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11: 28.15 Expansions for Small $q$

… ►

28.15.3 $${\mathrm{me}}_{\nu }(z,q)={\mathrm{e}}^{\mathrm{i}\nu z}-\frac{q}{4}\left(\frac{1}{\nu +1}{\mathrm{e}}^{\mathrm{i}(\nu +2)z}-\frac{1}{\nu -1}{\mathrm{e}}^{\mathrm{i}(\nu -2)z}\right)+\frac{q^2}{32}\left(\frac{1}{(\nu +1)(\nu +2)}{\mathrm{e}}^{\mathrm{i}(\nu +4)z}+\frac{1}{(\nu -1)(\nu -2)}{\mathrm{e}}^{\mathrm{i}(\nu -4)z}-\frac{2({\nu }^2+1)}{{({\nu }^2-1)}^2}{\mathrm{e}}^{\mathrm{i}\nu z}\right)+\mathrm{\cdots };$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.3.17 (only two terms and without normalization)

Permalink:

http://dlmf.nist.gov/28.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(ii), §28.15 and Ch.28

…

12: 19.10 Relations to Other Functions

… ►If $y>0$ is assumed (without loss of generality), then …

13: Guide to Searching the DLMF

… ►

Table 1: Query Examples

► ►►►

Query	Matching records contain
…
int adj sin	$\int$ immediately followed by $\sin$ without any intervening terms.
…

►

…

14: 14.16 Zeros

… ►For all cases concerning ${𝖯}_{\nu }^{\mu }\left(x\right)$ and $P_{\nu }^{\mu }\left(x\right)$ we assume that $\nu \ge -\frac{1}{2}$ without loss of generality (see (14.9.5) and (14.9.11)). …

15: 21.3 Symmetry and Quasi-Periodicity

… ►Because of this property, the elements of $𝜶$ and $𝜷$ are usually restricted to $[0,1)$, without loss of generality. …

16: 26.3 Lattice Paths: Binomial Coefficients

… ► $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ is the number of ways of choosing $n$ objects from a collection of $m$ distinct objects without regard to order. …

17: 32.2 Differential Equations

… ►If $\gamma \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\gamma =1$ and $\delta =-1$, without loss of generality, by rescaling $w$ and $z$ if necessary. If $\gamma =0$ and $\alpha \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\alpha =1$ and $\delta =-1$, without loss of generality. Lastly, if $\delta =0$ and $\beta \gamma \ne 0$, then set $\beta =-1$ and $\gamma =1$, without loss of generality. ►If $\delta \ne 0$ in ${\text{P}}_{\text{V}}$, then set $\delta =-\frac{1}{2}$, without loss of generality. …

18: 8.2 Definitions and Basic Properties

… ►without restrictions on the integration paths. … ►without restriction on $a$. …

19: 10.38 Derivatives with Respect to Order

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10.38.1 $$\frac{\partial I_{\pm \nu }\left(z\right)}{\partial \nu }=\pm I_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.42

Referenced by:

§10.38, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.38.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial I_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.38.2).

See also:

Annotations for §10.38 and Ch.10

… ►For $\partial I_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.38.1), (10.38.2), and (10.38.4). ►

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

… ►

10.38.6 $${\frac{\partial I_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=-\frac{1}{\sqrt{2\pi x}}\left(E_1\left(2x\right){\mathrm{e}}^x\pm \mathrm{Ei}\left(2x\right){\mathrm{e}}^{-x}\right),$$

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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, $\mathrm{Ei}\left(x\right)$: exponential integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first 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:

10.2.32 and 10.2.33 (both corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

►

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

ⓘ

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

…

20: 14.13 Trigonometric Expansions

… ►

14.13.1 $${𝖯}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{2^{\mu +1}{(\sin \theta )}^{\mu }}{{\pi }^{1/2}}\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\sin \left((\nu +\mu +2k+1)\theta \right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Volkmer (2021)

A&S Ref:

8.7.1

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

►

14.13.2 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)={\pi }^{1/2}{2}^{\mu }{(\sin \theta )}^{\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\cos \left((\nu +\mu +2k+1)\theta \right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Cohl et al. (2021, Theorem 6.2)

A&S Ref:

8.7.2

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

… ►If then they converge, but, if $\theta \ne \frac{1}{2}\pi$, they do not converge absolutely. … ►

14.13.3 $${𝖯}_n\left(\cos \theta \right)=\frac{2^{2n+2}{(n!)}^2}{\pi (2n+1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{1\cdot 3\mathrm{\cdots }(2k-1)}{k!}\frac{(n+1)(n+2)\mathrm{\cdots }(n+k)}{(2n+3)(2n+5)\mathrm{\cdots }(2n+2k+1)}\sin \left((n+2k+1)\theta \right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $n$: nonnegative integer and : variable

A&S Ref:

8.7.3

Permalink:

http://dlmf.nist.gov/14.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

►

14.13.4 $${𝖰}_n\left(\cos \theta \right)=\frac{2^{2n+1}{(n!)}^2}{(2n+1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{1\cdot 3\mathrm{\cdots }(2k-1)}{k!}\frac{(n+1)(n+2)\mathrm{\cdots }(n+k)}{(2n+3)(2n+5)\mathrm{\cdots }(2n+2k+1)}\cos \left((n+2k+1)\theta \right),$$

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Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $n$: nonnegative integer and : variable

A&S Ref:

8.7.4

Permalink:

http://dlmf.nist.gov/14.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

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