# §19.10 Relations to Other Functions

## §19.10(i) Theta and Elliptic 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).

## §19.10(ii) Elementary Functions

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

 19.10.1 $\displaystyle\mathop{\ln\/}\nolimits\!\left(x/y\right)$ $\displaystyle=(x-y)\mathop{R_{C}\/}\nolimits\!\left(\tfrac{1}{4}(x+y)^{2},xy% \right),$ $\displaystyle\mathop{\mathrm{arctan}\/}\nolimits\!\left(x/y\right)$ $\displaystyle=x\mathop{R_{C}\/}\nolimits\!\left(y^{2},y^{2}+x^{2}\right),$ $\displaystyle\mathop{\mathrm{arctanh}\/}\nolimits\!\left(x/y\right)$ $\displaystyle=x\mathop{R_{C}\/}\nolimits\!\left(y^{2},y^{2}-x^{2}\right),$ $\displaystyle\mathop{\mathrm{arcsin}\/}\nolimits\!\left(x/y\right)$ $\displaystyle=x\mathop{R_{C}\/}\nolimits\!\left(y^{2}-x^{2},y^{2}\right),$ $\displaystyle\mathop{\mathrm{arcsinh}\/}\nolimits\!\left(x/y\right)$ $\displaystyle=x\mathop{R_{C}\/}\nolimits\!\left(y^{2}+x^{2},y^{2}\right),$ $\displaystyle\mathop{\mathrm{arccos}\/}\nolimits\!\left(x/y\right)$ $\displaystyle=(y^{2}-x^{2})^{1/2}\mathop{R_{C}\/}\nolimits\!\left(x^{2},y^{2}% \right),$ $\displaystyle\mathop{\mathrm{arccosh}\/}\nolimits\!\left(x/y\right)$ $\displaystyle=(x^{2}-y^{2})^{1/2}\mathop{R_{C}\/}\nolimits\!\left(x^{2},y^{2}% \right).$

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

For relations to the Gudermannian function $\mathop{\mathrm{gd}\/}\nolimits\!\left(x\right)$ and its inverse $\mathop{{\mathrm{gd}^{-1}}\/}\nolimits\!\left(x\right)$4.23(viii)), see (19.6.8) and

 19.10.2 $(\mathop{\sinh\/}\nolimits\phi)\mathop{R_{C}\/}\nolimits\!\left(1,{\mathop{% \cosh\/}\nolimits^{2}}\phi\right)=\mathop{\mathrm{gd}\/}\nolimits\!\left(\phi% \right).$