Maple style elliptic integrals

Maple style elliptic integrals#

Complete integral of the 1st kind, \(\mathrm{EllipticK}(k)\)#

math53.ellipticK(k)#

Returns the complete elliptic integral of the first kind K(k) and the real part if \(|k| > 1\). See also Maplesoft [385], Ehrhardt [309] (3.2.4.1).

\[\mathrm{EllipticK}(k) = \int_0^{1} \frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-k^2 t^2)}}\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticK(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticK('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticK(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticK('0.51')
Gpr('5.3518479027559984754E-1')

Complete integral of the 1st kind for imaginary modulus, \(\mathrm{EllipticKim}(k)\)#

math53.ellipticKim(k)#

Returns the complete elliptic integral of the first kind for the imaginary modulus \(ik\) with \(k \in \mathbb{R}\). See also Maplesoft [385], Ehrhardt [309] (3.2.4.2).

\[\mathrm{EllipticKim}(k) = \int_0^{1} \frac{\mathrm{d}t}{\sqrt{(1-t^2)(1+k^2 t^2)}}\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticKim(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticKim('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticKim(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticKim('0.51')
Gpr('5.3518479027559984754E-1')

Complementary complete integral of the 1st kind, \(\mathrm{EllipticCK}(k) = K'(k)\)#

math53.ellipticCK(k)#

Returns the complementary complete elliptic integral of the first kind with \(k \ne 0\). See also Maplesoft [385], Ehrhardt [309] (3.2.4.3), Ehrhardt [309] (4.2.28).

\[\mathrm{EllipticCK}(k) = \mathrm{EllipticK}(k_c) = \int_0^{1} \frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-k_c^2 t^2)}}\]

Note: the complex version is called Ellck(z) in Amath.

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticCK(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticCK('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticCK(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticCK('0.51')
Gpr('5.3518479027559984754E-1')

Complete integral of the 2nd kind, \(\mathrm{EllipticEC}(k)\)#

math53.ellipticEC(k)#

Returns the complete elliptic integral of the second kind \(E(k)\) and the real part if \(|k| > 1\). See also Maplesoft [384], Ehrhardt [309] (3.2.4.4).

\[\mathrm{EllipticEC}(k) = \int_0^{1} \frac{\sqrt{(1-k^2 t^2)}}{\sqrt{(1-t^2)}} \mathrm{d}t\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticEC(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticEC('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticEC(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticEC('0.51')
Gpr('5.3518479027559984754E-1')

Complete integral of the 2nd kind for imaginary modulus, \(\mathrm{EllipticECim}(k)\)#

math53.ellipticECim(k)#

Returns the complete elliptic integral of the second kind for the imaginary modulus \(ik\) with \(k \in \mathbb{R}\). See also Maplesoft [384], Ehrhardt [309] (3.2.4.5).

\[\mathrm{EllipticECim}(k) = E(ik) = \int_0^{1} \frac{\sqrt{(1+k^2 t^2)}}{\sqrt{(1-t^2)}} \mathrm{d}t\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticECim(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticECim('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticECim(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticECim('0.51')
Gpr('5.3518479027559984754E-1')

Complementary complete integral of the 2nd kind, \(\mathrm{EllipticCE}(k)\)#

math53.ellipticCE(k)#

Returns the complementary complete elliptic integral of the second kind. See also Maplesoft [384], Ehrhardt [309] (3.2.4.6).

\[\mathrm{EllipticEC}(k) = \mathrm{EllipticEC}(k_c) = \int_0^{1} \frac{\sqrt{(1-k_c^2 t^2)}}{\sqrt{(1-t^2)}} \mathrm{d}t\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticCE(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticCE('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticCE(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticCE('0.51')
Gpr('5.3518479027559984754E-1')

Complete integral of the 3rd kind, \(\mathrm{EllipticPiC}(\nu, k)\)#

math53.ellipticPiC(nu, k)#

Returns the complete elliptic integral of the third kind with \(|k| \ne 1, \nu \ne 1\) (or its real part if \(|k| > 1\)). See also Maplesoft [386], Ehrhardt [309] (3.2.4.7).

\[\mathrm{EllipticPiC}(\nu, k) = \int_0^{1} \frac{\mathrm{d}t}{(1-\nu t^2)\sqrt{(1-t^2)(1-k^2 t^2)}}\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticPiC(0.8, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticPiC(0.8, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticPiC(0.8, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticPiC(0.8, '0.51')
Gpr('5.3518479027559984754E-1')

Complete integral of the 3rd kind for imaginary modulus, \(\mathrm{EllipticPiCim}(\nu, k)\)#

math53.ellipticPiCim(nu, k)#

Returns the complementary complete elliptic integral of the third kind with \(\nu \ne 1\). See also Maplesoft [384], Ehrhardt [309] (3.2.4.8).

\[\mathrm{EllipticPiCim}(\nu, k) = \mathrm{EllipticPiC}(\nu, k) = \int_0^{1} \frac{\mathrm{d}t}{(1-\nu t^2)\sqrt{(1-t^2)(1+k^2 t^2)}}\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticPiCim(0.8, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticPiCim(0.8, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticPiCim(0.8, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticPiCim(0.8, '0.51')
Gpr('5.3518479027559984754E-1')

Complementary complete integral of the 3rd kind, \(\mathrm{EllipticCPi}(\nu, k)\)#

math53.ellipticCPi(nu, x)#

Returns the complementary complete elliptic integral of the third kind with \(|k| \ne 0, \nu \ne 1\). See also Maplesoft [384], Ehrhardt [309] (3.2.4.9).

\[\mathrm{EllipticCPi}(\nu, k) = \mathrm{EllipticPiC}(\nu, k_c) = \int_0^{1} \frac{\mathrm{d}t}{(1-\nu t^2)\sqrt{(1-t^2)(1-k_c^2 t^2)}}\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticCPi(0.8, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticCPi(0.8, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticCPi(0.8, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticCPi(0.8, '0.51')
Gpr('5.3518479027559984754E-1')

Incomplete integral of the 1st kind, \(\mathrm{EllipticF}(z, k)\)#

math53.ellipticF(z, k)#

Returns the incomplete elliptic integral of the first kind with \(|z| \le 1, |kz| \le 1\). See also Maplesoft [385], Ehrhardt [309] (3.2.4.10).

\[\mathrm{EllipticF}(z, k) = \int_0^{z} \frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-k^2 t^2)}}\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticF(0.8, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticF(0.8, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticF(0.8, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticF(0.8, '0.51')
Gpr('5.3518479027559984754E-1')

Incomplete integral of the 2nd kind, \(\mathrm{EllipticE}(z, k)\)#

math53.ellipticE(z, k)#

Returns the incomplete elliptic integral of the second kind with \(|z| \le 1, |kz| \le 1\). See also Maplesoft [384], Ehrhardt [309] (3.2.4.11).

\[\mathrm{EllipticE}(z, k) = \int_0^{z} \frac{\sqrt{(1-k^2 t^2)}}{\sqrt{(1-t^2)}} \mathrm{d}t\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticE(0.8, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticE(0.8, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticE(0.8, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticE(0.8, '0.51')
Gpr('5.3518479027559984754E-1')

Incomplete integral of the 3rd kind, \(\mathrm{EllipticPi}(z, \nu, k)\)#

math53.ellipticPi(z, nu, k)#

Returns the incomplete elliptic integral of the third kind with \(|z| \le 1, |kz| \le 1\). See also Maplesoft [386], Ehrhardt [309] (3.2.4.12).

\[\mathrm{EllipticPi}(z, \nu, k) = \int_0^{z} \frac{\mathrm{d}t}{(1-\nu t^2)\sqrt{(1-t^2)(1-k_c^2 t^2)}}\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllipticPi(0.6, 0.8, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllipticPi(0.6, 0.8, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllipticPi(0.6, 0.8, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllipticPi(0.6, 0.8, '0.51')
Gpr('5.3518479027559984754E-1')
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