of the first and second kinds

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11: 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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$$E\left(1/k\right)=(1/k)\left(E\left(k\right)\pm \mathrm{i}E\left(k^{\prime }\right)-k_{}^{\prime }{}_{}{}^{2}K\left(k\right)\mp \mathrm{i}{k}^{2}K\left(k^{\prime }\right)\right),$$

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$$E\left(1/k^{\prime }\right)=(1/k^{\prime })\left(E\left(k^{\prime }\right)\mp \mathrm{i}E\left(k\right)-k^{2}K\left(k^{\prime }\right)\pm \mathrm{i}k_{}^{\prime }{}_{}{}^{2}K\left(k\right)\right),$$

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$E(\varphi ,k_1)=(E(\beta ,k)-k_{}^{\prime }{}_{}{}^{2}F(\beta ,k))/k,$

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12: 10.31 Power Series

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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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13: 19.6 Special Cases

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$K\left(0\right)=E\left(0\right)=K^{\prime }\left(1\right)=E^{\prime }\left(1\right)=\frac{1}{2}\pi ,$

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$$\mathrm{\Pi }({\alpha }^2,k)\to K\left(k\right)-\left(E\left(k\right)/k_{}^{\prime }{}_{}{}^2\right),$$ ${\alpha }^2\to 1+$,

… ►Exact values of $K\left(k\right)$ and $E\left(k\right)$ for various special values of $k$ are given in Byrd and Friedman (1971, 111.10 and 111.11) and Cooper et al. (2006). … ►

$$\mathrm{\Pi }(\varphi ,1,k)=F(\varphi ,k)-\frac{1}{k_{}^{\prime }{}_{}{}^2}(E(\varphi ,k)-\mathrm{\Delta }\tan \varphi ).$$

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14: 19.38 Approximations

… ►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …

15: 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 19.3.5: $\mathrm{\Pi }({\alpha }^2,k)$ 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 19.3.11: $\mathrm{\Re }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the branch cut ($k^2>1$) it has the value $kE\left(1/k\right)+(k_{}^{\prime }{}_{}{}^2/k)K\left(1/k\right)$, with limit 1 as $k^2\to 1+$. Magnify 3D Help

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Figure 19.3.12: $\mathrm{\Im }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the upper edge of the branch cut ($k^2>1$) it has the (negative) value $K\left(k^{\prime }\right)-E\left(k^{\prime }\right)$, with limit 0 as $k^2\to 1+$. Magnify 3D Help

16: 10.38 Derivatives with Respect to Order

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10.38.2 $$\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }=\frac{1}{2}\pi \csc \left(\nu \pi \right)\left(\frac{\partial I_{-\nu }\left(z\right)}{\partial \nu }-\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }\right)-\pi \cot \left(\nu \pi \right)K_{\nu }\left(z\right),$$ $\nu \notin \mathbb{Z}$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\mathbb{Z}$: set of all integers, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\notin$: not an element of, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.43

Referenced by:

(10.38.1), §10.38, §10.38

Permalink:

http://dlmf.nist.gov/10.38.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38 and Ch.10

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

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17: 10.27 Connection Formulas

… ►Other solutions of (10.25.1) are $I_{-\nu }\left(z\right)$ and $K_{-\nu }\left(z\right)$. … ►

10.27.2 $$I_{-\nu }\left(z\right)=I_{\nu }\left(z\right)+(2/\pi )\sin \left(\nu \pi \right)K_{\nu }\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.2

Permalink:

http://dlmf.nist.gov/10.27.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.4 $$K_{\nu }\left(z\right)=\frac{1}{2}\pi \frac{I_{-\nu }\left(z\right)-I_{\nu }\left(z\right)}{\sin \left(\nu \pi \right)}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.2

Referenced by:

§10.30(i), §10.31, §10.38, §10.45, §10.47(iv), §8.6(i)

Permalink:

http://dlmf.nist.gov/10.27.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.5 $$K_n\left(z\right)=\frac{{(-1)}^{n-1}}{2}\left({\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}+{\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =-n}\right),$$ $n=0,\pm 1,\pm 2,\mathrm{\dots }$.

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Symbols:

$I_{\nu }\left(z\right)$: 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, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.27.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.9 $$\pi \mathrm{i}{J}_{\nu }\left(z\right)={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{-\pi \mathrm{i}/2}\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\pi \mathrm{i}/2}\right),$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.67(i)

Permalink:

http://dlmf.nist.gov/10.27.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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18: 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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19: 10.30 Limiting Forms

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10.30.1 $$I_{\nu }\left(z\right)\sim {(\frac{1}{2}z)}^{\nu }/\mathrm{\Gamma }\left(\nu +1\right),$$ $\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

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

A&S Ref:

9.6.7

Referenced by:

§10.30(i)

Permalink:

http://dlmf.nist.gov/10.30.E1

Encodings:

pMML, png, TeX

See also:

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

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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).

20: 14.18 Sums

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14.18.3 $${𝖰}_{\nu }\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \varphi \right)={𝖯}_{\nu }\left(\cos {\theta }_1\right){𝖰}_{\nu }\left(\cos {\theta }_2\right)+2\sum _{m=1}^{\mathrm{\infty }}{(-1)}^m{𝖯}_{\nu }^{-m}\left(\cos {\theta }_1\right){𝖰}_{\nu }^m\left(\cos {\theta }_2\right)\cos \left(m\varphi \right).$$

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $\sin z$: sine function, $m$: nonnegative integer, $\nu$: general degree, ${\theta }_1$, ${\theta }_2$: real and $\varphi$: real

Referenced by:

§14.18(ii), Ch.14

Permalink:

http://dlmf.nist.gov/14.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.18(ii), §14.18 and Ch.14

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14.18.5 $$Q_{\nu }\left(\cosh {\xi }_1\cosh {\xi }_2-\sinh {\xi }_1\sinh {\xi }_2\cos \varphi \right)=P_{\nu }\left(\cosh {\xi }_1\right)Q_{\nu }\left(\cosh {\xi }_2\right)+2\sum _{m=1}^{\mathrm{\infty }}{(-1)}^m{P}_{\nu }^{-m}\left(\cosh {\xi }_1\right)Q_{\nu }^m\left(\cosh {\xi }_2\right)\cos \left(m\varphi \right).$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $m$: nonnegative integer, $\nu$: general degree, $\varphi$: real and ${\xi }_1$, ${\xi }_2$: positive

Referenced by:

§14.18(ii)

Permalink:

http://dlmf.nist.gov/14.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.18(ii), §14.18 and Ch.14

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14.18.7 $$(x-y)\sum _{k=0}^n(2k+1)P_k\left(x\right)Q_k\left(y\right)=(n+1)\left(P_{n+1}\left(x\right)Q_n\left(y\right)-P_n\left(x\right)Q_{n+1}\left(y\right)\right)-1.$$

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Symbols:

$P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind and $n$: nonnegative integer

A&S Ref:

8.9.2

Referenced by:

§14.18(iii), Erratum (V1.0.7) for Subsection 14.18(iii)

Permalink:

http://dlmf.nist.gov/14.18.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.18(iii), §14.18(iii), §14.18 and Ch.14

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