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11: 2.5 Mellin Transform Methods

… ►where … ►Since

\mathscr{M}\mskip-3.0muh_{1}\mskip 3.0mu\left(z\right)

is analytic for

\Re z>-c

by Table 2.5.1, the analytically-continued

\mathscr{M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)

allows us to extend the Mellin transform of

h

via … ►Since

\mathscr{M}\mskip-3.0mue^{-t}\mskip 3.0mu\left(z\right)=\Gamma\left(z\right)

, by the Parseval formula (2.5.5), there are real numbers

p_{1}

and

p_{2}

such that

-c<p_{1}<1

p_{2}<\min(1,\Re\beta_{0})

, and …Since

\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right)

is analytic for

\Re z>-c

, by (2.5.14), …Similarly, since

\mathscr{M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)

can be continued analytically to a meromorphic function (when

\kappa=0

) or to an entire function (when

\kappa\neq 0

), we can choose

\rho

so that

\mathscr{M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)

has no poles in

1<\Re z\leq\rho<2

. …

12: 2.11 Remainder Terms; Stokes Phenomenon

… ►Since the ray

\operatorname{ph}z=\frac{3}{2}\pi

is well away from the new boundaries, the compound expansion (2.11.7) yields much more accurate results when

\operatorname{ph}z\to\frac{3}{2}\pi

. … ►Here

\operatorname{erfc}

is the complementary error function (§7.2(i)), and … ► … ►The following example, based on Weniger (1996), illustrates their power. … ►Their extrapolation is based on assumed forms of remainder terms that may not always be appropriate for asymptotic expansions. …

13: 10.74 Methods of Computation

… ► … ►And since there are no error terms they could, in theory, be used for all values of

z

; however, there may be severe cancellation when

|z|

is not large compared with

n^{2}

. … ►For evaluation of the Hankel functions

{H^{(1)}_{\nu}}\left(z\right)

and

{H^{(2)}_{\nu}}\left(z\right)

for complex values of

\nu

and

z

based on the integral representations (10.9.18) see Remenets (1973). … ►The integral representation used is based on (10.32.8). … ► …

14: 16.11 Asymptotic Expansions

… ►

§16.11(i) Formal Series

… ►

§16.11(ii) Expansions for Large Variable

… ►

16.11.7

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q}F_{q}}% \left({a_{1},\dots,a_{q}\atop b_{1},\dots,b_{q}};z\right)\sim H_{q,q}(z{% \mathrm{e}}^{\mp\pi\mathrm{i}})+E_{q,q}(z).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $E_{p,q}(z)$ : formal infinite series and $H_{p,q}(z)$ : formal infinite series
Referenced by:: §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►(Either sign may be used when

\operatorname{ph}z=0

since the first term on the right-hand side becomes exponentially small compared with the second term.) … ►

§16.11(iii) Expansions for Large Parameters

…

15: 31.9 Orthogonality

… ►

§31.9(i) Single Orthogonality

… ►The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. … ►The right-hand side may be evaluated at any convenient value, or limiting value, of

\zeta

(0,1)

since it is independent of

\zeta

. ►For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64). ►

§31.9(ii) Double Orthogonality

…

16: 2.10 Sums and Sequences

… ►Then … ►Thus

R_{m}(n)

and

R_{m+1}(n)

are of opposite signs, and since their difference is the term corresponding to

s=m

in (2.10.4),

R_{m}(n)

is bounded in absolute value by this term and has the same sign. … ►Since … ►

§2.10(iii) Asymptotic Expansions of Entire Functions

… ►We need a “comparison function”

g(z)

with the properties: …

17: Guide to Searching the DLMF

… ►Wildcards allow matching patterns and marking parts of an expression that don’t matter (as for example, which variable name the author uses for a function): … ►The syntax of the special functions can be LaTeX-like or as employed in widely used computer algebra systems. … ►

•

All the inverse trigonometric functions (arcsin vs. Arcsin, etc.).

… ►Sometimes there are distinctions between various special function names based on font style, such as the use of bold or calligraphic letters. … ►For example, you may want equations that contain trigonometric functions, but you don’t care which trigonometric function. …

18: 3.11 Approximation Techniques

… ►Since

L_{0}=1

L_{n}

is a monotonically increasing function of

n

, and (for example)

L_{1000}=4.07\dots

, this means that in practice the gain in replacing a truncated Chebyshev-series expansion by the corresponding minimax polynomial approximation is hardly worthwhile. … ►

Example

… ►The rational function … ►Since

X_{k\ell}=X_{\ell k}

, the matrix is again symmetric. … ►For splines based on Bernoulli and Euler polynomials, see §24.17(ii). …

19: 2.6 Distributional Methods

… ►Since the functions

t^{-s-\alpha}

s=1,2,\dots

, are not locally integrable on

[0,\infty)

, we cannot assign distributions to them in a similar manner. …since the

n

th derivative of

f_{n,n}

f_{n}

. … ►Since the function

t^{\mu}\left(\ln t-\gamma-\psi\left(\mu+1\right)\right)

and all its derivatives are locally absolutely continuous in

(0,\infty)

, the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives. Furthermore, since

f_{n,n}^{(n)}(t)=f_{n}(t)

, it follows from (2.6.37) that the remainder terms

t^{\mu-1}\ast f_{n}

in the last two equations can be associated with a locally integrable function in

(0,\infty)

. … ►Since …

20: 18.39 Applications in the Physical Sciences

… ►Since the operators

T_{e}+V(r)

and

\mathrm{L}^{2}

commute and have simultaneous eigenfunctions (see §1.3(iv)), the wave function

\Psi(r,\theta,\phi)

separates as … ►

c) Spherical Radial Coulomb Wave Functions

… ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the

x_{i}

and

w_{i}

, from the J-matrix as in §3.5(vi), straightforward. … ►The technique to accomplish this follows the DVR idea, in which methods are based on finding tridiagonal representations of the co-ordinate,

x

. …