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Hypergeometric Equations and Modular Transformations

  • Jeremy Gray

Abstract

This chapter does three things. It gives a short account of the work of Euler, Gauss, Kummer, and Riemann on the hypergeometric equation, with some indication of its immediate antecedents and consequences. It therefore looks very briefly at some of the work of Gauss, Legendre, Abel and Jacobi on elliptic functions, in particular at their work on modular functions and modular transformations. It concludes with a description of the general theory of linear differential equations supplied by Cauchy and Weierstrass. There are many omissions, some of which are rectified elsewhere in the literature1. The sole aim of this chapter is to provide a setting for the work of Fuchs on linear ordinary differential equations, to be discussed in Chapter II. and for later work on modular functions, discussed in Chapter IV and V.

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Notes

  1. 1.
    The most thorough recent treatment of all these topics is Houzel’s essay in [Dieudonné, 1978, II]. Gauss’s work on elliptic functions and differential equations is treated in the essays by Schlesinger [Schlesinger 1909a,b] and in his Handbuch [Schlesinger 1898, Vol. II, 2]. Among many accounts of the mathematics, two which contain valuable historical remarks are Klein’s Vorlesungen über die hypergeometrische Function [1894] and, more recently, Hille’s Ordinary Differential Equations in the Complex Domain [1976].Google Scholar
  2. 2.
    An integral form of the solution was also to prove of interest to later mathematicians: \(f(x) = \int_0^1 {{u^{b - 1}}{{(1 - u)}^{c - b - 1}}{{(1 - xu)}^{ - a}}du} \) where the integral is taken as a function of the parameter. It is assumed that b > 0, c−b > 0, x < 1, so that the integral will converge. It is easy to see, by differentiating under the integral, that f(x) satisfies (1.1.1). To obtain f as a power series expand (1−xu)−a by the binomial theorem and replace each term in the resulting infinite series of integrals by the Eulerian Beta functions they represent. An Euler Beta function is defined by \(B\left( {b + r,c - b} \right) = \int_0^1 {u^{b + r} \left( {1 - u} \right)^{c - b - 1} du} \) it satisfies the function equation or recurrence relation \(B\left( {b + r,c - b} \right) = \int_0^1 {{u^{b + r}}{{\left( {1 - u} \right)}^{c - b - 1}}du} \) which yields precisely the relationship between successive terms of the hypergeometric series, and so f(x) is, up to a constant factor, represented by (1.1.2). This argument is given in Klein [1894, 11], see also [Whittaker and Watson, 1973, Ch. XII].Google Scholar
  3. 3.
    Biographical accounts of Gauss can be found in Sartorius von Waltershausen, Gauss zum Gedächtnis [1856], G. W. Dunnington [1955], and H. Wussing [1979]. Accounts of most aspects of his scientific work are contained in Materialien für eine wissenschaftlichen Biographie von Gauss, edited by Brendel, Klein, and Schlesinger 1911–1920 and mostly reprinted in [Gauss, Werke X.2, 1922–1933]. The best introductions in English are the article on Gauss in the Dictionary of Scientific Biography [May, 1972], which is weak mathematically, Dieudonné [ 1978], and W. K. Biihler [1981].Google Scholar
  4. 4.
    There are detailed discussions in Schlesinger [1898] and [Gauss, Werke, X.2, 1922–1933], [Krazer, 1909], [Geppert, 1927], and [Houzel in Dieudonne 1978 vol. II]. In particular Geppert supplies the interpretation of the fragments in [Gauss, Werke III, 1866, 361-490] on which this account is based.Google Scholar
  5. 5.
    Later Gauss calculated M(1, √2) to twenty decimal places, his usual level of detail, and found it to be 1.19814 02347 35592 20744 [Werke III, 364].Google Scholar
  6. 6.
    This is not contiguous in the sense of Arbogast [1791], which more or less means continuous.Google Scholar
  7. 7.
    The lemniscate has equation r2 = cos 2θ in polar coordinates, and arc-length \(\int_0^t {\frac{{bx}}{{\left( {1 - x^4 } \right)^{\frac{1}{2}} }}} \). Introduced by Jacob Bernoulli in 1694 it had been studied by Fagnano (1716, 1750) who established a formula for the duplication of arc, and by Euler (1752) who gave a formula for increasing the arc n times, n an integer. Euler considered the problem an intriguing one because it showed that the differential equation \(\frac{{dx}}{{\left( {1 - x^4 } \right)^{\frac{1}{2}} }} = \frac{{dy}}{{\left( {1 - y^4 } \right)^{\frac{1}{2}} }}\) had algebraic solutions: \(x^2 + y^2 = c^2 + 2xy\left( {1 - c^4 } \right)^{\frac{1}{2}} - c^2 x^2 y^2 \) The comparison with the equation \(\frac{{dx}}{{\left( {1 - x^2 } \right)^{\frac{1}{2}} }} = \frac{{dy}}{{\left( {1 - y^2 } \right)^{\frac{1}{2}} }}\)which leads to \(x^2 + y^2 = c^2 + 2xy\left( {1 - c^2 } \right)^{\frac{1}{2}} \)guided his researches. (It provides the algebraic duplication formula for sine and cosine.) Gauss observed that, whereas in the trigonometric case the formula for multiplication by n leads to an equation of degree n, for the lemniscatic case the formula is of degree n2, and invented double periodicity on March 19, 1797, see Diary entry #60, to cope with the extra roots. He was fond of this discovery, and hints dropped about it in his Disquisitiones Arithmeticae (§335) inspired Abel to make his own discovery of elliptic functions. An English translation of Gauss’s diary is available in [Gray, 1984b and corrigenda].Google Scholar
  8. 8.
    Gauss had discussed the multiple-valued function log x in a letter to Bessel the previous year [Werke II, 108] and also stated the residue theorem for integrals of complex functions around closed curves. From the discussion in Kline [1972, 632-642] it seems that Gauss was well ahead of the much younger Cauchy on this topic; see also Freudenthal [1971], and Bottazzini [1981, 133].Google Scholar
  9. 9.
    For a comparison of the work of Abel and Jacobi see [Krazer 1909] or [Houzel, Dieudonné, 1978, II]. It seems that Abel was ahead of Jacobi in discovering elliptic functions; there is no doubt he was the first to study the general problem of inverting integrals of all algebraic functions. Jacobi’s work was perhaps more influential because of his efforts as a teacher and an organizer of research, and also because Abel died in 1829 at the age of 26.Google Scholar
  10. 10.
    \(\frac{{\frac{{d\lambda }}{{dk}}\frac{{d^3 \lambda }}{{dk^3 }} - \frac{3}{2}\left( {\frac{{d^2 \lambda }}{{dk^2 }}} \right)^2 }}{{\left( {\frac{{d\lambda }}{{dk}}} \right)^2 }}\)following Cayley [1883], as the Schwarzian derivative of A with respect to k. Schwarz’s use of it is discussed in detail in Chapter III.Google Scholar
  11. 11.
    See [Edwards, 1977] for a discussion of Kummer’s number theory, and [Lampe 1892–3] and [Biermann 1973] for further biographical details about Kummer.Google Scholar
  12. 12.
    Kummer wrote this equation as \(2\frac{{d^3 z}}{{dxdx^2 }} - 3\left( {\frac{{d^2 z}}{{dzdx}}} \right)^2 = \left( {2\frac{{dp}}{{dz}} + p^2 - 4Q} \right)\left( {\frac{{dz}}{{dx}}} \right)^2 + \left( {2\frac{{dp}}{{dz}} + p^2 - 4q} \right)\) using the then customary notation for higher derivatives.Google Scholar
  13. 13.
    In section III Kummer studied the special cases when γ depends linearly on α and β and quadratic changes of variable produce other hypergeometric series. Typical of his results is his equation 53: \(F(\alpha ,\beta ,\alpha - \beta + 1,x) = {(1 - x)^{ - \alpha }}F(\frac{\alpha }{2},\frac{{\alpha - 2\beta + 1}}{2},\alpha - \beta + 1,\frac{{ - 4x}}{{{{\left( {1 - x} \right)}^2}}})\) His results were incomplete and were extended by Riemann [1857a, §5].Google Scholar
  14. 14.
    Neuenschwander [1979, 1981b] discusses the limited use Riemann had for the theory of analytic continuation, of which he was certainly aware, see e.g. [1857c, 88-89].Google Scholar
  15. 15.
    For a discussion of Riemann’s topological ideas and their implications for analysis see Pont [1974 Ch. II], Pont does not make as much as he should have done of the distinction between homotopy-and homology-theoretic ideas. A better discussion will be found in [Scholz, 1980, Chapter 2].Google Scholar
  16. 16.
    For a history of Cauchy’s Theorem see Brill and Noether [1892–93, Chapter II]. They also discuss Gauss’s independent discovery of it.Google Scholar
  17. 17.
    Riemann wrote (A), (B), and (C) for A, B, and C respectively.Google Scholar
  18. 18.
    Riemann wrote (b) for B′ and (c) for C′. I have introduced vector notation purely for brevity.Google Scholar
  19. 19.
    E. Scholz tells me there is evidence in the Riemann Nachlass to show that Riemann had read Puiseux. See also Neuenschwander, [1979, 7]Google Scholar

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© Springer Science+Business Media New York 1986

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  • Jeremy Gray

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