relation to inverse Gudermannian function

SearchAdvancedHelp

(0.003 seconds)

3 matching pages

1: 19.6 Special Cases

… ►

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

ⓘ

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

…

2: 19.10 Relations to Other Functions

… ►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 …

3: 22.16 Related Functions

… ►

§22.16(i) Jacobi’s Amplitude ($\mathrm{am}$) Function

… ►For the Gudermannian function $\mathrm{gd}\left(x\right)$ see §4.23(viii). … ►

§22.16(ii) Jacobi’s Epsilon Function

… ►

Relation to Theta Functions

… ►

§22.16(iii) Jacobi’s Zeta Function

…