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EllipticPi

Incomplete and complete elliptic integrals of the third kind

EllipticCPi

Complementary complete elliptic integral of the third kind

 Calling Sequence EllipticPi(z, nu, k) EllipticPi(nu, k) EllipticCPi(nu, k)

Parameters

 z - algebraic expression (the sine of the amplitude) nu - algebraic expression (the characteristic) k - algebraic expression (the parameter)

Description

 • The incomplete elliptic integral EllipticPi is defined by

$\mathrm{EllipticPi}\left(z,\mathrm{\nu },k\right)=\underset{0}{\overset{z}{\int }}\frac{1}{\left(1-\mathrm{\nu }{t}^{2}\right)\sqrt{1-{t}^{2}}\sqrt{1-{k}^{2}{t}^{2}}}ⅆt$

 • The complete elliptic integrals EllipticPi and EllipticCPi are defined by

$\mathrm{EllipticPi}\left(\mathrm{\nu },k\right)=\mathrm{EllipticPi}\left(1,\mathrm{\nu },k\right)$

$\mathrm{EllipticCPi}\left(\mathrm{\nu },k\right)=\mathrm{EllipticPi}\left(1,\mathrm{\nu },\sqrt{1-{k}^{2}}\right)$

Examples

 > $\mathrm{EllipticPi}\left(0.1,0.2,0.3\right)$
 ${0.1002494388}$ (1)
 > $\mathrm{EllipticPi}\left(0.2,0.3\right)$
 ${1.800217337}$ (2)
 > $\mathrm{EllipticCPi}\left(0.2,0.3\right)$
 ${3.032020785}$ (3)

References

 Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover, 1972.