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21: 13.7 Asymptotic Expansions for Large Argument

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13.7.2 $$𝐌(a,b,z)\sim \frac{{\mathrm{e}}^z{z}^{a-b}}{\mathrm{\Gamma }\left(a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{z}^{-s}+\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}{z}^{-a}}{\mathrm{\Gamma }\left(b-a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

A&S Ref:

13.5.1 (with corrected region of validity)

Referenced by:

§13.2(iv), §13.7(ii), §13.9(i)

Permalink:

http://dlmf.nist.gov/13.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(i), §13.7 and Ch.13

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Figure 13.7.1: Regions ${\text{R}}_1$, ${\text{R}}_2$, ${\overline{\text{R}}}_2$, ${\text{R}}_3$, and ${\overline{\text{R}}}_3$ are the closures of the indicated unshaded regions bounded by the straight lines and circular arcs centered at the origin, with $r=|b-2a|$. Magnify

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22: 28.33 Physical Applications

… ►Hence from §28.17 the corresponding Mathieu equation is stable or unstable according as $(q,a)$ is in the intersection of $ℒ$ with the colored or the uncolored open regions depicted in Figure 28.17.1. …

23: 5.11 Asymptotic Expansions

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5.11.1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)\sim \left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}$$

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.40

Referenced by:

§5.11(i), §5.11(i), §5.11(ii), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11 and Ch.5

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5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

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24: 3.5 Quadrature

… ►In more advanced methods points are sampled from a probability distribution, so that they are concentrated in regions that make the largest contribution to the integral. …

25: 19.33 Triaxial Ellipsoids

… ►If a conducting ellipsoid with semiaxes $a,b,c$ bears an electric charge $Q$, then the equipotential surfaces in the exterior region are confocal ellipsoids: …

26: 28.8 Asymptotic Expansions for Large $q$

… ►The approximations apply when the parameters $a$ and $q$ are real and large, and are uniform with respect to various regions in the $z$-plane. …

27: 9.17 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the defining differential equation (9.2.1) by direct numerical methods. … ►The former reference includes a parallelized version of the method. ►In the case of the Scorer functions, integration of the differential equation (9.12.1) is more difficult than (9.2.1), because in some regions stable directions of integration do not exist. … ►For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). … ►For quadrature methods for Scorer functions see Gil et al. (2001), Lee (1980), and Gordon (1970, Appendix A); but see also Gautschi (1983). …

28: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►For $Z_1{Z}_2=-1$ and $m=m_{\mathrm{e}}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_0=\mathrm{\hslash }/(m_{\mathrm{e}}c\alpha )$, and to a multiple of the Rydberg constant, … ►The relativistic motion of spinless particles in a Coulomb field, as encountered in pionic atoms and pion-nucleon scattering (Backenstoss (1970)) is described by a Klein–Gordon equation equivalent to (33.2.1); see Barnett (1981a). The motion of a relativistic electron in a Coulomb field, which arises in the theory of the electronic structure of heavy elements (Johnson (2007)), is described by a Dirac equation. … ►The Coulomb solutions of the Schrödinger and Klein–Gordon equations are almost always used in the external region, outside the range of any non-Coulomb forces or couplings. ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$, or $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). …

29: Bibliography G

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Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA.

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Notes:

CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26

External Links:

MathReview (O. Widlund), ZentralBlatt

Referenced by:

§18.38(i).

Encodings:

BibTeX

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30: 18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. …