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21: 14.13 Trigonometric Expansions

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14.13.1

\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{2^{\mu+1}(\sin\theta)^{\mu% }}{\pi^{1/2}}\*\sum_{k=0}^{\infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\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),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $\mu$ : general order, $\nu$ : general degree and $0<\theta<\pi$ : 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

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14.13.2

\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{1/2}2^{\mu}(\sin\theta)^{% \mu}\*\sum_{k=0}^{\infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\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:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $\mu$ : general order, $\nu$ : general degree and $0<\theta<\pi$ : 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

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14.13.3

\mathsf{P}_{n}\left(\cos\theta\right)=\frac{2^{2n+2}(n!)^{2}}{\pi(2n+1)!}\*% \sum_{k=0}^{\infty}\frac{1\cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k% )}{(2n+3)(2n+5)\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, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer and $0<\theta<\pi$ : 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

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22: 19.14 Reduction of General Elliptic Integrals

… ►Cases in which

\cos\phi<0

can be included by application of (19.2.10). …In (19.14.4)

0\leq y<x

, each quadratic polynomial is positive on the interval

(y,x)

, and

\alpha,\beta,\gamma

is a permutation of

0,a_{1}b_{2},a_{2}b_{1}

(not all 0 by assumption) such that

\alpha\leq\beta\leq\gamma

. … ►

19.14.5

{\sin}^{2}\phi=\frac{\gamma-\alpha}{U^{2}+\gamma},

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Symbols:: $\sin\NVar{z}$ : sine function and $\phi$ : real or complex argument
Permalink:: http://dlmf.nist.gov/19.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

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19.14.7

{\sin}^{2}\phi=\frac{(\gamma-\alpha)x^{2}}{a_{1}a_{2}+\gamma x^{2}}.

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Symbols:: $\sin\NVar{z}$ : sine function and $\phi$ : real or complex argument
Permalink:: http://dlmf.nist.gov/19.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

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19.14.8

{\sin}^{2}\phi=\frac{\gamma-\alpha}{b_{1}b_{2}y^{2}+\gamma}.

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function and $\phi$ : real or complex argument
Permalink:: http://dlmf.nist.gov/19.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

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23: 14.16 Zeros

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§14.16(ii) Interval $-1<x<1$

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(a)

$\mu\leq 0$ .

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(c)

$\mu>0$ , $n<m$ , and $m-n$ is odd.

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§14.16(iii) Interval $1<x<\infty$

… ►

(a)

$\mu>0$ , $\mu>\nu$ , $\mu\notin\mathbb{Z}$ , and $\sin\left((\mu-\nu)\pi\right)$ and $\sin\left(\mu\pi\right)$ have opposite signs.

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24: 19.30 Lengths of Plane Curves

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x=a\sin\phi,

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19.30.2

s=a\int_{0}^{\phi}\sqrt{1-k^{2}{\sin}^{2}\theta}\,\mathrm{d}\theta.

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $s$ : arclength
Permalink:: http://dlmf.nist.gov/19.30.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(i), §19.30 and Ch.19

►When

0\leq\phi\leq\tfrac{1}{2}\pi

, … ►

y=b\sqrt{t}

0\leq t<\infty

… ►For

0\leq\theta\leq\tfrac{1}{4}\pi

, the arclength

s

of Bernoulli’s lemniscate …

25: 5.9 Integral Representations

… ►

5.9.7

\Gamma\left(z\right)\sin\left(\tfrac{1}{2}\pi z\right)=\int_{0}^{\infty}t^{z-1% }\sin t\,\mathrm{d}t,

-1<\Re z<1

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $z$ : complex variable
Referenced by:: §5.9(i)
Permalink:: http://dlmf.nist.gov/5.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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26: 8.21 Generalized Sine and Cosine Integrals

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8.21.4

\operatorname{si}\left(a,z\right)=\int_{z}^{\infty}t^{a-1}\sin t\,\mathrm{d}t,

\Re a<1

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $a$ : parameter
A&S Ref:: 6.5.8 (This function is called $S(z,a)$ in AMS 55.)
Referenced by:: §8.21(iii), §8.21(iii), §8.21(v), §8.21(vii)
Permalink:: http://dlmf.nist.gov/8.21.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §8.21(iii), §8.21 and Ch.8

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27: 2.5 Mellin Transform Methods

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2.5.9

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(1-z\right)=\frac{\pi}{\sin\left(\pi z% \right)},

0<\Re z<1

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Symbols:: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $f(t)=1/(1+t)$ : function
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/2.5.E9
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{M}(f;z)$ .
See also:: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

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2.5.10

\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right)=\frac{2^{z-1}\Gamma\left(% \nu+\frac{1}{2}z\right)}{{\Gamma}^{2}\left(1-\frac{1}{2}z\right)\Gamma\left(1+% \nu-\frac{1}{2}z\right)\Gamma\left(z\right)}\frac{\pi}{\sin\left(\pi z\right)},

-2\nu<\Re z<1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $h(t)={J_{\nu}}^{2}\left(t\right)$ : function and $f(t)=1/(1+t)$ : function
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/2.5.E10
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{M}(f;z)$ .
See also:: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

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28: 5.13 Integrals

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5.13.2

{\frac{1}{2\pi}\int_{-\infty}^{\infty}|\Gamma\left(a+it\right)|^{2}e^{(2b-\pi)% t}\,\mathrm{d}t=\frac{\Gamma\left(2a\right)}{(2\sin b)^{2a}}},

a>0

0<b<\pi

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.13 and Ch.5

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29: 4.45 Methods of Computation

… ►and since

|\theta|\leq\frac{1}{2}\pi=1.57\dots

\sin\theta

and

\cos\theta

can be computed straightforwardly from (4.19.1) and (4.19.2). …

30: 24.7 Integral Representations

… ►

24.7.11

B_{n}\left(x\right)=\frac{1}{2\pi i}\int_{-c-i\infty}^{-c+i\infty}(x+t)^{n}% \left(\frac{\pi}{\sin\left(\pi t\right)}\right)^{2}\,\mathrm{d}t,

0<c<1

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : integer, $t$ : real or complex and $x$ : real or complex
Keywords:: Mellin transform
Referenced by:: §24.7(ii)
Permalink:: http://dlmf.nist.gov/24.7.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.7(ii), §24.7(ii), §24.7 and Ch.24

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