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Cornelis Simon Meijer 1904-1974

Cornelis Simon Meijer (Pieterburen 17 August 1904 -- Glimmen 12 April 1974) was a professor in Groningen from 1946 till 1972.

[Boele L.J. Braaksma, Rijksuniversiteit Groningen 1975, 2002]

Cornelis Meijer was born into the family of Remt Meijer (Hornhuizen 9 March 1874 -- Pieterburen 10 April 1912) and Magdalena Frederika van Ham (Pieterburen 7 April 1879 -- Groningen 26 1945), who were married 23 May 1901.

Meijer studied mathematics at the university of Groningen from 1924 until 1929 under the direction of professors Van der Corput and Van der Waerden. In 1933 he presented his doctor’s thesis: Asymptotische Entwicklungen Besselscher, Hankelscher und verwandter Funktionen, Bestimmung von numerischen oberen Schranken für das Restglied mittels der Methode der Sattelpunkte, with prof. Van der Corput as promotor. He was professor of mathematics at the university of Groningen from 1946 until his retirement in 1972. He died on April 12, 1974 at the age of 69. He was married to Andrea Jacoba Kapteijn (1912-2005).

His scientific work consists of asymptopic expansions with error estimates and integral representations for special functions of mathematical physics, in particular Bessel and Whittaker functions, the generalizations of the Laplace transform which carry his name, and the ”Meijer $G$-function”.

His first papers [1-3] and his thesis [4] are concerned with asymptotic expansions of Bessel, Hankel and related functions. A special feature is the precise numerical bounds for the remainder term in these expansions. Earlier more restrictive results were given by Schläfli, Hankel, Weber, Watson and Van Veen. Meijer attacked these problems anew by means of the saddle point method of Debije. At that time most applications of this method did not involve numerical bounds for remainder terms with exception of the work of Van Veen. Meijer succeeded in obtaining precise numerical bounds valid for larger regions of the variables and parameters than previous writers by using a version of Lagrange’s theorem on the expansion of implicitly given function with a remainder term. His method runs as follows.

Suppose the behaviour of

$\displaystyle{I(\lambda)=\int_a^b e^{\lambda\phi(z)}\psi(z)dz}$

is to be investigated as $\lambda\to\infty$. Suppose $\psi$ and $\phi$ are analytic functions, $\phi'(a)=0$, $\phi''(a)\neq 0$, $\phi(a)-\phi(z)>0$ if $a<z\leq b$. Then one substitutes $\xi=\phi(a)-\phi(z)$, so that

$\displaystyle{I(\lambda)=e^{\lambda \phi(a)}\int_0^\beta e^{-\lambda \xi}\psi(z) \frac{dz}{d\xi}d\xi.}$

Now one may expand $\psi(z) \frac{dz}{d\xi}$ in power series of $\xi$ according to Lagrange. However, Meijer writes $\psi(z) \frac{dz}{d\xi}$ as a partial sum of this series plus a remainder term in the form of an integral which may be suitably estimated. Thus he obtains sharp bounds for the error term in the asymptotic expansion of $I(\lambda)$. His results on asymptotic expansions with error estimates of Bessel and Hankel functions were for a long time the most complete in this field and are of importance for numerical purposes. Recent results in this area have been given by Olver using asymptotic methods of differential equations.

In his thesis Meijer exploited suitable integral representations of Bessel and Hankel functions in order to obtain asymptotic expansions. Integral representations of special functions of mathematical physics was his main field of interest in subsequent years. During the period 1934 - 1941 Meijer and Erdélyi developed the theory of integral representations of Whittaker functions and their products (cf. H. Buchholz, The confluent hypergeometric functions, Springer Verlag, Berlin, 1969).

The most important tools in his investigations in this area are Barnes integrals for these functions which can be expressed in terms of generalized hypergeometric functions. In 1936 Meijer [13] defined the $G$-function which is the most general useful function of this kind. If $m$, $n$, $p$, $q$ are integers, $0\leq n\leq p$, $0\leq m\leq q$ then he defined

$\displaystyle G_{p,q}^{m,n} \left[ z \left| \begin{matrix} a_1,\ldots,a_p\\ b_1,\ldots,b_q \end{matrix} \right. \right] = \frac{1}{2\pi}\int_C \frac{\Pi_{h=1}^m \Gamma(b_h-s)\Pi_{j=1}^n\Gamma(1-a_j+s)}{\Pi_{h=m+1}^q \Gamma(1-b_h+s)\Pi_{j=n+1}^p\Gamma(a_j-s)}z^sds, $(1)

where $C$ is a contour in the complex $s$-plane from $\infty-i\alpha$ to $\infty+i\beta$ such that the poles of $\Pi_{h=1}^m \Gamma(b_h-s)$ are to the right of $C$ and those of $\Pi_{j=1}^n\Gamma(1-a_j+s)$ are to the left of $C$. In order that this last situation is possible for the $a_j$ and $b_h$ and the real numbers $\alpha$ and $\beta$ have to satisfy certain inequalities.

By means of residue calculus one shows that the $G$-function is a linear combination of generalized hypergeometric functions. In fact, this was Meijer’s original definition of the $G$-function. The $G$-function satisfies the generalized hypergeometric differential equation and all significant particular solutions of this equation may be expressed in terms of the $G$-function.

Barnes (Proc. London Math. Soc. (2), 59-116 (1907)) considered the asymptotic behaviour of a fundamental system of solutions of the generalized hypergeometric differential equation, consisting of the $G$-functions corresponding to $m=1$, $n=p$; $m=q$, $n=1$ ; and $m=q$, $n=0$. In his fundamental paper On the $G$-function of 1946 [34] Meijer deduced the asymptotic expansions of $G$-functions in the case $p < q$ by expressing them in terms of the Barnes fundamental system of solutions. To this end he wrote the integrand in (1) as the product of the integrand for $m = q$, $n = 0$ and

$\displaystyle{\pi^{m-q+n}\left\{ \Pi_{j=1}^n\left\{\sin\pi(a_j-s)\right\}^{-1}\Pi_{h=m+1}^q\sin\pi(b_h-s) \right\}}$

and expanded the last quotient as

$e^{\lambda\pi i s}\sum_{j=1}^nc_j\left\{\sin\pi(a_j-s)\right\}^{-1}+ \mbox{a linear combination of terms} \,\, e^{\lambda_k\pi i s}$.

Then one gets an expansion formula for the $G$-function in terms of the functions whose asymptotic behaviour has been given by Barnes. Similarly, he derived analytic continuations of the $G$-function in the case $p = q$.

The papers [5-28, 30-32, 35] contain integral relations for the functions of Bessel, Hankel, Legendre, Whittaker, Lommel and Struve and hypergeometric functions or products thereof which are derived from Barnes integrals. They are special cases of the following type integrals

$\displaystyle \int_L e^{-t}t^{\alpha-1}G(zt)dt$,(2)
$\displaystyle \int_L t^{\alpha-1}G(zt)G_1(\zeta t)dt$,(3)
$\displaystyle \int_0^1 G(zt)_2F_1(\alpha,\beta;\gamma;1-t)(1-t)^{\gamma-1}t^\sigma dt$,(4)
$\displaystyle \int_0^1 G(zt)_2F_1(\alpha,\beta;\gamma;1-t)(t-1)^{\gamma-1}t^\sigma dt$,(5)

where $G$ and $G_1$ are suitable $G$-functions and $L$ is the positive real axis or a Hankel contour from $\infty$ to $\infty$ around $t = 0$, or a path from $t = −\infty i$ to $t = -\infty i$. The evaluation of these integrals proceeds in general by substitution of the Barnes integral for $G(zt)$ and inverting the order of integration. The remaining integrals are either elementary as in (2) or can be calculated by expanding the hypergeometric function and using Gauss' formula for $_2F_1(1)$ as in the cases (4) and (5) or by using Mellin inversion theorem as in the second integral (3). The Mellin inversion theorem may be applied to the $G$-function in case the path in (1) may be replaced by a path from $−\infty i$ to $+\infty i$. In this last case one may also use the product formula for the Mellin transform.

In [33, 36, 37] Meijer deduced several expansion formulae for the $G$-function. The simplest cases of these are Taylor expansions of $G(z)$ around a point $z = w$. For example one has

$\displaystyle G_{p,q}^{m,n} \left[ \lambda w \left| \begin{matrix} a_1,\ldots,a_p\\ b_1,\ldots,b_q \end{matrix} \right. \right] = \sum_{r=0}^\infty \frac{(\lambda-1)^r}{r!} G_{p,q+1}^{m,n+1} \left[ w \left| \begin{matrix} 0,a_1,\ldots,a_p\\ b_1,\ldots,b_q,r \end{matrix} \right. \right] $(6)

if certain conditions are satisfied. Meijer replaces $\lambda$ by $\lambda t$, multiplies both sides by $e^{-t}t^{\alpha-1}$ or $t^{\alpha-1}$ times a suitable generalized hypergeometric function $_k\phi_j(\lambda)$, and integrates both sides over a suitable contour in the $t$-plane using the results concerning (2)-(5). In this way he deduces expansions of $G$-functions $G(\lambda w)$ in terms of a sum of generalized hypergeometric functions $_h\phi_j(\lambda)$ times $G$-functions $G(w)$.

Many known relations for special functions, for example generating functions for orthogonal polynomials and generalized hypergeometric functions, are special cases of these expansion theorems.

Meijer’s name is also connected with two generalizations of the Laplace and Fourier transforms and their inversion theorems. He found these results in 1940 and 1941 [29], [32]. The first integral fransform of Meijer is connected with the Hankel transform. The corresponding inversion theorem gives conditions for the validity of the pair of formulae

$\displaystyle{ f(s)=\sqrt{\frac{2}{\pi}}\int_0^\infty K_\nu(st)(st)^{\frac{1}{2}}F(t)dt }$
$\displaystyle{ F(t)=\frac{1}{i\sqrt{2\pi}}\int_{\beta-i\infty}^{\beta+i\infty} I_\nu(ts)(ts)^{\frac{1}{2}}f(s)ds }$

where $K_\nu$ is the Bessel function of the third kind and $I_\nu$ is a Bessel function with imaginary argument. The second integral transform of Meijer concerns the formulae

$\displaystyle{ f(s) = \int_0^\infty e^{\frac{1}{2}st}W_{k+\frac{1}{2},m}(st)(st)^{-k-\frac{1}{2}}F(t)dt }$
$\displaystyle{ F(t) = \frac{\Gamma(1-k+m)}{2\pi i \Gamma(1+2m)} \int_{\beta-i\infty}^{\beta+i\infty} e^{\frac{1}{2}ts}M_{k-\frac{1}{2},m}(ts)(ts)^{k-\frac{1}{2}}f(s)ds }$

where $W$ and $M$ are Whittaker functions. Important special cases of these formulae are already given in previous papers by Meijer. The proofs of these beautiful inversion theorems are related to Titchmarsh’ treatment of the inversion theorem of Hankel. Erdélyi showed that these results may be deduced from the corresponding theorems for Laplace transforms by means of fractional integrations.

The last years of his life Meijer worked on asymptotic expansions of $G$-functions with large parameters. Although he obtained many special results in this area, he delayed publication seeking a more complete and comprehensive treatment.

In his teaching as well as in his scientific work he was very scrupulous. His lectures were always well prepared and he devoted much of his time to students. He was very modest, much esteemed by his colleagues and students.

Meijer’s work has been continued by several people. In particular his students B.L.J. Braaksma, W. Knol and U.J. Knottnerus wrote their PhD thesis about G-functions. In 1956 J. Boersma wrote an essay On a function, which is a special case of Meijer's G-function which won a prize from the University of Groningen; his interest in the G-function was to last throughout his life. In later years several authors, for instance Y.L. Luke, J.L. Fields and J. Wimp, extended Meijer's work. In particular they considered expansions of Meijer's G-function in other G-functions. These contain, among others, expansions of hypergeometric functions in Jacobi, Laguerre and Hermite polynomials and expansions of the sine and cosine integrals in squares of Bessel functions. Surveys of Meijer’s work on the $G$-function may be found in the books by A. Erdélyi et al., Y.L. Luke, and F.W.J. Olver et al., see below. Finally it should be mentioned that the computer algebra programs Mathematica and Maple have incorporated the Meijer G-function.

More about the work of C.S. Meijer:

R. Beals and J. Szmigielski, Meijer G-functions: a gentle introduction. Notices AMS 60 (2013), 866-871.

J. Boersma, On a function, which is a special case of Meijer's G-function. Compositio Mathematica, 15 (1962), 34-63

A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions Vols I, II and III; Tables of integral transforms Vols I and II (Bateman Manuscript Project). McGraw-Hill Book Company 1953-1955.

Y.L. Luke, The Special Functions and Their Approximations Vols I and II. Academic Press 1969.

W. Miller Jr, Lie theory and Meijer's G-function. SIAM J. Math. Anal. 5 (1974), 309-318.

F.W.J Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Mathematical Functions. Cambridge University Press 2010.

Publications by C.S. Meijer:

[1] Asymtotische Entwicklungen von Besselschen, Hankelschen und ver- wandten Functionen. Proc. Kon. Akad. Wet. Amsterdam 35 (1932), 656-667, 852-866, 948-958, 1079-1090.
[2] Über die asymptotische Entwicklung von
$\displaystyle{\displaystyle \int_0^{\infty-i( \text{arg} w-\mu )} e^{\nu z-w\sin hz}dz \qquad \left( -\frac{\pi}{2}<\mu<\frac{\pi}{2} \right)}$
für grosse Werte von $|w|$ und $|ν|$. Proc. Kon. Akad. Wet. Amsterdam 35 (1932), 1170-1180, 1291-1303.
[3] Asymptotische Entwicklungen von Besselschen und Hankelschen Functionen für grosse Werte des Arguments und Ordung. Math. Annalen 108 (1933), 321-359.
[4] Asymptotische Entwicklungen Besselscher, Hankelscher und verwandter Functionen. Bestimmung von numerischen oberen Schranken für das Restgied mittels der Methode der Sattelpunkte. Thesis, Groningen, 1933.
[5] Über die Integraldarstellungen der Whittakerschen Funktion $W_{k,m}(z)$ und der Hankelschen und Besselschen Funktionen. Nieuw Archief Wisk. (2) 18 (1934), 35-37.
[6] Einige Integraldarstellungen für Whittakersche und Besselsche Funktionen. Proc. Kon. Akad. Wet. Amsterdam 37 (1934), 805-812.
[7] Noch einige Integraldarstellungen für die Whittakersche Funktion. Proc. Kon. Akad. Wet. Amsterdam 38 (1935), 528-535.
[8] Integraalvoorstellingen van producten van Hankelsche functies. Handelingen XXVe Ned. Natuur- en Geneesk. Congres, 1935.
[9] Integraldarstellungen aus der Theorie der Besselschen Funktionen. Proc. Lond. Math. Soc. (2) 40 (1936), 1-22.
[10] Einige Integraldarstellungen für Produkte von Whittakerschen Functionen. Quart. J. Math. Oxf. 6 (1935), 241-248.
[11] Integraldarstellungen für Lommelshe and Struvesche Funktionen. Proc. Kon. Akad. Wet. Amsterdam 38 (1935), 628-634, 744-749.
[12] Neue Integraldarstellungen aus der Theorie der Whittakerschen und Hankelschen Funktionen. Math. Annalen 112 (1936), 469-489.
[13] Über Wittakersche Bezw. Besselsche Funktionen und deren Produkte. Nieuw. Archief Wisk. (2) 18 (1936), 10-39.
[14] Einige Integraldarstellungen aus der Theorie der Besselschen und Whittakerschen Funktionen. Proc. Kon. Akad. Wet. Amsterdam 39 (1936), 394-403, 519-527.
[15] Über Produkte von Whittakerschen Funktionen. Proc. Kon. Akad. Wet. Amsterdam 40 (1937), 133-141, 259-262.
[16] Noch einige Integraldarstellungen für Produkte von Whittakerschen Funktionen. Proc. Kon. Akad. Wet. Amsterdam 40 (1937), 871-879.
[17] Über eine Integraldarstellung der Whittaker Funktionschen. Proc. Kon. Akad. Wet. Amsterdam 41 (1938), 42-44.
[18] Einige Inegraldarstellungen für die Lommenshe Funktion $S_{\mu,\nu} (z)$. Proc. Kon. Akad. Wet. Amsterdam 41 (1938), 151-154.
[19] Integraldarstellungen für Produkte von Legendreschen Funktionen. Nieuw Archief Wisk. (2) 19 (1938), 207-234.
[20] Note über das Produkt $M_{k,m}(z)M_{−k,m}(z)$. Proc. Kon. Akad. Wet. Amsterdam 41 (1938), 275-277.
[21] Beiträge zur Theorie der Whittakerschen Funktionen. Proc. Kon. Akad. Wet. Amsterdam 41 (1938), 624-633, 744-755, 879-888.
[22] Über die Kummersche Funktion $_1 F_1(a;b;z)$. Proc. Kon. Akad. Wet. Amsterdam 41 (1938), 1108-1114.
[23] Integraldarstellungen Whittakerscher Funktionen. Proc. Kon. Akad. Wet. Amsterdam 41 (1938), 1096-1107, 42 (1939), 141-146.
[24] Zur Theorie der hypergeometrischen Funktionen. Proc. Kon. Akad. Wet. Amsterdam 42 (1939), 355-369.
[25] Über Besselsche, Lommelsche und Whittakershe Funktionen.Proc. Kon. Akad. Wet. Amsterdam 42 (1939), 872-879, 938-947.
[26] Über Produkte von Legendreschen Funktionen. Proc. Kon. Akad. Wet. Amsterdam 42 (1939), 930-937.
[27] Integraldarstellungen für Struvesche und Besselsche Funktionen. Compositio Math. 6 (1939), 348-367.
[28] Über Besselsche, Struvesche und Lommelsche Funktionen. Proc. Kon. Akad. Wet. Amsterdam 43 (1940), 198-210, 366-378.
[29] Über eine Erweiterung der Laplace-Transformation. Proc. Kon. Akad. Wet. Amsterdam 43 (1940), 599-608, 702-711.
[30] Neue Integralstellungen für Whittakersche Funktionen. Proc. Kon. Akad. Wet. Amsterdam 44 (1941), 81-92, 186-194, 298-307, 442-451, 590-598.
[31] Integraldarstellungen für Whittakersche Funktionen und ihre Produkte. Proc. Kon. Akad. Wet. Amsterdam 44 (1941), 435-441, 599-605.
[32] Eine neue Erweiterung der Laplace-Transformation. Proc. Kon. Akad. Wet. Amsterdam 44 (1941), 727-737, 831-839.
[33] Multiplikationstheoreme für die Funktion $G^{m,n}_{p,q}(z)$. Proc. Kon. Akad. Wet. Amsterdam 44 (1941), 1062-1070.
[34] On the $G$-function. Proc. Kon. Akad. Wet. Amsterdam 49 (1946), 227-237, 344-356, 457-469, 632-641, 765-772, 936-943, 1063-1072, 1165-1175.
[35] Convergentie en divergentie. Inaugular lecture. P. Noordhoff, Groningen, 1946.
[36] Neue Integraldarstellungen für Besselsche Funktionen. Compositio Math. 8 (1951), 49-60.
[37] Expansion theorems for the $G$-function, Proc. Kon. Akad. Wet. Amsterdam Ser. A 55 (1952), 369-379, 483-487, 56 (1953), 43-49, 187-193, 349-357, 57 (1954), 77-82, 83-91, 273-279, 58 (1955), 243- 251, 309-314, 59 (1956), 70-82.
[38] Ontwikkelingen van gegenraliseerde hypergeometrische functies. Simon Stevin 31 (1956), 117-139.


P. C. S i k k e m a, Differential operators and differential equations of infinite order with constant coefficients, Researches in connection with integral functions of finite order. P. Noordhoff, Groningen, 1953.

U. J. K n o t t n e r u s, Approximation formulae for generalized hyper-geometric functions for large values of the parameters with applications to expansion theorems for the function $G^{m,n}_{p,q}(z)$. Wolters, Groningen, 1960.

B. L. J. B r a a k s m a, Asymptotic expansions and analytic continuations for a class of Barnes integrals. P. Noordhoff, Groningen, 1963.

W. K n o l, Generalizations of two relations in Bessel function theory. V. R. B. Groningen, 1970.

Publications of C.S. Meijer, see: MathScinet

Mathematics Genealogy Project for C.S. Meijer