asymptotic expansions as ϵ→0

SearchAdvancedHelp

(0.013 seconds)

21—30 of 140 matching pages

21: 10.21 Zeros

… ►

10.21.22 $${\rho }_{\nu }(t)\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $\nu$: complex parameter, ${\rho }_{\nu }(t)$: zero of cylinder function and $\alpha$: coefficients

Permalink:

http://dlmf.nist.gov/10.21.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

… ►

10.21.27 $${\sigma }_{\nu }(t)\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k^{\prime }}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $\nu$: complex parameter, ${\sigma }_{\nu }(t)$: zero of cylinder function and ${\alpha }^{\prime }$: coefficients

Permalink:

http://dlmf.nist.gov/10.21.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

… ►

10.21.32 $$j_{\nu ,m}\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$ , $m$: integer, $k$: nonnegative integer, $\nu$: complex parameter and $\alpha$: coefficients

Referenced by:

§10.21(vii), §10.21(vii)

Permalink:

http://dlmf.nist.gov/10.21.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

►

10.21.33 $$y_{\nu ,m}\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$ , $m$: integer, $k$: nonnegative integer, $\nu$: complex parameter and $\alpha$: coefficients

Permalink:

http://dlmf.nist.gov/10.21.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

… ►

10.21.36 $$j_{\nu ,m}^{\prime }\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k^{\prime }}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$ , $m$: integer, $k$: nonnegative integer, $\nu$: complex parameter and ${\alpha }^{\prime }$: coefficients

Referenced by:

§10.21(vii)

Permalink:

http://dlmf.nist.gov/10.21.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

…

22: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►

8.18.3 $$I_x(a,b)=\frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}\left(\sum _{k=0}^{n-1}{d}_k{F}_k+O\left(a^{-n}\right)F_0\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $n$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $F_k$: functions and $d_k$: coefficients

Referenced by:

Erratum (V1.0.14) for Equation (8.18.3)

Permalink:

http://dlmf.nist.gov/8.18.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Previously this equation appeared without the order estimate as $I_x(a,b)\sim \frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}{\sum }_{k=0}^{\mathrm{\infty }}{d}_k{F}_k$. The range of $x$ was extended to include $1$.

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

… ►

8.18.9 $$I_x(a,b)\sim \frac{1}{2}\mathrm{erfc}\left(-\eta \sqrt{b/2}\right)+\frac{1}{\sqrt{2\pi (a+b)}}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{c}_k(\eta )}{{(a+b)}^k},$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

… ►

8.18.14 $$I_x(a,b)\sim Q(b,a\zeta )-\frac{{(2\pi b)}^{-1/2}}{{\mathrm{\Gamma }}^{\ast }\left(b\right)}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{h_k(\zeta ,\mu )}{a^k},$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $Q(a,z)$: normalized incomplete gamma function, ${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $\mu$ , $\zeta$: variable and $h_k(\zeta ,\mu )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

…

23: 2.5 Mellin Transform Methods

… ►

2.5.17 $$f(t)\sim \sum _{s=0}^{\mathrm{\infty }}{a}_s{t}^{{\alpha }_s},$$ $t\to 0+$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $f(x)$: locally integrable function and $a_s$: coefficients

Referenced by:

§2.5(ii)

Permalink:

http://dlmf.nist.gov/2.5.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

… ►

2.5.18 $$h(t)\sim \exp \left(\mathrm{i}\kappa t^p\right)\sum _{s=0}^{\mathrm{\infty }}{b}_s{t}^{-{\beta }_s},$$ $t\to +\mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $h(x)$: locally integrable function, $\kappa$: real, $p$: positive and $b$: right endpoint

Referenced by:

§2.5(iii), §2.5(ii), §2.5(iii), §2.5(iii)

Permalink:

http://dlmf.nist.gov/2.5.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

… ►

2.5.44 $$ℒh\left(\zeta \right)\sim \sum _{n=0}^{\mathrm{\infty }}{b}_n\mathrm{\Gamma }\left(1-{\beta }_n\right){\zeta }^{{\beta }_n-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-\zeta )}^n}{n!}ℳh\left(n+1\right),$$ $\zeta \to 0+$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $ℒ\left(f\right)\left(s\right)$: Laplace transform, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $h(x)$: locally integrable function and $b$: right endpoint

Keywords:

Laplace transform, Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E44

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒf\left(z\right)$ from $ℒ(f;z)$. In addition, the notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(iii), §2.5 and Ch.2

… ►

2.5.48 $$ℒh\left(\zeta \right)\sim (-\ln \zeta )\sum _{k=0}^{\mathrm{\infty }}\frac{{\zeta }^k}{k!}+\sum _{k=0}^{\mathrm{\infty }}\psi \left(k+1\right)\frac{{\zeta }^k}{k!},$$ $\zeta \to 0+$.

ⓘ

Symbols:

$ℒ\left(f\right)\left(s\right)$: Laplace transform, $\sim$: Poincaré asymptotic expansion, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function and $h(x)$: locally integrable function

Keywords:

Laplace transform

Referenced by:

§2.5(iii)

Permalink:

http://dlmf.nist.gov/2.5.E48

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒf\left(z\right)$ from $ℒ(f;z)$.

See also:

Annotations for §2.5(iii), §2.5(iii), §2.5 and Ch.2

…

24: 9.12 Scorer Functions

… ►

9.12.25 $$\mathrm{Gi}\left(z\right)\sim \frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

See Olver (1997b, pp. 431–432).

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E25

Encodings:

pMML, png, TeX

See also:

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

… ►

9.12.27 $$\mathrm{Hi}\left(z\right)\sim -\frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}\left(-z\right)|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

See Olver (1997b, pp. 431–432).

Referenced by:

(9.12.31)

Permalink:

http://dlmf.nist.gov/9.12.E27

Encodings:

pMML, png, TeX

See also:

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

… ►

9.12.29 $$\mathrm{Hi}\left(z\right)\sim -\frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k}+\frac{{\mathrm{e}}^{\zeta }}{\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$.

ⓘ

Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Permalink:

http://dlmf.nist.gov/9.12.E29

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

►

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

ⓘ

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

…

25: 2.6 Distributional Methods

… ►

2.6.6 $$S(x)\sim \frac{2\pi }{\sqrt{3}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{3}}{s}\right)x^{-s-(1/3)},$$ $x\to \mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter and $S(x)$: integral

Referenced by:

§2.6(i)

Permalink:

http://dlmf.nist.gov/2.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(i), §2.6 and Ch.2

… ►

2.6.7 $$S(x)\sim \frac{2\pi }{\sqrt{3}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{3}}{s}\right)x^{-s-(1/3)}-\sum _{s=1}^{\mathrm{\infty }}\frac{3^s(s-1)!}{2\cdot 5\mathrm{\cdots }(3s-1)}{x}^{-s};$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $S(x)$: integral

Referenced by:

§2.6(i), §2.6(ii)

Permalink:

http://dlmf.nist.gov/2.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(i), §2.6 and Ch.2

… ►

2.6.9 $$f(t)\sim \sum _{s=0}^{\mathrm{\infty }}{a}_s{t}^{-s-\alpha },$$ $t\to +\mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $f(t)$: locally integrable function and $a_n$: coefficients

Referenced by:

§2.6(iii), §2.6(ii), §2.6(iii), §2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.31 $$f(t)\sim {\mathrm{e}}^{\mathrm{i}ct}\sum _{s=0}^{\mathrm{\infty }}{a}_s{t}^{-s-\alpha },$$ $t\to +\mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $c\ne 0$: real, $f(t)$: locally integrable function and $a_n$: coefficients

Permalink:

http://dlmf.nist.gov/2.6.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.32 $${\int }_0^{\mathrm{\infty }}\frac{f(t)}{{(t+z)}^{\rho }}dt,$$ $\rho >0$,

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral and $f(t)$: locally integrable function

Permalink:

http://dlmf.nist.gov/2.6.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

…

26: 12.14 The Function $W(a,x)$

… ►

12.14.23 $$s_1(a,x)+\mathrm{i}{s}_2(a,x)\sim \sum _{r=0}^{\mathrm{\infty }}{(-\mathrm{i})}^r\frac{{\left(\frac{1}{2}+\mathrm{i}a\right)}_{2r}}{2^{r}r!x^{2r}}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $x$: real variable, $a$: real or complex parameter and $s_n(a,x)$: coefficients

A&S Ref:

19.21.8 (modification of)

Permalink:

http://dlmf.nist.gov/12.14.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(viii), §12.14 and Ch.12

… ►

12.14.26 $$W(\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})\sim \frac{2^{\frac{1}{2}}{\mathrm{e}}^{\frac{1}{4}\pi {\mu }^2}l(\mu )}{{(t^2-1)}^{\frac{1}{4}}}\left(\sin \sigma \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{𝒜}_{2s}(t)}{{\mu }^{4s}}+\cos \sigma \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{𝒜}_{2s+1}(t)}{{\mu }^{4s+2}}\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, ${𝒜}_s(t)$: coefficients, $\sigma$ and $l(\mu )$

Referenced by:

§12.14(ix)

Permalink:

http://dlmf.nist.gov/12.14.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

… ►

12.14.29 $$l(\mu )\sim \frac{2^{\frac{1}{4}}}{{\mu }^{\frac{1}{2}}}\sum _{s=0}^{\mathrm{\infty }}\frac{l_s}{{\mu }^{4s}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $l(\mu )$ and $l_s$: coefficient

Permalink:

http://dlmf.nist.gov/12.14.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

… ►

12.14.34 $$W(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{l(\mu )}{{(t^2+1)}^{\frac{1}{4}}}\left(\cos \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{u}}_{2s}(t)}{{(t^2+1)}^{3s}{\mu }^{4s}}-\sin \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{u}}_{2s+1}(t)}{{(t^2+1)}^{3s+\frac{3}{2}}{\mu }^{4s+2}}\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, ${\overline{u}}_s(t)$: solution, $l(\mu )$ and $\overline{\sigma }$

Permalink:

http://dlmf.nist.gov/12.14.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

►

12.14.35 $$W^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim -\frac{\mu }{\sqrt{2}}l(\mu ){(t^2+1)}^{\frac{1}{4}}\left(\sin \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{v}}_{2s}(t)}{{(t^2+1)}^{3s}{\mu }^{4s}}+\cos \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{v}}_{2s+1}(t)}{{(t^2+1)}^{3s+\frac{3}{2}}{\mu }^{4s+2}}\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, ${\overline{v}}_s(t)$: solution, $l(\mu )$ and $\overline{\sigma }$

Permalink:

http://dlmf.nist.gov/12.14.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

…

27: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

►Let $s=2m+1$, $m=0,1,2,\mathrm{\dots }$, and $\nu$ be fixed with . … ►

28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

ⓘ

Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

…

28: 30.9 Asymptotic Approximations and Expansions

… ►

30.9.1 $${\lambda }_n^m\left({\gamma }^2\right)\sim -{\gamma }^2+\gamma q+{\beta }_0+{\beta }_1{\gamma }^{-1}+{\beta }_2{\gamma }^{-2}+\mathrm{\cdots },$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $q$ and ${\beta }_n$: coeffients

A&S Ref:

21.7.6

Referenced by:

item

Permalink:

http://dlmf.nist.gov/30.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.9(i), §30.9 and Ch.30

… ►

30.9.4 $${\lambda }_n^m\left({\gamma }^2\right)\sim 2q|\gamma |+c_0+c_1{|\gamma |}^{-1}+c_2{|\gamma |}^{-2}+\mathrm{\cdots },$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $q$ and $c_n$: coeffients

A&S Ref:

21.8.2

Referenced by:

item

Permalink:

http://dlmf.nist.gov/30.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.9(ii), §30.9 and Ch.30

…

29: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►

11.11.4 $${𝐀}_{\nu }\left(z\right)\sim \frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu }{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}}.$$

ⓘ

Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions

Permalink:

http://dlmf.nist.gov/11.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.11(i), §11.11 and Ch.11

… ►

11.11.8 $${𝐀}_{\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$,

ⓘ

Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Referenced by:

(11.11.18), (11.11.19), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to \mathrm{\infty }$, , has been replaced to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

… ►

11.11.10 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim -\frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(-\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

ⓘ

Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Proof sketch:

Apply Laplace’s method to the integral (11.10.4).

Referenced by:

(11.11.18), (11.11.19), §11.11(iii), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

… ►When $\nu$ is real and positive, all of (11.11.10)–(11.11.17) can be regarded as special cases of two asymptotic expansions given in Olver (1997b, pp. 352–360) for ${𝐀}_{-\nu }\left(\lambda \nu \right)$ as $\nu \to +\mathrm{\infty }$, one being uniform for , and the other being uniform for $\lambda \ge 1$. … ►Lastly, corresponding asymptotic approximations and expansions for ${𝐉}_{\nu }\left(\lambda \nu \right)$ and ${𝐄}_{\nu }\left(\lambda \nu \right)$, with or $\lambda >1$, follow from (11.10.15) and (11.10.16) and the corresponding asymptotic expansions for the Bessel functions $J_{\nu }\left(z\right)$ and $Y_{\nu }\left(z\right)$; see §10.19(ii). …

30: 12.11 Zeros

… ►

12.11.4 $$u_{a,s}\sim 2^{\frac{1}{2}}\mu \left(p_0(\alpha )+\frac{p_1(\alpha )}{{\mu }^4}+\frac{p_2(\alpha )}{{\mu }^8}+\mathrm{\cdots }\right),$$

ⓘ

Defines:

$u_{a,s}$: zeros (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter, $\alpha$ and $p_n(\zeta )$: coefficients

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.7 $$u_{a,s}^{\prime }\sim 2^{\frac{1}{2}}\mu \left(q_0(\beta )+\frac{q_1(\beta )}{{\mu }^4}+\frac{q_2(\beta )}{{\mu }^8}+\mathrm{\cdots }\right),$$

ⓘ

Defines:

$u_{a,s}^{\prime }$: zeros (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter and $\beta$

Permalink:

http://dlmf.nist.gov/12.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

…