The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
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162
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A Treatise on the Theory of Bessel Functions
– Watson
- 1952
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110
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Generalized hypergeometric functions
– Slater
- 1966
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35
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Special functions: An introduction to the classical functions of mathematical physics
– Temme
- 1996
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28
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A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions with an account of the principal transcendental functions. Reprint of the 4-th
– Whittaker, Watson
- 1927
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16
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The asymptotic expansions of the incomplete gamma functions
– Temme
- 1979
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14
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On some inequalities for the gamma and psi functions
– Alzer
- 1997
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11
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A computational procedure for incomplete gamma functions
– Gautschi
- 1979
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9
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A stochastic approach to the Gamma function
– Gordon
- 1994
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9
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Some extensions of W. Gautschi’s inequalities for the gamma function
– Kershaw
- 1983
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9
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Numerical evaluation of special functions
– Lozier, Olver
- 1994
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9
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Handbuch der Theorie der Gammafunktion
– Nielsen
- 1906
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8
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Many server queueing processes with Poisson input and exponential service times
– Karlin, McGregor
- 1958
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7
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Nicholson-type integrals for products of Gegenbauer functions and related topics, Theory and Applications of Special Functions
– Durand
- 1975
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7
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Analytic computations in number theory
– Odlyzko
- 1994
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7
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An asymptotic representation for the Riemann zeta function on the critical line
– Paris
- 1994
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7
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Scientific computation on mathematical problems and conjectures
– Varga
- 1990
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6
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Some gamma function inequalities
– Alzer
- 1993
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6
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On some inequalities for the incomplete gamma function
– Alzer
- 1997
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6
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Uniform asymptotic smoothing of Stokes’s discontinuities
– Berry
- 1989
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6
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On some polynomials of
– Carlitz
- 1958
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6
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Computation of the incomplete gamma function ratios and their inverse
– DiDonato, Morris
- 1986
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6
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Complex zeros of the error function and of the complementary error function
– Fettis, Caslin, et al.
- 1973
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6
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Some inequalities involving (r!) 1=r
– Minc, Sathre
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6
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Hyperasymptotic solutions of second-order linear differential equations
– Daalhuis, B, et al.
- 1995
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5
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The Riemann-Siegel expansion for the zeta function: high orders and remainders
– Berry
- 1995
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5
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On gamma function inequalities
– Bustoz, Ismail
- 1986
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5
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On a Class of Incomplete Gamma Functions with Applications
– Chaudhry, Zubair
- 1994
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5
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Mathematical Problems of Radiative Equilibrium
– Hopf
- 1934
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5
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Inequalities for ultraspherical polynomials and the gamma function
– Lorch
- 1984
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4
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Some elementary inequalities relating to the gamma and incomplete gamma function
– Gautschi
- 1959
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4
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On the asymptotics of the Tricomi-Carlitz polynomials and their zero distributions
– Goh, Wimp
- 1994
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4
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The zero distribution of the Tricomi-Carlitz polynomials
– Goh, Wimp
- 1997
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4
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Completely monotonic functions associated with the gamma function and its q-analogues
– Ismail, Lorch, et al.
- 1986
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4
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Monotonicity results for the gamma function. Atti
– Kershaw, Laforgia
- 1985
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4
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Further inequalities for the gamma function
– Laforgia
- 1984
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4
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Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms
– Olver
- 1991
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4
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Exponentially-improved asymptotic solutions of ordinary differential equations. I: The confluent hypergeometric function
– Olver
- 1993
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4
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On the calculation of the inverse of the error function
– Strecok
- 1968
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4
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On the computation of the incomplete gamma functions for large values of the parameters, 479--489 in: Algorithms for approximation
– Temme
- 1987
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4
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Uniform asymptotics for the incomplete gamma function starting from negative values of the parameters
– Temme
- 1996
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4
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Asymptotische Eigenschaften der unvollstandigen Gammafunktion
– Tricomi
- 1950
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3
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Inequalities for zero-balanced hypergeometric functions
– Anderson, Barnard, et al.
- 1995
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3
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Rational Chebyshev approximations for the inverse of the error function
– Blair, Edwards, et al.
- 1976
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3
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Chebyshev approximations for the exponential integral Ei(x
– Cody, Thatcher
- 1969
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3
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Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. To appear in: Royal Society of London Proceedings: Philosophical Transactions
– Dunster
- 1994
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3
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Error bounds for asymptotic expansions of the ratio of two gamma functions
– Frenzen
- 1987
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3
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Inequalities and monotonicity properties for gamma and q-gamma functions, in Approximation and computation: A festschrift in honor of Walter Gautschi
– Ismail, Muldoon
- 1994
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3
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The generalized exponential integral. In: Approximation and Computation: A Festschrift
– Olver
- 1994
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3
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Asymptotic inversion of incomplete gamma functions
– Temme
- 1992
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3
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Asymptotics of zeros of incomplete gamma functions
– Temme
- 1995
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