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31: 18.17 Integrals

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18.17.17 $${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n}^{(\lambda )}\left(x\right)\cos \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda \right)J_{\lambda +2n}\left(y\right)}{(2n)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }},$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

►

18.17.18 $${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n+1}^{(\lambda )}\left(x\right)\sin \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda +1\right)J_{2n+\lambda +1}\left(y\right)}{(2n+1)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }}.$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

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18.17.19 $${\int }_{-1}^1{P}_n\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx={\mathrm{i}}^n\sqrt{\frac{2\pi }{y}}{J}_{n+\frac{1}{2}}\left(y\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

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18.17.20 $${\int }_0^1{P}_n\left(1-2x^2\right)\cos \left(xy\right)dx={(-1)}^n\frac{1}{2}\pi J_{n+\frac{1}{2}}\left(\frac{1}{2}y\right)J_{-n-\frac{1}{2}}\left(\frac{1}{2}y\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

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18.17.21 $${\int }_0^1{P}_n\left(1-2x^2\right)\sin \left(xy\right)dx=\frac{1}{2}\pi {\left(J_{n+\frac{1}{2}}\left(\frac{1}{2}y\right)\right)}^2.$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

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32: 18.10 Integral Representations

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Laguerre

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18.10.9 $$L_n^{(\alpha )}\left(x\right)=\frac{{\mathrm{e}}^x{x}^{-\frac{1}{2}\alpha }}{n!}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{n+\frac{1}{2}\alpha }{J}_{\alpha }\left(2\sqrt{xt}\right)dt,$$ $\alpha >-1$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $t$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.10.14

Referenced by:

§18.10(iv)

Permalink:

http://dlmf.nist.gov/18.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(iv), §18.10(iv), §18.10 and Ch.18

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33: 18.18 Sums

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Laguerre

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18.18.27 $$\sum _{n=0}^{\mathrm{\infty }}\frac{n!L_n^{(\alpha )}\left(x\right)L_n^{(\alpha )}\left(y\right)}{{\left(\alpha +1\right)}_n}{z}^n=\frac{\mathrm{\Gamma }\left(\alpha +1\right){(xyz)}^{-\frac{1}{2}\alpha }}{1-z}\exp \left(\frac{-(x+y)z}{1-z}\right)I_{\alpha }\left(\frac{2{(xyz)}^{\frac{1}{2}}}{1-z}\right),$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\exp z$: exponential function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $y$: real variable, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(vii), §18.18(vii)

Permalink:

http://dlmf.nist.gov/18.18.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(vii), §18.18(vii), §18.18 and Ch.18

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34: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

►For the polynomials $P_l$ see §18.3, and for the function $Y_{l,m}$ see §14.30. ►

34.3.19

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§14.18(iv), §14.30(iii), §34.3(vii)

Permalink:

http://dlmf.nist.gov/34.3.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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34.3.21

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§14.17(iv)

Permalink:

http://dlmf.nist.gov/34.3.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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35: 18.12 Generating Functions

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18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n,$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (9.8.12))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) for Equation (18.12.2)

Permalink:

http://dlmf.nist.gov/18.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.6 $$\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right){\mathrm{e}}^{z\cos \theta }{(\frac{1}{2}z\sin \theta )}^{\frac{1}{2}-\lambda }{J}_{\lambda -\frac{1}{2}}\left(z\sin \theta \right)=\sum _{n=0}^{\mathrm{\infty }}\frac{C_n^{(\lambda )}\left(\cos \theta \right)}{{\left(2\lambda \right)}_n}{z}^n,$$ $0\le \theta \le \pi$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.12 $${\mathrm{e}}^{xz}{J}_0\left(z\sqrt{1-x^2}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n\left(x\right)}{n!}{z}^n.$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.14 $$\mathrm{\Gamma }\left(\alpha +1\right){(xz)}^{-\frac{1}{2}\alpha }{\mathrm{e}}^z{J}_{\alpha }\left(2\sqrt{xz}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(\alpha )}\left(x\right)}{{\left(\alpha +1\right)}_n}{z}^n.$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.16

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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36: 7.6 Series Expansions

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7.6.9 $$\mathrm{erf}\left(az\right)=\frac{2z}{\sqrt{\pi }}{\mathrm{e}}^{(\frac{1}{2}-a^2)z^2}\sum _{n=0}^{\mathrm{\infty }}{T}_{2n+1}\left(a\right){𝗂}_n^{(1)}\left(\frac{1}{2}{z}^2\right),$$ $-1\le a\le 1$.

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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37: 10.41 Asymptotic Expansions for Large Order

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10.41.3 $$I_{\nu }\left(\nu z\right)\sim \frac{{\mathrm{e}}^{\nu \eta }}{{(2\pi \nu )}^{\frac{1}{2}}{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}\frac{U_k(p)}{{\nu }^k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.7

Referenced by:

§10.41(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

►

10.41.4 $$K_{\nu }\left(\nu z\right)\sim {\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{\mathrm{e}}^{-\nu \eta }}{{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k(p)}{{\nu }^k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.8

Permalink:

http://dlmf.nist.gov/10.41.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

►

10.41.5 $$I_{\nu }^{\prime }\left(\nu z\right)\sim \frac{{(1+z^2)}^{\frac{1}{4}}{\mathrm{e}}^{\nu \eta }}{{(2\pi \nu )}^{\frac{1}{2}}z}\sum _{k=0}^{\mathrm{\infty }}\frac{V_k(p)}{{\nu }^k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $V_k(p)$: polynomial coefficient

A&S Ref:

9.7.9

Referenced by:

§10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

►

10.41.6 $$K_{\nu }^{\prime }\left(\nu z\right)\sim -{\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{(1+z^2)}^{\frac{1}{4}}{\mathrm{e}}^{-\nu \eta }}{z}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{V_k(p)}{{\nu }^k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $V_k(p)$: polynomial coefficient

A&S Ref:

9.7.10

Referenced by:

§10.41(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

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10.41.12 $$I_{\nu }\left(\nu z\right)=\frac{{\mathrm{e}}^{\nu \eta }}{{(2\pi \nu )}^{\frac{1}{2}}{(1+z^2)}^{\frac{1}{4}}}\left(\sum _{k=0}^{\mathrm{\ell }-1}\frac{U_k(p)}{{\nu }^k}+O\left(\frac{1}{z^{\mathrm{\ell }}}\right)\right),$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: 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

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38: 10.19 Asymptotic Expansions for Large Order

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10.19.9 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim \frac{2^{\frac{4}{3}}}{{\nu }^{\frac{1}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu }{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel 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, $k$: nonnegative integer, $\nu$: complex parameter, $P_k(a)$: polynomial coefficient and $Q_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.27):

Originally the polynomials $P_k(a)$, $Q_k(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively.

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

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10.19.13 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim -\frac{2^{\frac{5}{3}}}{{\nu }^{\frac{2}{3}}}{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{R_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{4}{3}}}{{\nu }^{\frac{4}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{S_k(a)}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel 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, $k$: nonnegative integer, $\nu$: complex parameter, $R_k(a)$: polynomial coefficient and $S_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

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39: 10.50 Wronskians and Cross-Products

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10.50.4 $${𝗃}_0\left(z\right){𝗃}_n\left(z\right)+{𝗒}_0\left(z\right){𝗒}_n\left(z\right)=\cos \left(\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor n/2\rfloor }{(-1)}^k\frac{a_{2k}(n+\frac{1}{2})}{z^{2k+2}}+\sin \left(\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{(-1)}^k\frac{a_{2k+1}(n+\frac{1}{2})}{z^{2k+3}},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\lfloor x\rfloor$: floor of $x$ , $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.1.33 (modified)

Referenced by:

§10.50, §10.50

Permalink:

http://dlmf.nist.gov/10.50.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.50 and Ch.10

…

40: 10.40 Asymptotic Expansions for Large Argument

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10.40.1 $$I_{\nu }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.1

Referenced by:

§10.30(ii), §10.40(i), §10.40(i), §10.40(ii), §10.40(iii), §10.45, §10.67(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/10.40.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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10.40.2 $$K_{\nu }\left(z\right)\sim {\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.2

Referenced by:

§10.40(i), §10.40(i), §10.40(ii), §10.40(iii), §10.41(iv), §10.45

Permalink:

http://dlmf.nist.gov/10.40.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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10.40.3 $$I_{\nu }^{\prime }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

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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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.3

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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10.40.4 $$K_{\nu }^{\prime }\left(z\right)\sim -{\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

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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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.4

Referenced by:

§10.40(i), §10.40(i), §10.67(i)

Permalink:

http://dlmf.nist.gov/10.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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10.40.5 $$I_{\nu }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_k(\nu )}{z^k}\pm \mathrm{i}{\mathrm{e}}^{\pm \nu \pi \mathrm{i}}\frac{{\mathrm{e}}^{-z}}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$.

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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, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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