# §19.7 Connection Formulas

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

### Legendre’s Relation

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

Also,

 19.7.2 $\displaystyle K\left(ik/k^{\prime}\right)$ $\displaystyle=k^{\prime}K\left(k\right),$ $\displaystyle K\left(-ik^{\prime}/k\right)$ $\displaystyle=kK\left(k^{\prime}\right),$ $\displaystyle E\left(ik/k^{\prime}\right)$ $\displaystyle=(1/k^{\prime})E\left(k\right),$ $\displaystyle E\left(-ik^{\prime}/k\right)$ $\displaystyle=(1/k)E\left(k^{\prime}\right).$ ⓘ Symbols: $K\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the second kind, $\mathrm{i}$: imaginary unit, $k$: real or complex modulus and $k^{\prime}$: complementary modulus Referenced by: §19.15, Erratum (V1.0.17) for Equation (19.7.2) Permalink: http://dlmf.nist.gov/19.7.E2 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png Clarification (effective with 1.0.17): The argument $k^{\prime}/ik$ has been replaced with $-ik^{\prime}/k$ twice to clarify the meaning. See also: Annotations for §19.7(i), §19.7(i), §19.7 and Ch.19
 19.7.3 $\displaystyle K\left(1/k\right)$ $\displaystyle=k(K\left(k\right)\mp\mathrm{i}K\left(k^{\prime}\right)),$ $\displaystyle K\left(1/k^{\prime}\right)$ $\displaystyle=k^{\prime}(K\left(k^{\prime}\right)\pm\mathrm{i}K\left(k\right)),$ $\displaystyle E\left(1/k\right)$ $\displaystyle=(1/k)\left(E\left(k\right)\pm\mathrm{i}E\left(k^{\prime}\right)-% {k^{\prime}}^{2}K\left(k\right)\mp\mathrm{i}k^{2}K\left(k^{\prime}\right)% \right),$ $\displaystyle E\left(1/k^{\prime}\right)$ $\displaystyle=(1/k^{\prime})\left(E\left(k^{\prime}\right)\mp\mathrm{i}E\left(% k\right)-k^{2}K\left(k^{\prime}\right)\pm\mathrm{i}{k^{\prime}}^{2}K\left(k% \right)\right),$

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 $\displaystyle F\left(\phi,k_{1}\right)$ $\displaystyle=kF\left(\beta,k\right),$ $\displaystyle E\left(\phi,k_{1}\right)$ $\displaystyle=(E\left(\beta,k\right)-{k^{\prime}}^{2}F\left(\beta,k\right))/k,$ $\displaystyle\Pi\left(\phi,\alpha^{2},k_{1}\right)$ $\displaystyle=k\Pi\left(\beta,k^{2}\alpha^{2},k\right),$ $k_{1}=1/k$, $\sin\beta=k_{1}\sin\phi\leq 1$. ⓘ Defines: $k_{1}$: change of variable (locally) and $\beta$: change of variable (locally) Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the second kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the third kind, $\sin\NVar{z}$: sine function, $\phi$: real or complex argument, $k$: real or complex modulus, $k^{\prime}$: complementary modulus and $\alpha^{2}$: real or complex parameter Referenced by: §19.15, Figure 19.3.6, Figure 19.3.6, Figure 19.3.6, §19.7(ii) Permalink: http://dlmf.nist.gov/19.7.E4 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.7(ii), §19.7(ii), §19.7 and Ch.19

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

where

 19.7.6 $\displaystyle\kappa$ $\displaystyle=\frac{k}{\sqrt{1+k^{2}}},$ $\displaystyle\kappa^{\prime}$ $\displaystyle=\frac{1}{\sqrt{1+k^{2}}},$ $\displaystyle\sin\theta$ $\displaystyle=\frac{\sqrt{1+k^{2}}\sin\phi}{\sqrt{1+k^{2}{\sin}^{2}\phi}},$ $\displaystyle\alpha_{1}^{2}$ $\displaystyle=\frac{\alpha^{2}+k^{2}}{1+k^{2}}.$ ⓘ Defines: $\kappa$: modulus, change of variable (locally), $\kappa^{\prime}$: complementary modulus, change of variable (locally), $\theta$: change of variable (locally) and $\alpha_{1}$: change of variable (locally) Symbols: $\sin\NVar{z}$: sine function, $\phi$: real or complex argument, $k$: real or complex modulus and $\alpha^{2}$: real or complex parameter Permalink: http://dlmf.nist.gov/19.7.E6 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §19.7(ii), §19.7(ii), §19.7 and Ch.19

### Imaginary-Argument Transformation

With $\sinh\phi=\tan\psi$,

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

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

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

There are three relations connecting $\Pi\left(\phi,\alpha^{2},k\right)$ and $\Pi\left(\phi,\omega^{2},k\right)$, 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 $\Pi\left(\phi,\alpha^{2},k\right)$ when $\alpha^{2}>{\csc}^{2}\phi$ (see (19.6.5) for the complete case). Let $c={\csc}^{2}\phi\neq\alpha^{2}$. Then

 19.7.8 $\Pi\left(\phi,\alpha^{2},k\right)+\Pi\left(\phi,\omega^{2},k\right)=F\left(% \phi,k\right)+\sqrt{c}R_{C}\left((c-1)(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})% \right),$ $\alpha^{2}\omega^{2}=k^{2}$. ⓘ Defines: $\omega^{2}$: change of variable (locally) Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the third kind, $\phi$: real or complex argument, $k$: real or complex modulus, $\alpha^{2}$: real or complex parameter and $c$: change of variable Referenced by: §19.25(i), §19.6(i) Permalink: http://dlmf.nist.gov/19.7.E8 Encodings: TeX, pMML, png See also: Annotations for §19.7(iii), §19.7 and Ch.19

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

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