Are there any identities for incomplete elliptic integrals of the third kind with complex arguments?
$begingroup$
Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments:
For $tantheta = sinhphi$
17.4.8
$F(iphi | alpha) = iF(theta | frac{pi}{2} - alpha)$
17.4.9
$E(iphi | alpha) = -i E(theta | frac{pi}{2} - alpha) + i F(theta frac{pi}{2} - alpha) + itantheta(1-cos^2alpha sin^2theta)^{1/2}$
Are there any useful identities for incomplete elliptic integrals of the third kind, $Pi(n;phi | alpha)$?
I would like to evaluate such integrals numerically, but many libraries such as GSL only handle real arguments.
complex-numbers elliptic-integrals
asked Jan 11 at 19:22
DannyDanny
634
$endgroup$
add a comment |
$begingroup$
Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments:
For $tantheta = sinhphi$
17.4.8
$F(iphi | alpha) = iF(theta | frac{pi}{2} - alpha)$
17.4.9
$E(iphi | alpha) = -i E(theta | frac{pi}{2} - alpha) + i F(theta frac{pi}{2} - alpha) + itantheta(1-cos^2alpha sin^2theta)^{1/2}$
Are there any useful identities for incomplete elliptic integrals of the third kind, $Pi(n;phi | alpha)$?
I would like to evaluate such integrals numerically, but many libraries such as GSL only handle real arguments.
complex-numbers elliptic-integrals
asked Jan 11 at 19:22
DannyDanny
634
$endgroup$
add a comment |
$begingroup$
Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments:
For $tantheta = sinhphi$
17.4.8
$F(iphi | alpha) = iF(theta | frac{pi}{2} - alpha)$
17.4.9
$E(iphi | alpha) = -i E(theta | frac{pi}{2} - alpha) + i F(theta frac{pi}{2} - alpha) + itantheta(1-cos^2alpha sin^2theta)^{1/2}$
Are there any useful identities for incomplete elliptic integrals of the third kind, $Pi(n;phi | alpha)$?
I would like to evaluate such integrals numerically, but many libraries such as GSL only handle real arguments.
complex-numbers elliptic-integrals
asked Jan 11 at 19:22
DannyDanny
634
$endgroup$
Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments:
For $tantheta = sinhphi$
17.4.8
$F(iphi | alpha) = iF(theta | frac{pi}{2} - alpha)$
17.4.9
$E(iphi | alpha) = -i E(theta | frac{pi}{2} - alpha) + i F(theta frac{pi}{2} - alpha) + itantheta(1-cos^2alpha sin^2theta)^{1/2}$
Are there any useful identities for incomplete elliptic integrals of the third kind, $Pi(n;phi | alpha)$?
I would like to evaluate such integrals numerically, but many libraries such as GSL only handle real arguments.
complex-numbers elliptic-integrals
complex-numbers elliptic-integrals
asked Jan 11 at 19:22
DannyDanny
634
asked Jan 11 at 19:22
DannyDanny
634
asked Jan 11 at 19:22
DannyDanny
634
asked Jan 11 at 19:22
DannyDanny
634
asked Jan 11 at 19:22
DannyDanny
634
634
add a comment |
add a comment |
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