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21: 13.9 Zeros

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13.9.8

\phi_{r}=\frac{j_{b-1,r}^{2}}{2b-4a}\left(1+\frac{2b(b-2)+j_{b-1,r}^{2}}{3(2b-% 4a)^{2}}\right)+O\left(\frac{1}{a^{5}}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $r=1,2,3,\dots$ , $\phi_{r}$ : positive zeros and $j_{b,r}$ : positive zero of Bessel
Referenced by:: §13.22, §13.9(i)
Permalink:: http://dlmf.nist.gov/13.9.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §13.9(i), §13.9 and Ch.13

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13.9.9

z=\pm(2n+a)\pi\mathrm{i}+\ln\left(-\frac{\Gamma\left(a\right)}{\Gamma\left(b-a% \right)}\left(\pm 2n\pi\mathrm{i}\right)^{b-2a}\right)+O\left(n^{-1}\ln n% \right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.9(i)
Permalink:: http://dlmf.nist.gov/13.9.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §13.9(i), §13.9 and Ch.13

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13.9.10

a=-\frac{\pi^{2}}{4z}\left(n^{2}+(b-\tfrac{3}{2})n\right)-\frac{1}{16z}\left((% b-\tfrac{3}{2})^{2}\pi^{2}+\tfrac{4}{3}z^{2}-8b(z-1)-4b^{2}-3\right)+O\left(n^% {-1}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.9(i)
Permalink:: http://dlmf.nist.gov/13.9.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §13.9(i), §13.9 and Ch.13

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13.9.16

a=-n-\frac{2}{\pi}\sqrt{zn}-\frac{2z}{\pi^{2}}+\tfrac{1}{2}b+\tfrac{1}{4}+% \frac{z^{2}\left(\frac{1}{3}-4\pi^{-2}\right)+z-(b-1)^{2}+\frac{1}{4}}{4\pi% \sqrt{zn}}+O\left(\frac{1}{n}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.9(ii), DLMF Update; Version 1.0.13, Erratum (V1.0.12) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16)
Permalink:: http://dlmf.nist.gov/13.9.E16
Encodings:: pMML, png, TeX
Errata (effective with 1.0.13):: In applying changes in Version 1.0.12 to this equation, an editing error was made; it has been corrected.
Reported 2016-09-12 by Adri Olde Daalhuis
Errata (effective with 1.0.12):: Originally this equation was expressed in terms of the asymptotic symbol $\sim$ . As a consequence of the use of the $O$ order symbol on the right hand side, $\sim$ was replaced by $=$ .
Reported 2016-07-11 by Rudi Weikard
See also:: Annotations for §13.9(ii), §13.9 and Ch.13

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22: 14.15 Uniform Asymptotic Approximations

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14.15.1

\mathsf{P}^{-\mu}_{\nu}\left(\pm x\right)=\left(\frac{1\mp x}{1\pm x}\right)^{% \mu/2}\left(\sum_{j=0}^{J-1}\frac{{\left(\nu+1\right)_{j}}{\left(-\nu\right)_{% j}}}{j!\Gamma\left(j+1+\mu\right)}\left(\frac{1\mp x}{2}\right)^{j}+O\left(% \frac{1}{\Gamma\left(J+1+\mu\right)}\right)\right)

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $J$ : nonnegative integer
Referenced by:: §14.15(i)
Permalink:: http://dlmf.nist.gov/14.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.15(i), §14.15 and Ch.14

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14.15.2

P^{-\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(\mu+1\right)}\left(\frac{2% \mu u}{\pi}\right)^{1/2}K_{\nu+\frac{1}{2}}\left(\mu u\right)\*\left(1+O\left(% \frac{1}{\mu}\right)\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $u$
Referenced by:: §14.15(i)
Permalink:: http://dlmf.nist.gov/14.15.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §14.15(i), §14.15 and Ch.14

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14.15.3

\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{1}{\mu^{\nu+(1/2)}}\left(\frac{% \pi u}{2}\right)^{1/2}I_{\nu+\frac{1}{2}}\left(\mu u\right)\left(1+O\left(% \frac{1}{\mu}\right)\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $u$
Referenced by:: §14.15(i)
Permalink:: http://dlmf.nist.gov/14.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.15(i), §14.15 and Ch.14

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14.15.13

P^{-\mu}_{\nu}\left(\cosh\xi\right)=\frac{1}{\nu^{\mu}}\left(\frac{\xi}{\sinh% \xi}\right)^{1/2}I_{\mu}\left(\left(\nu+\tfrac{1}{2}\right)\xi\right)\*\left(1% +O\left(\frac{1}{\nu}\right)\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\mu$ : general order, $\nu$ : general degree and $\xi$ : variable
Permalink:: http://dlmf.nist.gov/14.15.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §14.15(iii), §14.15 and Ch.14

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14.15.17

P^{-\mu}_{\nu}\left(x\right)=\beta\left(\frac{\alpha^{2}-y}{x^{2}-1+\alpha^{2}% }\right)^{1/4}I_{\mu}\left(\left(\nu+\tfrac{1}{2}\right)|y|^{1/2}\right)\*% \left(1+O\left(\frac{1}{\nu}\right)\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree, $\beta$ , $y$ and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §14.15(iv), §14.15 and Ch.14

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23: 15.12 Asymptotic Approximations

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15.12.2

F\left(a,b;c;z\right)=\sum_{s=0}^{m-1}\frac{{\left(a\right)_{s}}{\left(b\right% )_{s}}}{{\left(c\right)_{s}}s!}z^{s}+O\left(c^{-m}\right),

|c|\to\infty

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $m$ : integer
Referenced by:: §15.19(iv)
Permalink:: http://dlmf.nist.gov/15.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(ii), §15.12 and Ch.15

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15.12.5

\mathbf{F}\left({a+\lambda,b-\lambda\atop c};\tfrac{1}{2}-\tfrac{1}{2}z\right)% =2^{(a+b-1)/2}\frac{(z+1)^{(c-a-b-1)/2}}{(z-1)^{c/2}}\sqrt{\zeta\sinh\zeta}% \left(\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b\right)^{1-c}\left(I_{c-1}\left((% \lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)(1+O(\lambda^{-2}))+\frac{I_{c% -2}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)}{2\lambda+a-b}\left% (\left(c-\tfrac{1}{2}\right)\left(c-\tfrac{3}{2}\right)\left(\frac{1}{\zeta}-% \coth\zeta\right)+\tfrac{1}{2}(2c-a-b-1)(a+b-1)\tanh\left(\tfrac{1}{2}\zeta% \right)+O(\lambda^{-2})\right)\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/15.12.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

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15.12.7

F\left({a,b-\lambda\atop c+\lambda};-z\right)=2^{b-c+(1/2)}\left(\frac{z+1}{2% \sqrt{z}}\right)^{\lambda}\left(\lambda^{a/2}U\left(a-\tfrac{1}{2},-\alpha% \sqrt{\lambda}\right)\left((1+z)^{c-a-b}z^{1-c}\left(\frac{\alpha}{z-1}\right)% ^{1-a}+O(\lambda^{-1})\right)+\frac{\lambda^{\ifrac{(a-1)}{2}}}{\alpha}U\left(% a-\tfrac{3}{2},-\alpha\sqrt{\lambda}\right)\left((1+z)^{c-a-b}z^{1-c}\left(% \frac{\alpha}{z-1}\right)^{1-a}-2^{c-b-(\ifrac{1}{2})}\left(\frac{\alpha}{z-1}% \right)^{a}+O(\lambda^{-1})\right)\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $\alpha$
Permalink:: http://dlmf.nist.gov/15.12.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

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15.12.9

(z+1)^{3\lambda/2}(2\lambda)^{c-1}\mathbf{F}\left({a+\lambda,b+2\lambda\atop c% };-z\right)={\lambda^{-1/3}\left({\mathrm{e}}^{\pi\mathrm{i}(a-c+\lambda+(1/3)% )}\operatorname{Ai}\left({\mathrm{e}}^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{% \ifrac{2}{3}}\beta^{2}\right)+{\mathrm{e}}^{\pi\mathrm{i}(c-a-\lambda-(1/3))}% \operatorname{Ai}\left({\mathrm{e}}^{\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac% {2}{3}}\beta^{2}\right)\right)\left(a_{0}(\zeta)+O(\lambda^{-1})\right)}+% \lambda^{-2/3}\left({\mathrm{e}}^{\pi\mathrm{i}(a-c+\lambda+(2/3))}% \operatorname{Ai}'\left({\mathrm{e}}^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{% \ifrac{2}{3}}\beta^{2}\right)+{\mathrm{e}}^{\pi\mathrm{i}(c-a-\lambda-(2/3))}% \operatorname{Ai}'\left({\mathrm{e}}^{\ifrac{2\pi\mathrm{i}}{3}}\lambda^{% \ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{1}(\zeta)+O(\lambda^{-1})\right),

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24: 2.10 Sums and Sequences

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2.10.12

|S(\alpha,\beta,n)|\leq\sum_{j=1}^{n-1}j^{\alpha}=O\left(1\right),\;O\left(\ln n% \right),\text{ or }O\left(n^{\alpha+1}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\ln\NVar{z}$ : principal branch of logarithm function and $S(\alpha,\beta,n)$ : sum
Referenced by:: §2.10(ii)
Permalink:: http://dlmf.nist.gov/2.10.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

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2.10.17

S(\alpha,\beta,n)=O\left(n^{\alpha}\right)+O\left(1\right).

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding and $S(\alpha,\beta,n)$ : sum
Permalink:: http://dlmf.nist.gov/2.10.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

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2.10.18

S(\alpha,\beta,n)=\frac{e^{in\beta}}{e^{i\beta}-1}n^{\alpha}+O\left(n^{\alpha-% 1}\right)+O\left(1\right),

n\to\infty

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $S(\alpha,\beta,n)$ : sum
Permalink:: http://dlmf.nist.gov/2.10.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

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(b´)

On the circle $|z|=r$ , the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_{j}$ , say,

2.10.30

f(z)-g(z)=O\left((z-z_{j})^{\sigma_{j}-1}\right),

z\to z_{j}

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $f(x)$ : analytic function, $g(z)$ : comparison function and $\sigma_{j}$ : positive constant
Permalink:: http://dlmf.nist.gov/2.10.E30
Encodings:: pMML, png, TeX
See also:: Annotations for §2.10(iv), §2.10 and Ch.2

where $\sigma_{j}$ is a positive constant.

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2.10.32

f^{(m)}(z)-g^{(m)}(z)=O\left((z-z_{j})^{\sigma_{j}-1}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $f(x)$ : analytic function, $g(z)$ : comparison function and $\sigma_{j}$ : positive constant
Permalink:: http://dlmf.nist.gov/2.10.E32
Encodings:: pMML, png, TeX
See also:: Annotations for §2.10(iv), §2.10 and Ch.2

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25: 18.15 Asymptotic Approximations

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18.15.7

\varepsilon_{M}(\rho,\theta)=\begin{cases}\theta O\left(\rho^{-2M-(3/2)}\right% ),&c\rho^{-1}\leq\theta\leq\pi-\delta,\\ \theta^{\alpha+(5/2)}O\left(\rho^{-2M+\alpha}\right),&0\leq\theta\leq c\rho^{-% 1},\end{cases}

ⓘ

Defines:: $\varepsilon_{M}(\rho,\theta)$ (locally)
Symbols:: $O\left(\NVar{x}\right)$ : 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

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18.15.12

P_{n}\left(\cos\theta\right)=\left(\frac{2}{\sin\theta}\right)^{\frac{1}{2}}% \sum_{m=0}^{M-1}\genfrac{(}{)}{0.0pt}{}{-\tfrac{1}{2}}{m}\genfrac{(}{)}{0.0pt}% {}{m-\frac{1}{2}}{n}\frac{\cos\alpha_{n,m}}{(2\sin\theta)^{m}}+O\left(\frac{1}% {n^{M+\frac{1}{2}}}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $n$ : nonnegative integer and $\alpha_{n,m}$
Referenced by:: §18.15(iii)
Permalink:: http://dlmf.nist.gov/18.15.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §18.15(iii), §18.15 and Ch.18

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18.15.14

L^{(\alpha)}_{n}\left(x\right)=\frac{n^{\frac{1}{2}\alpha-\frac{1}{4}}{\mathrm% {e}}^{\frac{1}{2}x}}{{\pi}^{\frac{1}{2}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}% \left(\cos\theta_{n}^{(\alpha)}(x)\left(\sum_{m=0}^{M-1}\frac{a_{m}(x)}{n^{% \frac{1}{2}m}}+O\left(\frac{1}{n^{\frac{1}{2}M}}\right)\right)+\sin\theta_{n}^% {(\alpha)}(x)\left(\sum_{m=1}^{M-1}\frac{b_{m}(x)}{n^{\frac{1}{2}m}}+O\left(% \frac{1}{n^{\frac{1}{2}M}}\right)\right)\right),

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18.15.22

L^{(\alpha)}_{n}\left(\nu x\right)=(-1)^{n}\frac{{\mathrm{e}}^{\frac{1}{2}\nu x% }}{2^{\alpha-\frac{1}{2}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}\*\left(\frac{\zeta% }{x-1}\right)^{\frac{1}{4}}\left(\frac{\operatorname{Ai}\left(\nu^{\frac{2}{3}% }\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{m=0}^{M-1}\frac{E_{m}(\zeta)}{\nu^{2m}}% +\frac{\operatorname{Ai}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}% }}\sum_{m=0}^{M-1}\frac{F_{m}(\zeta)}{\nu^{2m}}+\operatorname{envAi}\left(\nu^% {\frac{2}{3}}\zeta\right)O\left(\frac{1}{\nu^{2M-\frac{2}{3}}}\right)\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\sim$ : Poincaré asymptotic expansion, $\operatorname{envAi}\left(\NVar{x}\right)$ : envelope of Airy function $\operatorname{Ai}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $m$ : nonnegative integer, $n$ : nonnegative integer, $\nu$ , $\zeta$ , $E_{m}$ : coefficient, $F_{m}$ : coefficient and $x$ : real variable
Referenced by:: Erratum (V1.0.11) for Equation (18.15.22)
Permalink:: http://dlmf.nist.gov/18.15.E22
Encodings:: pMML, png, TeX
Errata (effective with 1.0.11):: Originally this equation was expressed in terms of the asymptotic symbol $\sim$ . As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$ .
See also:: Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

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18.15.27

H_{n}\left(x\right)=\lambda_{n}{\mathrm{e}}^{\frac{1}{2}x^{2}}\left(\sum_{m=0}% ^{M-1}\frac{u_{m}(x)\cos\omega_{n,m}(x)}{\mu^{\frac{1}{2}m}}+O\left(\frac{1}{% \mu^{\frac{1}{2}M}}\right)\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $m$ : nonnegative integer, $n$ : nonnegative integer, $\omega_{n,m}(x)$ and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.15.E27
Encodings:: pMML, png, TeX
See also:: Annotations for §18.15(v), §18.15 and Ch.18

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26: 13.2 Definitions and Basic Properties

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13.2.13

M\left(a,b,z\right)=1+O\left(z\right).

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §13.2(iii), §13.2 and Ch.13

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13.2.14

U\left(-n,b,z\right)=(-1)^{n}{\left(b\right)_{n}}+O\left(z\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §13.2(iii), §13.2 and Ch.13

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13.2.15

U\left(-n+b-1,b,z\right)=(-1)^{n}{\left(2-b\right)_{n}}z^{1-b}+O\left(z^{2-b}% \right).

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §13.2(iii), §13.2 and Ch.13

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13.2.17

U\left(a,2,z\right)=\frac{1}{\Gamma\left(a\right)}z^{-1}+O\left(\ln z\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 13.5.7
Permalink:: http://dlmf.nist.gov/13.2.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §13.2(iii), §13.2 and Ch.13

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13.2.21

U\left(a,0,z\right)=\frac{1}{\Gamma\left(a+1\right)}+O\left(z\ln z\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 13.5.11
Permalink:: http://dlmf.nist.gov/13.2.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §13.2(iii), §13.2 and Ch.13

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27: 10.41 Asymptotic Expansions for Large Order

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10.41.12

I_{\nu}\left(\nu z\right)=\frac{e^{\nu\eta}}{(2\pi\nu)^{\frac{1}{2}}(1+z^{2})^% {\frac{1}{4}}}\left(\sum_{k=0}^{\ell-1}\frac{U_{k}(p)}{\nu^{k}}+O\left(\frac{1% }{z^{\ell}}\right)\right),

|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant, $\eta$ , $p$ and $U_{k}(p)$ : polynomial coefficient
Referenced by:: §10.41(iv)
Permalink:: http://dlmf.nist.gov/10.41.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §10.41(iv), §10.41 and Ch.10

►

10.41.13

K_{\nu}\left(\nu z\right)=\left(\frac{\pi}{2\nu}\right)^{\frac{1}{2}}\frac{e^{% -\nu\eta}}{(1+z^{2})^{\frac{1}{4}}}\*\left(\sum_{k=0}^{\ell-1}(-1)^{k}\frac{U_% {k}(p)}{\nu^{k}}+O\left(\frac{1}{z^{\ell}}\right)\right),

|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant, $\eta$ , $p$ and $U_{k}(p)$ : polynomial coefficient
Referenced by:: §10.41(iv), §10.41(iv)
Permalink:: http://dlmf.nist.gov/10.41.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §10.41(iv), §10.41 and Ch.10

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10.41.14

J_{\nu}\left(\nu z\right)=\left(\frac{4\zeta}{1-z^{2}}\right)^{\frac{1}{4}}% \left(\frac{\operatorname{Ai}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1% }{3}}}\left(\sum_{k=0}^{\ell}\frac{A_{k}(\zeta)}{\nu^{2k}}+O\left(\frac{1}{% \zeta^{3\ell+3}}\right)\right)+\frac{\operatorname{Ai}'\left(\nu^{\frac{2}{3}}% \zeta\right)}{\nu^{\frac{5}{3}}}\left(\sum_{k=0}^{\ell-1}\frac{B_{k}(\zeta)}{% \nu^{2k}}+O\left(\frac{1}{\zeta^{3\ell+1}}\right)\right)\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $O\left(\NVar{x}\right)$ : order not exceeding, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $A_{k}(\zeta)$ : coefficients and $B_{k}(\zeta)$ : coefficients
Referenced by:: §10.41(v)
Permalink:: http://dlmf.nist.gov/10.41.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §10.41(v), §10.41 and Ch.10

►

10.41.15

Y_{\nu}\left(\nu z\right)=-\left(\frac{4\zeta}{1-z^{2}}\right)^{\frac{1}{4}}% \left(\frac{\operatorname{Bi}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1% }{3}}}\left(\sum_{k=0}^{\ell}\frac{A_{k}(\zeta)}{\nu^{2k}}+O\left(\frac{1}{% \zeta^{3\ell+3}}\right)\right)+\frac{\operatorname{Bi}'\left(\nu^{\frac{2}{3}}% \zeta\right)}{\nu^{\frac{5}{3}}}\left(\sum_{k=0}^{\ell-1}\frac{B_{k}(\zeta)}{% \nu^{2k}}+O\left(\frac{1}{\zeta^{3\ell+1}}\right)\right)\right),

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $O\left(\NVar{x}\right)$ : order not exceeding, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $A_{k}(\zeta)$ : coefficients and $B_{k}(\zeta)$ : coefficients
Referenced by:: §10.41(v)
Permalink:: http://dlmf.nist.gov/10.41.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §10.41(v), §10.41 and Ch.10

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28: 2.8 Differential Equations with a Parameter

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2.8.11

W_{n,1}(u,\xi)=e^{u\xi}\left(\sum_{s=0}^{n-1}\frac{A_{s}(\xi)}{u^{s}}+O\left(% \frac{1}{u^{n}}\right)\right),

\xi\in\mathbf{\Delta}_{1}(\alpha_{1})

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $n$ : positive integer, $W_{n,j}(u,\xi)$ : solution, $A_{s}(\xi)$ : coefficients, $\mathbf{\Delta}_{j}(\alpha_{j})$ : domain and $u$ : large real or complex parameter
Referenced by:: §2.8(ii), §2.8(ii)
Permalink:: http://dlmf.nist.gov/2.8.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(ii), §2.8 and Ch.2

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2.8.12

W_{n,2}(u,\xi)=e^{-u\xi}\left(\sum_{s=0}^{n-1}(-1)^{s}\frac{A_{s}(\xi)}{u^{s}}% +O\left(\frac{1}{u^{n}}\right)\right),

\xi\in\mathbf{\Delta}_{2}(\alpha_{2})

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $n$ : positive integer, $W_{n,j}(u,\xi)$ : solution, $A_{s}(\xi)$ : coefficients, $\mathbf{\Delta}_{j}(\alpha_{j})$ : domain and $u$ : large real or complex parameter
Referenced by:: §2.8(ii), §2.8(ii)
Permalink:: http://dlmf.nist.gov/2.8.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(ii), §2.8 and Ch.2

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2.8.15

W_{n,1}(u,\xi)=\operatorname{Ai}\left(u^{2/3}\xi\right)\left(\sum_{s=0}^{n-1}% \frac{A_{s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)+% \operatorname{Ai}'\left(u^{2/3}\xi\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi% )}{u^{2s+(4/3)}}+O\left(\frac{1}{u^{2n-1}}\right)\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $W_{n,j}(u,\xi)$ : solution, $n$ : positive integer, $A_{s}(\xi)$ : coefficients, $B_{s}(\xi)$ : coefficients and $u$ : large real or complex parameter
Referenced by:: §2.1(iii), §2.8(iii), §2.8(iii)
Permalink:: http://dlmf.nist.gov/2.8.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(iii), §2.8 and Ch.2

►

2.8.16

W_{n,2}(u,\xi)=\operatorname{Bi}\left(u^{2/3}\xi\right)\left(\sum_{s=0}^{n-1}% \frac{A_{s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)+% \operatorname{Bi}'\left(u^{2/3}\xi\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi% )}{u^{2s+(4/3)}}+O\left(\frac{1}{u^{2n-1}}\right)\right).

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $W_{n,j}(u,\xi)$ : solution, $n$ : positive integer, $A_{s}(\xi)$ : coefficients, $B_{s}(\xi)$ : coefficients and $u$ : large real or complex parameter
Referenced by:: §2.8(iii), §2.8(iii)
Permalink:: http://dlmf.nist.gov/2.8.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(iii), §2.8 and Ch.2

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2.8.29

W_{n,3}(u,\xi)=|\xi|^{1/2}J_{\nu}\left(u|\xi|^{1/2}\right)\left(\sum_{s=0}^{n-% 1}\frac{A_{s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)-|\xi|J_{% \nu+1}\left(u|\xi|^{1/2}\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi)}{u^{2s+1% }}+O\left(\frac{1}{u^{2n-2}}\right)\right),

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $O\left(\NVar{x}\right)$ : order not exceeding, $W_{n,j}(u,\xi)$ : solution, $\nu$ : real nonnegative constant, $n$ : positive integer, $A_{s}(\xi)$ : coefficients, $B_{s}(\xi)$ : coefficients and $u$ : large real or complex parameter
Referenced by:: §2.8(iv), §2.8(iv)
Permalink:: http://dlmf.nist.gov/2.8.E29
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(iv), §2.8 and Ch.2

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29: 10.52 Limiting Forms

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\mathsf{j}_{n}\left(z\right)=z^{-1}\sin\left(z-\tfrac{1}{2}n\pi\right)+{% \mathrm{e}}^{|\Im z|}O\left(z^{-2}\right),

►

\mathsf{y}_{n}\left(z\right)=-z^{-1}\cos\left(z-\tfrac{1}{2}n\pi\right)+{% \mathrm{e}}^{|\Im z|}O\left(z^{-2}\right),

…

30: 10.68 Modulus and Phase Functions

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10.68.16

M_{\nu}\left(x\right)=\frac{e^{x/\sqrt{2}}}{(2\pi x)^{\frac{1}{2}}}\left(1-% \frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)^{2}}{256}\frac{1}{x^{2}}-% \frac{(\mu-1)(\mu^{2}+14\mu-399)}{6144\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}% {x^{4}}\right)\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.21
Permalink:: http://dlmf.nist.gov/10.68.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

►

10.68.17

\ln M_{\nu}\left(x\right)=\frac{x}{\sqrt{2}}-\frac{1}{2}\ln\left(2\pi x\right)% -\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}-\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1% }{x^{3}}-\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+O\left(\frac{1}{x^{5}}% \right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.22
Permalink:: http://dlmf.nist.gov/10.68.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

►

10.68.18

\theta_{\nu}\left(x\right)=\frac{x}{\sqrt{2}}+\left(\frac{1}{2}\nu-\frac{1}{8}% \right)\pi+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}-% \frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{5}}% \right).

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.23
Referenced by:: §10.68(i), §10.70
Permalink:: http://dlmf.nist.gov/10.68.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

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10.68.20

\ln N_{\nu}\left(x\right)=-\frac{x}{\sqrt{2}}+\frac{1}{2}\ln\left(\frac{\pi}{2% x}\right)+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)(\mu-25)}{384\sqrt{2% }}\frac{1}{x^{3}}-\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+O\left(\frac{1}{x% ^{5}}\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.25
Permalink:: http://dlmf.nist.gov/10.68.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

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10.68.21

\phi_{\nu}\left(x\right)=-\frac{x}{\sqrt{2}}-\left(\frac{1}{2}\nu+\frac{1}{8}% \right)\pi-\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}+% \frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{5}}% \right).

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.26
Referenced by:: §10.68(i), §10.70
Permalink:: http://dlmf.nist.gov/10.68.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

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