of two (resp. three) equations in two (resp. three variables) power series, and re-obtained Lagrange series as a particular case.

For an account in modern terms of Volterra’s work, including his (1890, 1913, 1917) and some related problems, see Vesentini (1992).

We will see in (Sect. 8.​6) how Arzelà’s work connected to Montel’s concept of normal family of functions.

See Rüdenberg and Zassenhaus (1973, 119) for the letter from Minkowski to Hilbert, 5 January 1900.

An extended version of it was later published as (Hilbert 1904).

The edition of their correspondence (Gauss 1860–1865) runs to six volumes.

By the standards of his day, but see Gray and Micallef (to appear).

A surface is said to be developable if there are curves on it along which the tangent spaces have a line in common.

The only minimal surface that is a surface of revolution is the catenoid.

Weingarten (1863) obtained similar results.

We omit his complicated definitions, see Gray and Micallef (to appear).

See Enneper (1864, 108). For a picture of it and an instructive commentary, see Fischer (1986).

Weierstrass (1866a) and two papers (Weierstrass 1866b) and 1867a) only published for the first time in his Werke, 3.

Weierstrass omitted the trivial case \(f^{\prime} = ig^{\prime},h^{\prime} = 0\), which yields the plane as a minimal surface.

Schwarz made such models all his life, as numerous remarks in his papers attest. He brought them with him and proudly presented them to his French colleagues when he visited Paris in 1883. See also Hamel’s obituary of him (Hamel 1923).

Morera was born in Novara in 1856 and studied mathematics at the University of Turin, where he was a pupil of Siacci’s. From there he went to Pisa, then Pavia, then Berlin, where he met Kronecker, and Leipzig, where he had discussions with Klein and Mayer. Most of his work was on applied topics, including the theory of harmonic functions and dynamics. He became a professor of rational mechanics at Genoa in 1885, and in 1900 he moved to the Faculty of Sciences at the University of Turin, where he held various positions culminating in the Presidency of the Faculty. Shortly before his death in 1910 he was elected a Corresponding Member of the Mathematical Society of Kharkov (see Maggi 1910 and Somigliana 1910.)

In response to Morera’s paper, and by resorting to his theory of functions of lines Volterra (1887c, 348–350) proved a three-dimensional extension of Morera’s theorem, and claimed that it could be easily extended to n-dimensional spaces. Indeed, if the integral \(\int f(z_{1},\ldots,z_{n})dz_{1}\ldots dz_{n}\) of a continuous function on the boundary of any polycylinder lying in a domain B vanishes, then the function is analytic. (See Behnke and Thullen (1934, 41)).

See Darboux (1887–1896, 1, 490).

For his contributions, Douglas was awarded one of the first Fields medals in 1936; the other went to Ahlfors. The work of Radò, Douglas, and Courant is briefly described in (Gray and Micallef 2008) and in more detail in the forthcoming study (Gray and Micallef, to appear).

For a more detailed account of the history of minimal surfaces, see Gray and Micallef (to appear).

Among his pupils was Fujisawa, one of the first Japanese mathematicians to study in the West after the Meiji revolution and later the teacher of Takagi.

For an account of Christoffel’s life and work, see Butzer (1978) and Butzer (1981).

The map has a branch point of order two at each corner of the original square, thus converting the angle of π ∕ 2 there to an angle of π, as it should.

See, for example, Chap.​ 4, Classical Aerofoil Theory, in (Acheson 1990).

Unfortunately, most of the literature on Zhukovsky is in Russian, but we can mention (Grigorian 1981a,b, 1983) in addition to his (1929). However, Zhukovsky’s Collected Papers (1936–1937), carry articles in English, French, and German as well as Russian (the commentaries are in Russian).

One of the outstanding Italian mathematicians of the second half of the nineteenth century, Beltrami completed his studies at the secondary school (Liceo) in his native town Cremona. Then he entered the University of Pavia to get a mathematical degree, but he was soon expelled, accused of having taking part in riots against the Director. Without any doctoral degree, he continued his studies privatim under the direction of Brioschi and Cremona. Beltrami began his academic career in 1862, when he was named professor of Algebra and analytic Geometry at the University of Bologna. In 1864 he moved to Pisa, Pavia and eventually to Rome. Like Betti, Brioschi, Cremona, and many other mathematicians of his generation, Beltrami was named Senator of the Italian Kingdom, and after Brioschi’s death he succeeded him to the Presidency of the Accademia dei Lincei in 1898. In addition to his celebrated essay of interpretation of non-Euclidean geometry, Beltrami made substantial contributions to differential geometry and mathematical physics. For an account of his life and work, see Loria (1901).

Puiseux’s inequality can also be derived that way; the result is stated incorrectly in (Weinstein 1942). For an account employing the Weierstrass ℘-function, see Whittaker (1904, 104–106).

For an account of the circumstances surrounding the prize, which several mathematicians seemed keen for her to win, see (Cooke 1984, 110–115). We have also followed Cooke’s account of Kovalevskaya’s work (Cooke 1984, 151–159).

See also Zhukovsky (1891).

See also Diacu and Holmes (1996) and Andersen (1994).

See the long letter he sent on August 7, 1885 to Mittag-Leffler in (Mittag-Leffler 1912, 37–46).

See Kovalevskaya (1875) and Poincaré (1879). Later reproved and extended in (Koenigsberger 1894).

For a more detailed account, see Cooke (1984, 30–34).

Cooke (1984, 33) supplies a physically plausible initial condition with the same effect: \(u(x, 0) = {(1 + {x}^{2})}^{-1}\).

Not to say great initial embarrassment, see Barrow–Green (1997).

Anders Lindstedt was a Swedish astronomer graduated from Lund in 1877 and worked for a time in Dorpat (now Tartu, Estonia, but then a Russian province). In 1886 he moved to the Royal Technological Institute in Stockholm.

The matter proved understandably controversial, and a number of papers were exchanged before it was resolved.

Bruns (1887), see Whittaker (1904, 4th ed. 1937, Chap. 14).

It was shown in (Xia 1992) that this can happen in the five-body problem.

See Sundman (1912) and Birkhoff (1927, 260). We recommend the short account (Saari 1990, 105–119), and for information about Sundman, (Barrow–Green 2010).

It is usually claimed that their rate of convergence is too slow to prevent them being of any use, but lately this claim has been contested. See the references in (Barrow–Green 2010).