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11: 14.27 Zeros

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, $\mu \notin \mathbb{Z}$, $\nu \in \mathbb{Z}$, and $\sin \left((\mu -\nu )\pi \right)$ and $\sin \left(\mu \pi \right)$ have opposite signs.

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12: 10.7 Limiting Forms

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$$Y_{\nu }\left(z\right)=\sqrt{2/(\pi z)}\left(\sin \left(z-\frac{1}{2}\nu \pi -\frac{1}{4}\pi \right)+{\mathrm{e}}^{|\mathrm{\Im }z|}o\left(1\right)\right),$$ .

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13: 19.2 Definitions

… ►If , then $k_c$ is pure imaginary. …

14: 4.26 Integrals

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4.26.6 $$\int \cot xdx=\ln \left(\sin x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.118

Permalink:

http://dlmf.nist.gov/4.26.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.12

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $m$: integer

A&S Ref:

4.3.142

Referenced by:

§4.26(iii)

Permalink:

http://dlmf.nist.gov/4.26.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

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15: 4.43 Cubic Equations

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4.43.2 $$z^3+pz+q=0$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $p$: real constant and $q$: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=C$, when and $C\le 1$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=C$, when and $C>1$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=D$, when $p>0$.

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

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$A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=4q/A^3$, when $4p^3+27q^2\le 0$.

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16: 2.10 Sums and Sequences

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2.10.36 $$P_n\left(\cos \alpha \right)={\left(\frac{2}{\pi n\sin \alpha }\right)}^{1/2}\cos \left(n\alpha +\frac{1}{2}\alpha -\frac{1}{4}\pi \right)+o\left(n^{-1}\right).$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $o\left(x\right)$: order less than and $\sin z$: sine function

Permalink:

http://dlmf.nist.gov/2.10.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10(iv), §2.10 and Ch.2

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17: 14.5 Special Values

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14.5.11 $${𝖯}_{\nu }^{1/2}\left(\cos \theta \right)={\left(\frac{2}{\pi \sin \theta }\right)}^{1/2}\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right),$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.12

Permalink:

http://dlmf.nist.gov/14.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

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14.5.12 $${𝖯}_{\nu }^{-1/2}\left(\cos \theta \right)={\left(\frac{2}{\pi \sin \theta }\right)}^{1/2}\frac{\sin \left(\left(\nu +\frac{1}{2}\right)\theta \right)}{\nu +\frac{1}{2}},$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.14

Permalink:

http://dlmf.nist.gov/14.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

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14.5.18 $${𝖯}_{\nu }^{-\nu }\left(\cos \theta \right)=\frac{{(\sin \theta )}^{\nu }}{2^{\nu }\mathrm{\Gamma }\left(\nu +1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.17

Permalink:

http://dlmf.nist.gov/14.5.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iv), §14.5 and Ch.14

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14.5.20 $${𝖯}_{\frac{1}{2}}\left(\cos \theta \right)=\frac{2}{\pi }\left(2E\left(\sin \left(\frac{1}{2}\theta \right)\right)-K\left(\sin \left(\frac{1}{2}\theta \right)\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function and : variable

A&S Ref:

8.13.11 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

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14.5.21 $${𝖯}_{-\frac{1}{2}}\left(\cos \theta \right)=\frac{2}{\pi }K\left(\sin \left(\frac{1}{2}\theta \right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function and : variable

A&S Ref:

8.13.9

Permalink:

http://dlmf.nist.gov/14.5.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

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18: 4.40 Integrals

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4.40.9 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{ax}}{{\left(\cosh \left(\frac{1}{2}x\right)\right)}^2}dx=\frac{4\pi a}{\sin \left(\pi a\right)},$$ ,

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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, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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19: 1.14 Integral Transforms

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Table 1.14.1: Fourier transforms.

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$f(t)$	${\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(t){\mathrm{e}}^{\mathrm{i}xt}dt$
	$\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{\sin \left(ax\right)}{x}}$
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${\displaystyle \frac{\sinh \left(at\right)}{\sinh \left(\pi t\right)}}$	${\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle \frac{\sin a}{\cosh x+\cos a}}$,
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Table 1.14.2: Fourier cosine transforms.

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$f(t)$	$\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle {\int }_0^{\mathrm{\infty }}}f(t)\cos \left(xt\right)dt$,	$x>0$
	$\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{\sin \left(ax\right)}{x}}$
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Table 1.14.5: Mellin transforms.

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$f(x)$	${\displaystyle {\int }_0^{\mathrm{\infty }}}{x}^{s-1}f(x)dx$
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${\displaystyle \frac{1+x\cos \theta }{1+2x\cos \theta +x^2}}$	${\displaystyle \frac{\pi \cos \left(s\theta \right)}{\sin \left(s\pi \right)}}$,	,
${\displaystyle \frac{x\sin \theta }{1+2x\cos \theta +x^2}}$	${\displaystyle \frac{\pi \sin \left(s\theta \right)}{\sin \left(s\pi \right)}}$,	,

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20: 25.8 Sums

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25.8.8 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{k}{z}^{2k}=\ln \left(\frac{\pi z}{\sin \left(\pi z\right)}\right),$$ .

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $k$: nonnegative integer and $z$: complex variable

Keywords:

infinite series

Proof sketch:

Derivable from (25.8.6) by dividing by $z$ and integrating.

Permalink:

http://dlmf.nist.gov/25.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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