scaled gamma function
(0.008 seconds)
31—40 of 40 matching pages
31: 14.13 Trigonometric Expansions
32: 13.16 Integral Representations
…
►
►
§13.16(ii) Contour Integrals
… ►where the contour of integration separates the poles of from those of . … ►where the contour of integration separates the poles of from those of . …where the contour of integration passes all the poles of on the right-hand side.33: 3.5 Quadrature
…
►The given quantities follow from (18.2.5), (18.2.7), Table 18.3.1, and the relation .
…
►The are the monic Legendre polynomials, that is, the polynomials (§18.3) scaled so that the coefficient of the highest power of in their explicit forms is unity.
…
►In the case of Chebyshev weight functions
on , with , the nodes , weights , and error constant , are as follows:
…
►Furthermore, .
…
►
Example
…34: 14.15 Uniform Asymptotic Approximations
§14.15 Uniform Asymptotic Approximations
►§14.15(i) Large , Fixed
… ►In other words, the convergent hypergeometric series expansions of are also generalized (and uniform) asymptotic expansions as , with scale , ; compare §2.1(v). … ►(The inverse hyperbolic functions again take their principal values.) … ►35: Bibliography W
…
►
Evaluating elliptic functions and their inverses.
Comput. Math. Appl. 39 (3-4), pp. 131–136.
…
►
Infinitely differentiable generalized logarithmic and exponential functions.
Math. Comp. 57 (196), pp. 723–733.
…
►
Reduction formulae for products of theta functions.
J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
…
►
Concerning two series for the gamma function.
Math. Comp. 22 (103), pp. 617–626.
…
►
Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region.
Phys. Rev. B 13, pp. 316–374.
…
36: Bibliography G
…
►
Evaluation of -gamma function and -analogues by iterative algorithms.
Numer. Algorithms 49 (1-4), pp. 159–168.
…
►
Algorithm 542: Incomplete gamma functions.
ACM Trans. Math. Software 5 (4), pp. 482–489.
…
►
Some elementary inequalities relating to the gamma and incomplete gamma function.
J. Math. Phys. 38 (1), pp. 77–81.
…
►
A harmonic mean inequality for the gamma function.
SIAM J. Math. Anal. 5 (2), pp. 278–281.
…
►
A computational procedure for incomplete gamma functions.
ACM Trans. Math. Software 5 (4), pp. 466–481.
…
37: 15.12 Asymptotic Approximations
…
►
§15.12(i) Large Variable
►For the asymptotic behavior of as with , , fixed, combine (15.2.2) with (15.8.2) or (15.8.8). ►§15.12(ii) Large
… ►Also, … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).38: Philip J. Davis
…
►After being asked by Milton Abramowitz to work on the project, he chose to write the Chapter “Gamma Function and Related Functions.
…
►Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958.
…
►DLMF users can rotate, rescale, zoom and otherwise explore mathematical function surfaces.
The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling.
Moreover, a cutting plane feature allows users to track curves of intersection produced as a moving plane cuts through the function surface.
…
39: 18.39 Applications in the Physical Sciences
…
►The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by () with corresponding eigenfunctions expressed in terms of Whittaker functions (13.14.3).
The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV).
…
►The fact that both the eigenvalues of (18.39.31) and the scaling of the co-ordinate in the eigenfunctions, (18.39.30), depend on the sum leads to the substitution
…
►
c) Spherical Radial Coulomb Wave Functions
… ►where is a real, positive, scaling factor, and a non-negative integer. …40: Bibliography F
…
►
Incomplete gamma functions for large values of their variables.
Adv. in Appl. Math. 34 (3), pp. 467–485.
…
►
An asymptotic expansion of the double gamma function.
J. Approx. Theory 111 (2), pp. 298–314.
…
►
A note on the asymptotic expansion of a ratio of gamma functions.
Proc. Edinburgh Math. Soc. (2) 15, pp. 43–45.
…
►
Algorithm 309. Gamma function with arbitrary precision.
Comm. ACM 10 (8), pp. 511–512.
…
►
Application of the -function theory of Painlevé equations to random matrices: , the JUE, CyUE, cJUE and scaled limits.
Nagoya Math. J. 174, pp. 29–114.
…