§19.7 Connection Formulas

Keywords:: Legendre’s elliptic integrals
Permalink:: http://dlmf.nist.gov/19.7

§19.7(i) Complete Integrals of the First and Second Kinds
§19.7(ii) Change of Modulus and Amplitude
§19.7(iii) Change of Parameter of $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$

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

Notes:: Three proofs of (19.7.1) are given in Duren (1991). To prove it from (19.21.1) put $z+1=1/k^{2}$ , use homogeneity, and apply the penultimate equation in (19.25.1) twice.
Permalink:: http://dlmf.nist.gov/19.7.i

¶ Legendre’s Relation

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

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{{E^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the second kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind and $k$ : real or complex modulus
Referenced by:: §19.21(i), §19.35(i), §19.7(i)
Permalink:: http://dlmf.nist.gov/19.7.E1
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Also,

19.7.2

$\mathop{K\/}\nolimits\!\left(ik/k^{{\prime}}\right)=k^{{\prime}}\mathop{K\/}\nolimits\!\left(k\right),$

$\mathop{K\/}\nolimits\!\left(k^{{\prime}}/ik\right)=k\mathop{K\/}\nolimits\!\left(k^{{\prime}}\right),$

$\mathop{E\/}\nolimits\!\left(ik/k^{{\prime}}\right)=(1/k^{{\prime}})\mathop{E\/}\nolimits\!\left(k\right),$

$\mathop{E\/}\nolimits\!\left(k^{{\prime}}/ik\right)=(1/k)\mathop{E\/}\nolimits\!\left(k^{{\prime}}\right).$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §19.15
Permalink:: http://dlmf.nist.gov/19.7.E2
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19.7.3

$\mathop{K\/}\nolimits\!\left(1/k\right)=k(\mathop{K\/}\nolimits\!\left(k\right)\mp i\mathop{K\/}\nolimits\!\left(k^{{\prime}}\right)),$

$\mathop{K\/}\nolimits\!\left(1/k^{{\prime}}\right)=k^{{\prime}}(\mathop{K\/}\nolimits\!\left(k^{{\prime}}\right)\pm i\mathop{K\/}\nolimits\!\left(k\right)),$

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

$\mathop{E\/}\nolimits\!\left(1/k^{{\prime}}\right)=(1/k^{{\prime}})\left(\mathop{E\/}\nolimits\!\left(k^{{\prime}}\right)\mp i\mathop{E\/}\nolimits\!\left(k\right)-k^{2}\mathop{K\/}\nolimits\!\left(k^{{\prime}}\right)\pm i{k^{{\prime}}}^{2}\mathop{K\/}\nolimits\!\left(k\right)\right),$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.7.E3
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where upper signs apply if $\imagpart{k^{2}}>0$ and lower signs if $\imagpart{k^{2}}<0$ . This dichotomy of signs (missing in several references) is due to Fettis (1970).

§19.7(ii) Change of Modulus and Amplitude

Notes:: See the penultimate paragraph in §19.25(i).
Keywords:: Legendre’s elliptic integrals
Referenced by:: §19.25(i)
Permalink:: http://dlmf.nist.gov/19.7.ii

¶ Reciprocal-Modulus Transformation

Keywords:: Legendre’s elliptic integrals

19.7.4

$\mathop{F\/}\nolimits\!\left(\phi,k_{1}\right)=k\mathop{F\/}\nolimits\!\left(\beta,k\right),$

$\mathop{E\/}\nolimits\!\left(\phi,k_{1}\right)=(\mathop{E\/}\nolimits\!\left(\beta,k\right)-{k^{{\prime}}}^{2}\mathop{F\/}\nolimits\!\left(\beta,k\right))/k,$

$\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k_{1}\right)=k\mathop{\Pi\/}\nolimits\!\left(\beta,k^{2}\alpha^{2},k\right),$ $k_{1}=1/k$ , $\mathop{\sin\/}\nolimits\beta=k_{1}\mathop{\sin\/}\nolimits\phi\leq 1$ .

¶ Imaginary-Modulus Transformation

Keywords:: Legendre’s elliptic integrals

19.7.5

$\mathop{F\/}\nolimits\!\left(\phi,ik\right)=\kappa^{{\prime}}\mathop{F\/}\nolimits\!\left(\theta,\kappa\right),$

$\mathop{E\/}\nolimits\!\left(\phi,ik\right)=(1/\kappa^{{\prime}})\left(\mathop{E\/}\nolimits\!\left(\theta,\kappa\right)-\kappa^{2}\*(\mathop{\sin\/}\nolimits\theta\mathop{\cos\/}\nolimits\theta)\*(1-\kappa^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta)^{{-\ifrac{1}{2}}}\right),$

$\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},ik\right)=(\kappa^{{\prime}}/\alpha _{1}^{2})\left(\kappa^{2}\mathop{F\/}\nolimits\!\left(\theta,\kappa\right)+{\kappa^{{\prime}}}^{2}\alpha^{2}\mathop{\Pi\/}\nolimits\!\left(\theta,\alpha _{1}^{2},\kappa\right)\right),$

where

19.7.6

$\kappa=\frac{k}{\sqrt{1+k^{2}}},$

$\kappa^{{\prime}}=\frac{1}{\sqrt{1+k^{2}}},$

$\mathop{\sin\/}\nolimits\theta=\frac{\sqrt{1+k^{2}}\mathop{\sin\/}\nolimits\phi}{\sqrt{1+k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\phi}},$

$\alpha _{1}^{2}=\frac{\alpha^{2}+k^{2}}{1+k^{2}}.$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\kappa$ : modulus and $\kappa^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.7.E6
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¶ Imaginary-Argument Transformation

Keywords:: Legendre’s elliptic integrals

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

19.7.7

$\mathop{F\/}\nolimits\!\left(i\phi,k\right)=i\mathop{F\/}\nolimits\!\left(\psi,k^{{\prime}}\right),$

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

$\mathop{\Pi\/}\nolimits\!\left(i\phi,\alpha^{2},k\right)=i\left(\mathop{F\/}\nolimits\!\left(\psi,k^{{\prime}}\right)-\alpha^{2}\mathop{\Pi\/}\nolimits\!\left(\psi,1-\alpha^{2},k^{{\prime}}\right)\right)/{(1-\alpha^{2})}.$

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

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

Notes:: All three relations follow from the change of parameter for the symmetric integral of the third kind; see §19.21(iii) and (19.25.14).
Keywords:: Legendre’s elliptic integrals, Legendre’s relation
Referenced by:: §19.15, §19.25(i), §19.6(iv)
Permalink:: http://dlmf.nist.gov/19.7.iii

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

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : 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 $\omega^{2}$
Referenced by:: §19.25(i), §19.6(i)
Permalink:: http://dlmf.nist.gov/19.7.E8
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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 $(k^{2}-\alpha^{2})\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)+(k^{2}-\omega^{2})\mathop{\Pi\/}\nolimits\!\left(\phi,\omega^{2},k\right)=k^{2}\mathop{F\/}\nolimits\!\left(\phi,k\right)-\alpha^{2}\omega^{2}\sqrt{c-1}\mathop{R_{C}\/}\nolimits\!\left(c(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})\right),$ $(1-\alpha^{2})(1-\omega^{2})=1-k^{2}$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : 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 $\omega^{2}$
Permalink:: http://dlmf.nist.gov/19.7.E9
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : 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 $\omega^{2}$
Permalink:: http://dlmf.nist.gov/19.7.E10
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