relation to Gudermannian function

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1: 19.10 Relations to Other Functions

… ►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 $\mathrm{gd}\left(x\right)$ and its inverse ${\mathrm{gd}}^{-1}\left(x\right)$ (§4.23(viii)), see (19.6.8) and …

2: 19.6 Special Cases

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(ii), §19.6 and Ch.19

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3: 22.16 Related Functions

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22.16.4 $$\mathrm{am}(x,0)=x,$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function and $x$: real

Permalink:

http://dlmf.nist.gov/22.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

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22.16.8 $$\mathrm{am}(x,k)=\mathrm{gd}x-\frac{1}{4}k_{}^{\prime }{}_{}{}^2(x-\sinh x\cosh x)\mathrm{sech}x+O\left(k_{}^{\prime }{}_{}{}^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{gd}x$: Gudermannian function, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.4

Permalink:

http://dlmf.nist.gov/22.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

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