 EllipticCPi - Maple Help

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.