for large q

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11: 16.11 Asymptotic Expansions

… ►

16.11.10

{{}_{p+1}F_{p}}\left({a_{1}+r,\dots,a_{k-1}+r,a_{k},\dots,a_{p+1}\atop b_{1}+r% ,\dots,b_{k}+r,b_{k+1},\dots,b_{p}};z\right)=\sum_{n=0}^{m-1}\frac{{\left(a_{1% }+r\right)_{n}}\cdots{\left(a_{k-1}+r\right)_{n}}{\left(a_{k}\right)_{n}}% \cdots{\left(a_{p+1}\right)_{n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{k}% +r\right)_{n}}{\left(b_{k+1}\right)_{n}}\cdots{\left(b_{p}\right)_{n}}}\frac{z% ^{n}}{n!}+O\left(\frac{1}{r^{m}}\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : 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 and $r$ : large parameter
Permalink:: http://dlmf.nist.gov/16.11.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(iii), §16.11 and Ch.16

… ►

16.11.11

{{}_{p}F_{q}}\left({a_{1}+r,\dots,a_{p}+r\atop b_{1}+r,\dots,b_{q}+r};z\right)% =\sum_{n=0}^{m-1}\frac{{\left(a_{1}+r\right)_{n}}\cdots{\left(a_{p}+r\right)_{% n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{q}+r\right)_{n}}}\frac{z^{n}}{n% !}+O\left(\frac{1}{r^{(q-p)m}}\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : 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 and $r$ : large parameter
Permalink:: http://dlmf.nist.gov/16.11.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(iii), §16.11 and Ch.16

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12: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for

P^{-\mu}_{\nu}\left(x\right)

and

\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)

for

1<x<\infty

are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). …

13: 17.2 Calculus

§17.2 Calculus

►

§17.2(i) $q$ -Calculus

… ►

§17.2(iv) Derivatives

… ►These identities are the first in a large collection of similar results. …

14: 28.20 Definitions and Basic Properties

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28.20.3

\operatorname{Ce}_{\nu}\left(z,q\right)=\operatorname{ce}_{\nu}\left(\pm% \mathrm{i}z,q\right),

\nu\neq-1,-2,\dots

ⓘ

Defines:: $\operatorname{Ce}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function
Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathrm{i}$ : imaginary unit, $q=h^{2}$ : parameter, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.20(ii), §28.20 and Ch.28

… ►

28.20.5

\operatorname{Me}_{\nu}\left(z,q\right)=\operatorname{me}_{\nu}\left(-\mathrm{% i}z,q\right),

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Defines:: $\operatorname{Me}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function
Symbols:: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathrm{i}$ : imaginary unit, $q=h^{2}$ : parameter, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §28.20(ii), §28.20 and Ch.28

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§28.20(iii) Solutions ${\operatorname{M}^{(j)}_{\nu}}$

►Assume first that

\nu

is real,

q

is positive, and

a=\lambda_{\nu}\left(q\right)

; see §28.12(i). … ►

28.20.8

h=\sqrt{q}\;(>0).

ⓘ

Symbols:: $q=h^{2}$ : parameter and $h$ : parameter
Permalink:: http://dlmf.nist.gov/28.20.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §28.20(iii), §28.20 and Ch.28

…

15: 28.7 Analytic Continuation of Eigenvalues

… ►All the

a_{2n}\left(q\right)

n=0,1,2,\dots

, can be regarded as belonging to a complete analytic function (in the large). …

16: Bibliography J

… ►

S. Jorna and C. Springer (1971) Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions

\overline{ps}\,^{r}_{n}(\eta,h)

and

\overline{qs}\,^{r}_{n}(\eta,h)

for large

h

. Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §30.9(ii).
Encodings:: BibTeX

…

17: 30.9 Asymptotic Approximations and Expansions

… ►

§30.9(i) Prolate Spheroidal Wave Functions

►As

\gamma^{2}\to+\infty

, with

q=2(n-m)+1

, … ►The cases of large

m

, and of large

m

and large

|\gamma|

, are studied in Abramowitz (1949). …The behavior of

\lambda^{m}_{n}\left(\gamma^{2}\right)

for complex

\gamma^{2}

and large

|\lambda^{m}_{n}\left(\gamma^{2}\right)|

is investigated in Hunter and Guerrieri (1982).

18: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

… ►The last reference also includes an exponentially-improved version (§2.11(iii)) of the expansions (8.12.4) and (8.12.7) for

Q\left(a,z\right)

. … ►Lastly, a uniform approximation for

\Gamma\left(a,ax\right)

for large

a

, with error bounds, can be found in Dunster (1996a). … ►

Inverse Function

►For asymptotic expansions, as

a\to\infty

, of the inverse function

x=x(a,q)

that satisfies the equation …

19: 2.10 Sums and Sequences

… ►for large

n

. … ►As a first estimate for large

n

… ►(5.11.7) shows that the integrals around the large quarter circles vanish as

n\to\infty

. Hence … ►

Example

…

20: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►with

q(x)

real and continuous, unless otherwise noted. … ►

Example 1: Three Simple Cases where $q(x)=0$ , $X=[0,\pi]$

… ►This is accomplished by the variable change

x\to x{\mathrm{e}}^{\mathrm{i}\theta}

, in

\mathcal{L}

, which rotates the continuous spectrum

\boldsymbol{\sigma}_{c}\to\boldsymbol{\sigma}_{c}{\mathrm{e}}^{-2\mathrm{i}\theta}

and the branch cut of (1.18.66) into the lower half complex plain by the angle

-2\theta

, with respect to the unmoved branch point at

\lambda=0

; thus, providing access to resonances on the higher Riemann sheet should

\theta

be large enough to expose them. … ►

Example 1: In one and two dimensions any $q(x)$ with a ‘Dip, or Well’ has a partly discrete spectrum

… ►What then is the condition on

q(x)

to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if

q(x)\equiv 0

then there is only a continuous spectrum. …