trigonometric arguments

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21: 4.1 Special Notation

… ►The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. …

22: 19.2 Definitions

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19.2.4 $$F(\varphi ,k)={\int }_0^{\varphi }\frac{d\theta }{\sqrt{1-k^2{\sin }^2\theta }}={\int }_0^{\sin \varphi }\frac{dt}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}},$$

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Defines:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.5, §19.6(ii)

Permalink:

http://dlmf.nist.gov/19.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.5 $$E(\varphi ,k)={\int }_0^{\varphi }\sqrt{1-k^2{\sin }^2\theta }d\theta ={\int }_0^{\sin \varphi }\frac{\sqrt{1-k^2{t}^2}}{\sqrt{1-t^2}}dt.$$

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Defines:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.6(iii), §22.16(ii)

Permalink:

http://dlmf.nist.gov/19.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.6 $$D(\varphi ,k)={\int }_0^{\varphi }\frac{{\sin }^2\theta d\theta }{\sqrt{1-k^2{\sin }^2\theta }}={\int }_0^{\sin \varphi }\frac{t^{2}dt}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}}=(F(\varphi ,k)-E(\varphi ,k))/k^2.$$

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Defines:

$D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type

Symbols:

$dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

►

19.2.7 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)={\int }_0^{\varphi }\frac{d\theta }{\sqrt{1-k^2{\sin }^2\theta }(1-{\alpha }^2{\sin }^2\theta )}={\int }_0^{\sin \varphi }\frac{dt}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}(1-{\alpha }^2{t}^2)}.$$

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Defines:

$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

…

23: 19.23 Integral Representations

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19.23.6 $$4\pi R_F(x,y,z)={\int }_0^{2\pi }{\int }_0^{\pi }\frac{\sin \theta d\theta d\varphi }{{(x{\sin }^2\theta {\cos }^2\varphi +y{\sin }^2\theta {\sin }^2\varphi +z{\cos }^2\theta )}^{1/2}},$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.16(ii), §19.23

Permalink:

http://dlmf.nist.gov/19.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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19.23.6_5 $$R_G(x,y,z)=\frac{1}{4\pi }{\int }_0^{2\pi }{\int }_0^{\pi }{\left(x{\sin }^2\theta {\cos }^2\varphi +y{\sin }^2\theta {\sin }^2\varphi +z{\cos }^2\theta \right)}^{1/2}\sin \theta d\theta d\varphi ,$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

(19.16.3), (19.16.3), §19.16(i), §19.16(ii), §19.16(ii), §19.20(ii), §19.23, §19.23, §19.25(vi), Erratum (V1.1.0) for Rearrangement

Permalink:

http://dlmf.nist.gov/19.23.E6_5

Encodings:

pMML, png, TeX

Rearrangement (effective with 1.1.0):

This equation, which was originally (19.16.3), was moved here.

See also:

Annotations for §19.23 and Ch.19

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19.23.9 $$R_{-a}(𝐛;𝐳)=\frac{4\mathrm{\Gamma }\left(b_1+b_2+b_3\right)}{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\Gamma }\left(b_3\right)}{\int }_0^{\pi /2}{\int }_0^{\pi /2}{\left(\sum _{j=1}^3{z}_j{l}_j^2\right)}^{-a}\prod _{j=1}^3{l}_j^{2b_j-1}\sin \theta d\theta d\varphi ,$$ $b_j>0$, $\mathrm{\Re }{z}_j>0$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $l$: nonnegative integer and $\varphi$: real or complex argument

Referenced by:

§19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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24: 20.10 Integrals

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

►For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193). …

25: 19.16 Definitions

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19.16.12 $$R_{-a}(b_1,\mathrm{\dots },b_4;c-1,c-k^2,c,c-{\alpha }^2)=\frac{2{({\sin }^2\varphi )}^{1-a^{\prime }}}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\varphi }{(\sin \theta )}^{2a-1}{({\sin }^2\varphi -{\sin }^2\theta )}^{a^{\prime }-1}{(\cos \theta )}^{1-2b_1}{(1-k^2{\sin }^2\theta )}^{-b_2}{(1-{\alpha }^2{\sin }^2\theta )}^{-b_4}d\theta ,$$

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.16(ii), §19.25(i)

Permalink:

http://dlmf.nist.gov/19.16.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(ii), §19.16 and Ch.19

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26: 19.5 Maclaurin and Related Expansions

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19.5.4_1 $$F(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.8))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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19.5.4_2 $$E(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},-\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.9))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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19.5.4_3 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{F}_1(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};{\sin }^2\varphi ,{\alpha }^2{\sin }^2\varphi )k^{2m},$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Source:

Carlson (1961b, (2.11))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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27: Errata

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23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right)$$

Originally, the factor of 2 was missing from the denominator of the argument of the $\cot$ function.

Reported by Blagoje Oblak on 2019-05-27

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22.19.2 $$\sin \left(\frac{1}{2}\theta (t)\right)=\sin \left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\sin \left(\frac{1}{2}\alpha \right))$$

Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.

Reported 2014-03-05 by Svante Janson.

… ►

14.19.2 $$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;1-{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$

Originally the argument to $𝐅$ in this equation was incorrect (${\mathrm{e}}^{-2\xi }$, rather than $1-{\mathrm{e}}^{-2\xi }$), and the condition on $\mu$ was too weak ($\mu \ne \frac{1}{2}$, rather than $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$). Also, the factor multiplying $𝐅$ was rewritten to clarify the poles; originally it was $\frac{\mathrm{\Gamma }\left(1-2\mu \right)2^{2\mu }}{\mathrm{\Gamma }\left(1-\mu \right){\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}$.

Reported 2010-11-02 by Alvaro Valenzuela.

…

28: 19.25 Relations to Other Functions

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19.25.27 $$2{(z-x)}^{-1/2}{R}_G(x,y,z)=E(\varphi ,k)+{(\cot \varphi )}^{2}F(\varphi ,k)+(\cot \varphi )\sqrt{1-k^2{\sin }^2\varphi }.$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\cot z$: cotangent function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(iii)

Permalink:

http://dlmf.nist.gov/19.25.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(iii), §19.25 and Ch.19

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29: 10.67 Asymptotic Expansions for Large Argument

§10.67 Asymptotic Expansions for Large Argument

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§10.67(i) ${\mathrm{ber}}_{\nu }x,{\mathrm{bei}}_{\nu }x,{\ker }_{\nu }x,{\mathrm{kei}}_{\nu }x$, and Derivatives

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10.67.3 $${\mathrm{ber}}_{\nu }x\sim \frac{{\mathrm{e}}^{x/\sqrt{2}}}{{(2\pi x)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{3k}{4}-\frac{1}{8}\right)\pi \right)-\frac{1}{\pi }(\sin \left(2\nu \pi \right){\ker }_{\nu }x+\cos \left(2\nu \pi \right){\mathrm{kei}}_{\nu }x),$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.1, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(i), §10.67(i), §10.68(iii), §10.69

Permalink:

http://dlmf.nist.gov/10.67.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

… ► ►

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

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30: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

►

§4.23(i) General Definitions

… ►Graphs of the principal values for real arguments are given in §4.15. This section also includes conformal mappings, and surface plots for complex arguments. … ►

§4.23(iv) Logarithmic Forms

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