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31: 8.11 Asymptotic Approximations and Expansions

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8.11.3 $$R_n(a,z)=O\left(z^{-n}\right),$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter, $n$: nonnegative integer, $\delta$: small positive constant and $R_n(a,z)$: remainder

Permalink:

http://dlmf.nist.gov/8.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(i), §8.11 and Ch.8

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8.11.6 $$\gamma (a,z)\sim -z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ , $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Notes:

See Paris (2002b)

Referenced by:

§8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, . It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where and $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

See also:

Annotations for §8.11(iii), §8.11 and Ch.8

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8.11.7 $$\mathrm{\Gamma }(a,z)\sim z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ $\lambda >1$, $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Referenced by:

§8.11(iii), §8.11(iii), §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, $\lambda >1$. It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where $\lambda >1$ and $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$.

See also:

Annotations for §8.11(iii), §8.11 and Ch.8

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32: 9.12 Scorer Functions

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9.12.30 $${\int }_0^z\mathrm{Gi}\left(t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }-\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$.

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Integrate (9.12.25) and obtain the constant term by combining (9.12.12) and (9.12.31).(this equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.)

Permalink:

http://dlmf.nist.gov/9.12.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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9.12.31 $${\int }_0^z\mathrm{Hi}\left(-t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }+\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $k$: nonnegative integer, $x$: real variable, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Except for the constant term, this be verified by termwise integration of (9.12.27). To evaluate the constant term replace $z$ by $-x$ ($\le 0$) in (9.12.20) and integrate (§1.5(v)) to obtain $\pi {\int }_0^x\mathrm{Hi}\left(-t\right)dt={\int }_0^{\mathrm{\infty }}(1-{\mathrm{e}}^{-xt}){\mathrm{e}}^{-\frac{1}{3}{t}^3}{t}^{-1}dt$. Next, integrate the right-hand side of this equation by parts—integrating the factor $t^{-1}$ and differentiating the rest. As $x\to \mathrm{\infty }$ the asymptotic expansions of ${\int }_0^{\mathrm{\infty }}x{\mathrm{e}}^{-xt}{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ and ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ follow from (2.3.9). Also, ${\int }_0^{\mathrm{\infty }}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ can be found by replacing $\frac{1}{3}{t}^3$ by $t$ and referring to the first of (5.9.18). (This equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.)

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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33: 7.1 Special Notation

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$x$	real variable.
…
$\delta$	arbitrary small positive constant.
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… ►The main functions treated in this chapter are the error function $\mathrm{erf}z$; the complementary error functions $\mathrm{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$; the Voigt functions $𝖴(x,t)$ and $𝖵(x,t)$. …

34: 10.1 Special Notation

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$m,n$	integers. In §§10.47–10.71 $n$ is nonnegative.
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$x,y$	real variables.
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$\delta$	arbitrary small positive constant.
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$\psi \left(x\right)$	${\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$: logarithmic derivative of the gamma function (§5.2(i)).
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►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. …

35: 12.14 The Function $W(a,x)$

… ►For the modulus functions $\tilde{F}(a,x)$ and $\tilde{G}(a,x)$ see §12.14(x). … ►Here $w_1(a,x)$ and $w_2(a,x)$ are the even and odd solutions of (12.2.3): … ►When $x>0$ … ►Then as $x\to \mathrm{\infty }$ … ►In the following expansions, obtained from Olver (1959), $\mu$ is large and positive, and $\delta$ is again an arbitrary small positive constant. …

36: 18.28 Askey–Wilson Class

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18.28.19 $$R_n(x)=R_n(x;\alpha ,\beta ,\gamma ,\delta |q)=\sum _{\mathrm{\ell }=0}^n\frac{q^{\mathrm{\ell }}{(q^{-n},\alpha \beta q^{n+1};q)}_{\mathrm{\ell }}}{{(\alpha q,\beta \delta q,\gamma q,q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-q^{j}x+\gamma \delta q^{2j+1})={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},\alpha \beta q^{n+1},q^{-y},\gamma \delta q^{y+1}}{\alpha q,\beta \delta q,\gamma q};q,q),$$ $\alpha q$, $\beta \delta q$, or $\gamma q=q^{-N}$; $n=0,1,\mathrm{\dots },N$.

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Defines:

$R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$: $q$-Racah polynomial and $R_n(x)$ (locally)

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28 and Ch.18

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18.28.20 $$\sum _{y=0}^N{R}_n(q^{-y}+\gamma \delta q^{y+1})R_m(q^{-y}+\gamma \delta q^{y+1}){\omega }_y=h_n{\delta }_{n,m},$$ $n,m=0,1,\mathrm{\dots },N$,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer, $h_n$ and $R_n(x)$

Permalink:

http://dlmf.nist.gov/18.28.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28 and Ch.18

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18.28.23 $$R_n(q^{-y}+\gamma \delta q^{y+1};\alpha ,\beta ,\gamma ,\delta |q)=R_y(q^{-n}+\alpha \beta q^{n+1};\gamma ,\delta ,\alpha ,\beta |q),$$ $\alpha q$, $\beta \delta q$, or $\gamma q=q^{-N}$; $n,y=0,1,\mathrm{\dots },N$.

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$: $q$-Racah polynomial, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $q$: real variable and $n$: nonnegative integer

Referenced by:

§18.28(viii), §18.28(viii), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28(viii), §18.28 and Ch.18

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18.28.34 $$\underset{q\to 1}{lim}{R}_n(q^{-y}+q^{y+\gamma +\delta +1};q^{\alpha },q^{\beta },q^{\gamma },q^{\delta }|q)=R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta ).$$

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$: $q$-Racah polynomial, $y$: real variable, $\delta$: arbitrary small positive constant, $q$: real variable and $n$: nonnegative integer

Source:

Koekoek et al. (2010, (14.2.24))

Referenced by:

§18.28(x), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

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37: 1.5 Calculus of Two or More Variables

… ►Suppose also that ${\int }_c^{d}f(x,y)dy$ converges and ${\int }_c^d(\partial f/\partial x)dy$ converges uniformly on $a\le x\le b$, that is, given any positive number $ϵ$, however small, we can find a number $c_0\in [c,d)$ that is independent of $x$ and is such that …

38: 18.15 Asymptotic Approximations

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18.15.7

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Defines:

${\epsilon }_M(\rho ,\theta )$ (locally)

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\delta$: arbitrary small positive constant and $\rho$

Permalink:

http://dlmf.nist.gov/18.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

… ►For more powerful asymptotic expansions as $n\to \mathrm{\infty }$ in terms of elementary functions that apply uniformly when , $-1+\delta \le t\le 1-\delta$, or , where $t=x/\sqrt{2n+1}$ and $\delta$ is again an arbitrary small positive constant, see §§12.10(i)–12.10(iv) and 12.10(vi). …

39: 11.1 Special Notation

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$x$	real variable.
…
$\delta$	arbitrary small positive constant.

…

40: 10.7 Limiting Forms

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10.7.6 $$Y_{\mathrm{i}\nu }\left(z\right)=\frac{\mathrm{i}\mathrm{csch}\left(\nu \pi \right)}{\mathrm{\Gamma }\left(1-\mathrm{i}\nu \right)}{(\frac{1}{2}z)}^{-\mathrm{i}\nu }-\frac{\mathrm{i}\coth \left(\nu \pi \right)}{\mathrm{\Gamma }\left(1+\mathrm{i}\nu \right)}{(\frac{1}{2}z)}^{\mathrm{i}\nu }+{\mathrm{e}}^{|\nu \mathrm{ph}z|}o\left(1\right),$$ $\nu \in ℝ$ and $\nu \ne 0$.

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\coth z$: hyperbolic cotangent function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than, $\mathrm{ph}$: phase, $ℝ$: real line, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.7(i), §10.7(i)

Permalink:

http://dlmf.nist.gov/10.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.7(i), §10.7 and Ch.10

►See also §10.24 when $z=x$ $(>0)$. …