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21: 10.30 Limiting Forms

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10.30.2 $$K_{\nu }\left(z\right)\sim \frac{1}{2}\mathrm{\Gamma }\left(\nu \right){(\frac{1}{2}z)}^{-\nu },$$ $\mathrm{\Re }\nu >0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.9

Referenced by:

§10.30(i)

Permalink:

http://dlmf.nist.gov/10.30.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(i), §10.30 and Ch.10

►

10.30.3 $$K_0\left(z\right)\sim -\ln z.$$

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Symbols:

$\sim$: asymptotic equality, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.6.8

Referenced by:

§10.30(i)

Permalink:

http://dlmf.nist.gov/10.30.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(i), §10.30 and Ch.10

►For $K_{\nu }\left(x\right)$, when $\nu$ is purely imaginary and $x\to 0+$, see (10.45.2) and (10.45.7). … ►For $K_{\nu }\left(z\right)$ see (10.25.3).

22: 10.54 Integral Representations

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10.54.4 $${𝗃}_n\left(z\right)=\frac{{(-\mathrm{i})}^{n+1}}{2\pi }{\int }_{\mathrm{i}\mathrm{\infty }}^{(-1+,1+)}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$$

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Symbols:

$\pi$: 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, $(a,b)$: open interval, $\mathrm{ph}$: phase, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

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${𝗁}_n^{(1)}\left(z\right)={\displaystyle \frac{{(-\mathrm{i})}^{n+1}}{\pi }}{\displaystyle {\int }_{\mathrm{i}\mathrm{\infty }}^{(1+)}}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$

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${𝗁}_n^{(2)}\left(z\right)={\displaystyle \frac{{(-\mathrm{i})}^{n+1}}{\pi }}{\displaystyle {\int }_{\mathrm{i}\mathrm{\infty }}^{(-1+)}}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$

►For the Legendre polynomial $P_n$ and the associated Legendre function $Q_n$ see §§18.3 and 14.21(i), with $\mu =0$ and $\nu =n$. …

23: 10.39 Relations to Other Functions

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10.39.2 $$K_{\frac{1}{2}}\left(z\right)=K_{-\frac{1}{2}}\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}.$$

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Symbols:

$\pi$: 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 and $z$: complex variable

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/10.39.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.3 $$K_{\frac{1}{4}}\left(z\right)={\pi }^{\frac{1}{2}}{z}^{-\frac{1}{4}}U(0,2z^{\frac{1}{2}}),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.39.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.4 $$K_{\frac{3}{4}}\left(z\right)=\frac{1}{2}{\pi }^{\frac{1}{2}}{z}^{-\frac{3}{4}}\left(\frac{1}{2}U(1,2z^{\frac{1}{2}})+U(-1,2z^{\frac{1}{2}})\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.39.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.6 $$K_{\nu }\left(z\right)={\pi }^{\frac{1}{2}}{(2z)}^{\nu }{\mathrm{e}}^{-z}U(\nu +\frac{1}{2},2\nu +1,2z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: 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, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.40(iv), (9.6.21), (9.6.22)

Permalink:

http://dlmf.nist.gov/10.39.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.8 $$K_{\nu }\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{W}_{0,\nu }\left(2z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.48

Referenced by:

(9.6.21), (9.6.22)

Permalink:

http://dlmf.nist.gov/10.39.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

…

24: 19.13 Integrals of Elliptic Integrals

… ►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for $F(\varphi ,k)$ and $E(\varphi ,k)$, together with special cases. … ►For direct and inverse Laplace transforms for the complete elliptic integrals $K\left(k\right)$, $E\left(k\right)$, and $D\left(k\right)$ see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.

25: 10.25 Definitions

… ►The defining property of the second standard solution $K_{\nu }\left(z\right)$ of (10.25.1) is ►

10.25.3 $$K_{\nu }\left(z\right)\sim \sqrt{\pi /(2z)}{\mathrm{e}}^{-z},$$

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Defines:

$K_{\nu }\left(z\right)$: modified Bessel function of the second kind

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.25(ii), §10.25(iii), §10.30(ii)

Permalink:

http://dlmf.nist.gov/10.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.25(ii), §10.25 and Ch.10

… ►Both $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ are real when $\nu$ is real and $\mathrm{ph}z=0$. ►For fixed $z$ $(\ne 0)$ each branch of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ is entire in $\nu$. … ►Except where indicated otherwise it is assumed throughout the DLMF that the symbols $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ denote the principal values of these functions. …

26: 18.41 Tables

… ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_n\left(x\right)$, $U_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_n\left(x\right)$ and $U_n\left(x\right)$, and $0.5,1,3,5,10$ for $L_n\left(x\right)$ and $H_n\left(x\right)$. …

27: 14.6 Integer Order

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14.6.2 $${𝖰}_{\nu }^m\left(x\right)={(-1)}^m{\left(1-x^2\right)}^{m/2}\frac{d^m{𝖰}_{\nu }\left(x\right)}{{dx}^m}.$$

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Symbols:

: Ferrers function of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.6.7

Permalink:

http://dlmf.nist.gov/14.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(i), §14.6 and Ch.14

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14.6.4 $$Q_{\nu }^m\left(x\right)={\left(x^2-1\right)}^{m/2}\frac{d^m{Q}_{\nu }\left(x\right)}{{dx}^m},$$

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Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.6.7

Referenced by:

§14.6(i)

Permalink:

http://dlmf.nist.gov/14.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(i), §14.6 and Ch.14

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14.6.8 $$Q_{\nu }^{-m}\left(x\right)={(-1)}^m{\left(x^2-1\right)}^{-m/2}{\int }_x^{\mathrm{\infty }}\mathrm{\dots }{\int }_x^{\mathrm{\infty }}{Q}_{\nu }\left(x\right){\left(dx\right)}^m.$$

ⓘ

Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $dx$: differential of $x$ , $\int$: integral, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.14.18

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(ii), §14.6 and Ch.14

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28: 10.43 Integrals

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10.43.10 $${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^t{t}^{-\nu }{K}_{\nu }\left(t\right)dt=\frac{{\mathrm{e}}^x{x}^{-\nu +1}}{2\nu -1}(K_{\nu }\left(x\right)+K_{\nu -1}\left(x\right)),$$ $\mathrm{\Re }\nu >\frac{1}{2}$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: complex parameter

Referenced by:

§10.43(ii)

Permalink:

http://dlmf.nist.gov/10.43.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(ii), §10.43 and Ch.10

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10.43.14 $${\mathrm{Ki}}_0\left(x\right)=K_0\left(x\right),$$

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Symbols:

${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $x$: real variable

A&S Ref:

11.2.8

Permalink:

http://dlmf.nist.gov/10.43.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

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10.43.15 $${\mathrm{Ki}}_{-n}\left(x\right)={(-1)}^n\frac{d^n}{{dx}^n}{K}_0\left(x\right),$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $n$: integer and $x$: real variable

A&S Ref:

11.2.9

Permalink:

http://dlmf.nist.gov/10.43.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

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10.43.29 $${\int }_0^{\mathrm{\infty }}t\exp \left(-p^2{t}^2\right)I_0\left(at\right)K_0\left(at\right)dt=\frac{1}{4p^2}\exp \left(\frac{a^2}{2p^2}\right)K_0\left(\frac{a^2}{2p^2}\right),$$ $\mathrm{\Re }\left(p^2\right)>0$.

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $\mathrm{\Re }$: real part

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.43.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

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10.43.31 $$g(x)={\int }_0^{\mathrm{\infty }}f(y)K_{\mathrm{i}y}\left(x\right)dy,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable, $y$: real variable, $g(x)$: function and $f(x)$: Kontorovich–Lebedev transform of $g(x)$

Referenced by:

§10.43(v)

Permalink:

http://dlmf.nist.gov/10.43.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(v), §10.43 and Ch.10

…

29: 19.2 Definitions

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$E\left(k\right)=E(\pi /2,k),$

… ►The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions. … ►

19.2.8_2 $$E^{\prime }\left(k\right)={\int }_0^1\frac{\sqrt{1-(1-k^2)t^2}}{\sqrt{1-t^2}}dt,$$

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Defines:

$E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $k$: real or complex modulus

Referenced by:

§19.2(ii)

Permalink:

http://dlmf.nist.gov/19.2.E8_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.10):

This equation was added.

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

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$$E(m\pi \pm \varphi ,k)=2mE\left(k\right)\pm E(\varphi ,k),$$

… ►

$E\left(k\right)=\mathrm{cel}(k_c,1,1,k_c^2),$

…

30: 14.14 Continued Fractions

… ►

14.14.3 $$(\nu -\mu )\frac{Q_{\nu }^{\mu }\left(x\right)}{Q_{\nu -1}^{\mu }\left(x\right)}=\frac{x_0}{y_0-}\frac{x_1}{y_1-}\frac{x_2}{y_2-}\mathrm{\cdots },$$ $\nu \ne \mu$,

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Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

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