# §19.7 Connection Formulas

## §19.7(i) Complete Integrals of the First and Second Kinds

### Legendre’s Relation

 19.7.1 $\mathop{E\/}\nolimits\!\left(k\right)\mathop{{K^{\prime}}\/}\nolimits\!\left(k% \right)+\mathop{{E^{\prime}}\/}\nolimits\!\left(k\right)\mathop{K\/}\nolimits% \!\left(k\right)-\mathop{K\/}\nolimits\!\left(k\right)\mathop{{K^{\prime}}\/}% \nolimits\!\left(k\right)=\tfrac{1}{2}\pi.$

Also,

 19.7.2 $\displaystyle\mathop{K\/}\nolimits\!\left(ik/k^{\prime}\right)$ $\displaystyle=k^{\prime}\mathop{K\/}\nolimits\!\left(k\right),$ $\displaystyle\mathop{K\/}\nolimits\!\left(k^{\prime}/ik\right)$ $\displaystyle=k\mathop{K\/}\nolimits\!\left(k^{\prime}\right),$ $\displaystyle\mathop{E\/}\nolimits\!\left(ik/k^{\prime}\right)$ $\displaystyle=(1/k^{\prime})\mathop{E\/}\nolimits\!\left(k\right),$ $\displaystyle\mathop{E\/}\nolimits\!\left(k^{\prime}/ik\right)$ $\displaystyle=(1/k)\mathop{E\/}\nolimits\!\left(k^{\prime}\right).$
 19.7.3 $\displaystyle\mathop{K\/}\nolimits\!\left(1/k\right)$ $\displaystyle=k(\mathop{K\/}\nolimits\!\left(k\right)\mp\mathrm{i}\mathop{K\/}% \nolimits\!\left(k^{\prime}\right)),$ $\displaystyle\mathop{K\/}\nolimits\!\left(1/k^{\prime}\right)$ $\displaystyle=k^{\prime}(\mathop{K\/}\nolimits\!\left(k^{\prime}\right)\pm% \mathrm{i}\mathop{K\/}\nolimits\!\left(k\right)),$ $\displaystyle\mathop{E\/}\nolimits\!\left(1/k\right)$ $\displaystyle=(1/k)\left(\mathop{E\/}\nolimits\!\left(k\right)\pm\mathrm{i}% \mathop{E\/}\nolimits\!\left(k^{\prime}\right)-{k^{\prime}}^{2}\mathop{K\/}% \nolimits\!\left(k\right)\mp\mathrm{i}k^{2}\mathop{K\/}\nolimits\!\left(k^{% \prime}\right)\right),$ $\displaystyle\mathop{E\/}\nolimits\!\left(1/k^{\prime}\right)$ $\displaystyle=(1/k^{\prime})\left(\mathop{E\/}\nolimits\!\left(k^{\prime}% \right)\mp\mathrm{i}\mathop{E\/}\nolimits\!\left(k\right)-k^{2}\mathop{K\/}% \nolimits\!\left(k^{\prime}\right)\pm\mathrm{i}{k^{\prime}}^{2}\mathop{K\/}% \nolimits\!\left(k\right)\right),$

where upper signs apply if $\Im{k^{2}}>0$ and lower signs if $\Im{k^{2}}<0$. This dichotomy of signs (missing in several references) is due to Fettis (1970).

## §19.7(ii) Change of Modulus and Amplitude

### Reciprocal-Modulus Transformation

 19.7.4 $\displaystyle\mathop{F\/}\nolimits\!\left(\phi,k_{1}\right)$ $\displaystyle=k\mathop{F\/}\nolimits\!\left(\beta,k\right),$ $\displaystyle\mathop{E\/}\nolimits\!\left(\phi,k_{1}\right)$ $\displaystyle=(\mathop{E\/}\nolimits\!\left(\beta,k\right)-{k^{\prime}}^{2}% \mathop{F\/}\nolimits\!\left(\beta,k\right))/k,$ $\displaystyle\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k_{1}\right)$ $\displaystyle=k\mathop{\Pi\/}\nolimits\!\left(\beta,k^{2}\alpha^{2},k\right),$ $k_{1}=1/k$, $\mathop{\sin\/}\nolimits\beta=k_{1}\mathop{\sin\/}\nolimits\phi\leq 1$.

Provided the functions in these identities are correctly analytically continued in the complex $\beta$-plane, then the identities will also hold in the complex $\beta$-plane.

### Imaginary-Modulus Transformation

 19.7.5 $\displaystyle\mathop{F\/}\nolimits\!\left(\phi,ik\right)$ $\displaystyle=\kappa^{\prime}\mathop{F\/}\nolimits\!\left(\theta,\kappa\right),$ $\displaystyle\mathop{E\/}\nolimits\!\left(\phi,ik\right)$ $\displaystyle=(1/\kappa^{\prime})\left(\mathop{E\/}\nolimits\!\left(\theta,% \kappa\right)-\kappa^{2}\*(\mathop{\sin\/}\nolimits\theta\mathop{\cos\/}% \nolimits\theta)\*(1-\kappa^{2}{\mathop{\sin\/}\nolimits^{2}}\theta)^{-\ifrac{% 1}{2}}\right),$ $\displaystyle\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},ik\right)$ $\displaystyle=(\kappa^{\prime}/\alpha_{1}^{2})\left(\kappa^{2}\mathop{F\/}% \nolimits\!\left(\theta,\kappa\right)+{\kappa^{\prime}}^{2}\alpha^{2}\mathop{% \Pi\/}\nolimits\!\left(\theta,\alpha_{1}^{2},\kappa\right)\right),$

where

 19.7.6 $\displaystyle\kappa$ $\displaystyle=\frac{k}{\sqrt{1+k^{2}}},$ $\displaystyle\kappa^{\prime}$ $\displaystyle=\frac{1}{\sqrt{1+k^{2}}},$ $\displaystyle\mathop{\sin\/}\nolimits\theta$ $\displaystyle=\frac{\sqrt{1+k^{2}}\mathop{\sin\/}\nolimits\phi}{\sqrt{1+k^{2}{% \mathop{\sin\/}\nolimits^{2}}\phi}},$ $\displaystyle\alpha_{1}^{2}$ $\displaystyle=\frac{\alpha^{2}+k^{2}}{1+k^{2}}.$

### Imaginary-Argument Transformation

With $\mathop{\sinh\/}\nolimits\phi=\mathop{\tan\/}\nolimits\psi$,

 19.7.7 $\displaystyle\mathop{F\/}\nolimits\!\left(i\phi,k\right)$ $\displaystyle=i\mathop{F\/}\nolimits\!\left(\psi,k^{\prime}\right),$ $\displaystyle\mathop{E\/}\nolimits\!\left(i\phi,k\right)$ $\displaystyle=i\left(\mathop{F\/}\nolimits\!\left(\psi,k^{\prime}\right)-% \mathop{E\/}\nolimits\!\left(\psi,k^{\prime}\right)+(\mathop{\tan\/}\nolimits% \psi)\sqrt{1-{k^{\prime}}^{2}{\mathop{\sin\/}\nolimits^{2}}\psi}\right),$ $\displaystyle\mathop{\Pi\/}\nolimits\!\left(i\phi,\alpha^{2},k\right)$ $\displaystyle=i\left(\mathop{F\/}\nolimits\!\left(\psi,k^{\prime}\right)-% \alpha^{2}\mathop{\Pi\/}\nolimits\!\left(\psi,1-\alpha^{2},k^{\prime}\right)% \right)/{(1-\alpha^{2})}.$

For two further transformations of this type see Erdélyi et al. (1953b, p. 316).

## §19.7(iii) Change of Parameter of $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$

There are three relations connecting $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ and $\mathop{\Pi\/}\nolimits\!\left(\phi,\omega^{2},k\right)$, where $\omega^{2}$ is a rational function of $\alpha^{2}$. If $k^{2}$ and $\alpha^{2}$ are real, then both integrals are circular cases or both are hyperbolic cases (see §19.2(ii)).

The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ when $\alpha^{2}>{\mathop{\csc\/}\nolimits^{2}}\phi$ (see (19.6.5) for the complete case). Let $c={\mathop{\csc\/}\nolimits^{2}}\phi\neq\alpha^{2}$. Then

 19.7.8 $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)+\mathop{\Pi\/}% \nolimits\!\left(\phi,\omega^{2},k\right)=\mathop{F\/}\nolimits\!\left(\phi,k% \right)+\sqrt{c}\mathop{R_{C}\/}\nolimits\!\left((c-1)(c-k^{2}),(c-\alpha^{2})% (c-\omega^{2})\right),$ $\alpha^{2}\omega^{2}=k^{2}$.

Since $k^{2}\leq c$ we have $\alpha^{2}\omega^{2}\leq c$; hence $\alpha^{2}>c$ implies $\omega^{2}<1\leq c$.

The second relation maps each hyperbolic region onto itself and each circular region onto the other:

 19.7.9 $(k^{2}-\alpha^{2})\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)+(k^{% 2}-\omega^{2})\mathop{\Pi\/}\nolimits\!\left(\phi,\omega^{2},k\right)=k^{2}% \mathop{F\/}\nolimits\!\left(\phi,k\right)-\alpha^{2}\omega^{2}\sqrt{c-1}% \mathop{R_{C}\/}\nolimits\!\left(c(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})\right),$ $(1-\alpha^{2})(1-\omega^{2})=1-k^{2}$.

The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other:

 19.7.10 $(1-\alpha^{2})\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)+(1-% \omega^{2})\mathop{\Pi\/}\nolimits\!\left(\phi,\omega^{2},k\right)=\mathop{F\/% }\nolimits\!\left(\phi,k\right)+(1-\alpha^{2}-\omega^{2})\sqrt{c-k^{2}}\*% \mathop{R_{C}\/}\nolimits\!\left(c(c-1),(c-\alpha^{2})(c-\omega^{2})\right),$ $(k^{2}-\alpha^{2})(k^{2}-\omega^{2})=k^{2}(k^{2}-1)$.