§19.10 Relations to Other Functions

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).

If is assumed (without loss of generality), then

19.10.1

Symbols:

: Carlson’s combination of inverse circular and inverse hyperbolic functions, : inverse hyperbolic cosine function, : inverse hyperbolic sine function, : inverse hyperbolic tangent function, : arccosine function, : arcsine function, : arctangent function and : 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

In each case when , the quantity multiplying supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0.

For relations to the Gudermannian function and its inverse (§4.23(viii)), see (19.6.8) and

19.10.2

Symbols:

: Carlson’s combination of inverse circular and inverse hyperbolic functions, : Gudermannian function, : hyperbolic cosine function, : hyperbolic sine function and : real or complex argument

Referenced by:

§19.10(ii), §19.6(ii)

Permalink:

http://dlmf.nist.gov/19.10.E2

Encodings:

TeX, pMML, png