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21: 19.7 Connection Formulas

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§19.7(i) Complete Integrals of the First and Second Kinds

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19.7.1 $$E\left(k\right)K^{\prime }\left(k\right)+E^{\prime }\left(k\right)K\left(k\right)-K\left(k\right)K^{\prime }\left(k\right)=\frac{1}{2}\pi .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind and $k$: real or complex modulus

Referenced by:

§19.21(i), §19.35(i), §19.7(i)

Permalink:

http://dlmf.nist.gov/19.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(i), §19.7(i), §19.7 and Ch.19

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$E\left(\mathrm{i}k/k^{\prime }\right)=(1/k^{\prime })E\left(k\right),$

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$E\left(-\mathrm{i}{k}^{\prime }/k\right)=(1/k)E\left(k^{\prime }\right).$

… ►The second relation maps each hyperbolic region onto itself and each circular region onto the other: …

22: 10.38 Derivatives with Respect to Order

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10.38.3 $${{(-1)}^n\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=-K_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}{(-1)}^k\frac{{(\frac{1}{2}z)}^k{I}_k\left(z\right)}{k!(n-k)},$$

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Symbols:

$!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.44

Referenced by:

§10.38

Permalink:

http://dlmf.nist.gov/10.38.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

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10.38.4 $${\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{K}_k\left(z\right)}{k!(n-k)}.$$

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Symbols:

$!$: factorial (as in $n!$), $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.45

Referenced by:

§10.38

Permalink:

http://dlmf.nist.gov/10.38.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

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${{\displaystyle \frac{\partial I_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=-K_0\left(z\right),$

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${{\displaystyle \frac{\partial K_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=0.$

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10.38.7 $${\frac{\partial K_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=\pm \sqrt{\frac{\pi }{2x}}{E}_1\left(2x\right){\mathrm{e}}^x.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.34 (corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

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23: 10.31 Power Series

… ►When $\nu$ is not an integer the corresponding expansion for $K_{\nu }\left(z\right)$ is obtained from (10.25.2) and (10.27.4). … ►

10.31.1 $$K_n\left(z\right)=\frac{1}{2}{(\frac{1}{2}z)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{(-\frac{1}{4}{z}^2)}^k+{(-1)}^{n+1}\ln \left(\frac{1}{2}z\right)I_n\left(z\right)+{(-1)}^n\frac{1}{2}{(\frac{1}{2}z)}^n\sum _{k=0}^{\mathrm{\infty }}\left(\psi \left(k+1\right)+\psi \left(n+k+1\right)\right)\frac{{(\frac{1}{4}{z}^2)}^k}{k!(n+k)!},$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.11

Referenced by:

§10.30(i), §10.65(ii)

Permalink:

http://dlmf.nist.gov/10.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

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10.31.2 $$K_0\left(z\right)=-\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)I_0\left(z\right)+\frac{\frac{1}{4}{z}^2}{{(1!)}^2}+(1+\frac{1}{2})\frac{{(\frac{1}{4}{z}^2)}^2}{{(2!)}^2}+(1+\frac{1}{2}+\frac{1}{3})\frac{{(\frac{1}{4}{z}^2)}^3}{{(3!)}^2}+\mathrm{\cdots }.$$

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Symbols:

$\gamma$: Euler’s constant, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.6.13

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

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24: 19.3 Graphics

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►►

Figure 19.3.1: $K\left(k\right)$ and $E\left(k\right)$ as functions of $k^2$ for $-2\le k^2\le 1$. Graphs of $K^{\prime }\left(k\right)$ and $E^{\prime }\left(k\right)$ are the mirror images in the vertical line $k^2=\frac{1}{2}$. Magnify

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Figure 19.3.4: $E(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. If ${\sin }^2\varphi =1$ ($\ge k^2$), then the function reduces to $E\left(k\right)$, with value 1 at $k^2=1$. … Magnify 3D Help

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Figure 19.3.5: $\mathrm{\Pi }({\alpha }^2,k)$ as a function of $k^2$ and ${\alpha }^2$ for , $-2\le {\alpha }^2\le 2$. …As ${\alpha }^2\to 1+$ it has the limit $K\left(k\right)-(E\left(k\right)/k_{}^{\prime }{}_{}{}^2)$. … Magnify 3D Help

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Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . …Its value tends to $-\mathrm{\infty }$ as $k^2\to 1+$ by (19.6.6), and to the negative of the second lemniscate constant (see (19.20.22)) as $k^2(={\csc }^2\varphi )\to 2-$. Magnify 3D Help

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25: 10.32 Integral Representations

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10.32.5 $$K_0\left(z\right)=-\frac{1}{\pi }{\int }_0^{\pi }{\mathrm{e}}^{\pm z\cos \theta }\left(\gamma +\ln \left(2z{(\sin \theta )}^2\right)\right)d\theta .$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

9.6.17

Permalink:

http://dlmf.nist.gov/10.32.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32 and Ch.10

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10.32.6 $$K_0\left(x\right)={\int }_0^{\mathrm{\infty }}\cos \left(x\sinh t\right)dt={\int }_0^{\mathrm{\infty }}\frac{\cos \left(xt\right)}{\sqrt{t^2+1}}dt,$$ $x>0$.

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $x$: real variable

A&S Ref:

9.6.21

Permalink:

http://dlmf.nist.gov/10.32.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32 and Ch.10

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10.32.17 $$K_{\mu }\left(z\right)K_{\nu }\left(z\right)=2{\int }_0^{\mathrm{\infty }}{K}_{\mu \pm \nu }\left(2z\cosh t\right)\cosh \left((\mu \mp \nu )t\right)dt,$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.32.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32 and Ch.10

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10.32.18 $$K_{\nu }\left(z\right)K_{\nu }\left(\zeta \right)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{t}{2}-\frac{z^2+{\zeta }^2}{2t}\right)K_{\nu }\left(\frac{z\zeta }{t}\right)\frac{dt}{t},$$ , , .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.32.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32 and Ch.10

… ►For similar integrals for $J_{\nu }\left(z\right)K_{\nu }\left(z\right)$ and $I_{\nu }\left(z\right)K_{\nu }\left(z\right)$ see Paris and Kaminski (2001, p. 116). …

26: 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

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

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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: 24.15 Related Sequences of Numbers

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§24.15(iii) Stirling Numbers

►The Stirling numbers of the first kind $s(n,m)$, and the second kind $S(n,m)$, are as defined in §26.8(i). ►

24.15.6 $$B_n=\sum _{k=0}^n{(-1)}^k\frac{k!S(n,k)}{k+1},$$

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $!$: factorial (as in $n!$), $k$: integer and $n$: integer

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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24.15.7 $$B_n=\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n+1}{k+1}\right)S(n+k,k)/\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right),$$

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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24.15.9 $$p\frac{B_n}{n}\equiv S(p-1+n,p-1)(mod{p}^2),$$ $1\le n\le p-2$,

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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29: Brian R. Judd

… ►Judd’s books include Operator Techniques in Atomic Spectroscopy, published by McGraw-Hill in 1963 and reprinted by Princeton University Press in 1998, Second Quantization and Atomic Spectroscopy, published by Johns Hopkins in 1967, Topics in Atomic and Nuclear Theory (with J. …

30: 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

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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).