19 Elliptic Integrals Legendre’s Integrals 19.9 Inequalities 19.11 Addition Theorems

§19.10 Relations to Other Functions

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Permalink:: http://dlmf.nist.gov/19.10
See also:: Annotations for Ch.19

§19.10(i) Theta and Elliptic Functions

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Keywords:: Jacobian elliptic functions, Legendre’s elliptic integrals, Weierstrass elliptic functions, relations to other functions, theta functions
Permalink:: http://dlmf.nist.gov/19.10.i
See also:: Annotations for §19.10 and Ch.19

For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. See also Erdélyi et al. (1953b, Chapter 13).

§19.10(ii) Elementary Functions

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Keywords:: Gudermannian function, $R_{C}$ -function, elementary functions, relation to Gudermannian function, relation to $R_{C}$ -function, relation to elementary functions
Notes:: For (19.10.1) see (19.2.17). For (19.10.2) use (19.6.8).
Permalink:: http://dlmf.nist.gov/19.10.ii
See also:: Annotations for §19.10 and Ch.19

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

19.10.1	$\displaystyle\ln\left(x/y\right)$	$\displaystyle=(x-y)R_{C}\left(\tfrac{1}{4}(x+y)^{2},xy\right),$
	$\displaystyle\operatorname{arctan}\left(x/y\right)$	$\displaystyle=xR_{C}\left(y^{2},y^{2}+x^{2}\right),$
	$\displaystyle\operatorname{arctanh}\left(x/y\right)$	$\displaystyle=xR_{C}\left(y^{2},y^{2}-x^{2}\right),$
	$\displaystyle\operatorname{arcsin}\left(x/y\right)$	$\displaystyle=xR_{C}\left(y^{2}-x^{2},y^{2}\right),$
	$\displaystyle\operatorname{arcsinh}\left(x/y\right)$	$\displaystyle=xR_{C}\left(y^{2}+x^{2},y^{2}\right),$
	$\displaystyle\operatorname{arccos}\left(x/y\right)$	$\displaystyle=(y^{2}-x^{2})^{1/2}R_{C}\left(x^{2},y^{2}\right),$
	$\displaystyle\operatorname{arccosh}\left(x/y\right)$	$\displaystyle=(x^{2}-y^{2})^{1/2}R_{C}\left(x^{2},y^{2}\right).$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\operatorname{arctan}\NVar{z}$ : arctangent function and $\ln\NVar{z}$ : principal branch of logarithm function Referenced by: §19.10(ii) Permalink: http://dlmf.nist.gov/19.10.E1 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png See also: Annotations for §19.10(ii), §19.10 and Ch.19

In each case when $y=1$ , the quantity multiplying $R_{C}$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0.

For relations to the Gudermannian function $\operatorname{gd}\left(x\right)$ and its inverse ${\operatorname{gd}^{-1}}\left(x\right)$ (§4.23(viii)), see (19.6.8) and

19.10.2		$(\sinh\phi)R_{C}\left(1,{\cosh}^{2}\phi\right)=\operatorname{gd}\left(\phi% \right).$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\operatorname{gd}\NVar{x}$ : Gudermannian function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $\phi$ : real or complex argument Referenced by: §19.10(ii), §19.6(ii) Permalink: http://dlmf.nist.gov/19.10.E2 Encodings: TeX, pMML, png See also: Annotations for §19.10(ii), §19.10 and Ch.19

19.9 Inequalities 19.11 Addition Theorems

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§19.10 Relations to Other Functions

Contents

§19.10(i) Theta and Elliptic Functions

§19.10(ii) Elementary Functions

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