Bulirsch elliptic integrals

Bulirsch elliptic integrals#

See Carlson (1994)

Bulirsch elliptic integrals can be calculated by

\[\mathrm{cel1}(k_c) = \mathrm{cel}(k_c, 0, a, 0) = a R_F\left(0, k_c^2, 1 \right)\]
\[\mathrm{cel2}(k_c) = \mathrm{cel}(k_c, 0, a, b) = a R_F\left(0, k_c^2, 1 \right) + \frac{b}{3} R_J\left(0, k_c^2, 1, 0 \right)\]
\[\mathrm{cel}(k_c, p, a, b) = a R_F\left(0, k_c^2, 1 \right) + \frac{1}{3} (b-pa) R_J\left(0, k_c^2, 1, p \right)\]
\[\mathrm{el1}(x, k_c) = x R_F\left(1, 1+k_c^2 x^2, 1+x^2 \right)\]
\[\mathrm{el2}(x, k_c, a, b) = ax R_F\left(1, 1+k_c^2 x^2, 1+x^2 \right) + (b-a) \frac{x^3}{3} R_D\left(1, 1+k_c^2 x^2, 1+x^2 \right)\]
\[\mathrm{el3}(x, k_c, p) = x R_F\left(1, 1+k_c^2 x^2, 1+x^2 \right) + (1-p) \frac{x^3}{3} R_J\left(1, 1+k_c^2 x^2, 1+x^2, 1+px^2 \right)\]

Complete elliptic integral of the 1st kind \(\mathrm{cel1}(k_c)\)#

math53.cel1(kc)#

Returns Bulirsch’s complete elliptic integral of the first kind. See also Bulirsch [169], See also Bulirsch [170], NIST [180], Wikipedia [1403], Ehrhardt [309] (3.2.3.1).

\[\mathrm{cel1}(k_c) = \int_0^{\infty} \frac{\mathrm{d}t}{\sqrt{(1+t^2)(1+k_c^2 t^2)}}\]

with the complementary modulus \(K_c \ne 0\).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.Cel1(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Cel1('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Cel1(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Cel1('0.51')
Gpr('5.3518479027559984754E-1')

Complete elliptic integral of the 2nd kind \(\mathrm{cel2}(k_c, a, b)\)#

math53.cel2(kc, a, b)#

Returns Bulirsch’s complete elliptic integral of the second kind. See also Bulirsch [169], See also Bulirsch [170], NIST [180], Wikipedia [1403], Ehrhardt [309] (3.2.3.2).

\[\mathrm{cel2}(k_c, a, b) = \int_0^{\infty} \frac{a+bt^2}{(1+t^2)\sqrt{(1+t^2)(1+k_c^2 t^2)}} \mathrm{d}t\]

with the complementary modulus \(k_c \ne 0\).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.Cel2(0.5, 3, 4)
xreal('5.2359877559829887307E-1')
>>> xreal.Cel2('0.51', 3, 4)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Cel2(0.5, 3, 4)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Cel2('0.51', 3, 4)
Gpr('5.3518479027559984754E-1')

General complete elliptic integral \(\mathrm{cel}(k_c, p, a, b)\)#

math53.cel(kc, p, a, b)#

Returns Bulirsch’s general complete elliptic integral. See also Bulirsch [169], See also Bulirsch [170], NIST [180], Wikipedia [1403], Ehrhardt [309] (3.2.3.3).

\[\mathrm{cel}(k_c, p, a, b) = \int_0^{\infty} \frac{a+bt^2}{(1+pt^2)\sqrt{(1+t^2)(1+k_c^2 t^2)}} \mathrm{d}t\]

with the complementary modulus \(k_c \ne 0\).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.Cel(0.5, 2, 3, 4)
xreal('5.2359877559829887307E-1')
>>> xreal.Cel('0.51', 2, 3, 4)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Cel(0.5, 2, 3, 4)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Cel('0.51', 2, 3, 4)
Gpr('5.3518479027559984754E-1')

Incomplete elliptic integral of the 1st kind \(\mathrm{el1}(x, k_c)\)#

math53.el1(x, kc)#

Returns Bulirsch’s incomplete elliptic integral of the first kind. See also Bulirsch [169], See also Bulirsch [170], NIST [180], Wikipedia [1403], Ehrhardt [309] (3.2.3.4).

\[\mathrm{el1}(x, k_c) = \int_0^{x} \frac{\mathrm{d}t}{\sqrt{(1+t^2)(1+k_c^2 t^2)}}\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.El1(2, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.El1(2, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.El1(2, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.El1(2, '0.51')
Gpr('5.3518479027559984754E-1')

Incomplete elliptic integral of the 2nd kind \(\mathrm{el2}(x, k_c, a, b)\)#

math53.el2(x, kc, a, b)#

Returns Bulirsch’s incomplete elliptic integral of the second kind. See also Bulirsch [169], See also Bulirsch [170], NIST [180], Wikipedia [1403], Ehrhardt [309] (3.2.3.5).

\[\mathrm{el2}(x, k_c, a, b) = \int_0^{x} \frac{a+bt^2}{(1+t^2)\sqrt{(1+t^2)(1+k_c^2 t^2)}} \mathrm{d}t\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.El2(2, 0.5, 3, 4)
xreal('5.2359877559829887307E-1')
>>> xreal.El2(2, '0.51', 3, 4)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.El2(2, 0.5, 3, 4)
Gpr('5.2359877559829887307E-1')
>>> Gpr.El2(2, '0.51', 3, 4)
Gpr('5.3518479027559984754E-1')

Incomplete elliptic integral of the 3rd kind \(\mathrm{el3}(x, k_c, p)\)#

math53.el3(x, kc, p)#

Returns Bulirsch’s incomplete elliptic integral of the third kind. See also Bulirsch [169], See also Bulirsch [170], NIST [180], Wikipedia [1403], Ehrhardt [309] (3.2.3.6).

\[\mathrm{el3}(x, k_c, p) = \int_0^{x} \frac{1+t^2}{(1+pt^2)\sqrt{(1+t^2)(1+k_c^2 t^2)}} \mathrm{d}t\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.El3(2, 0.5, 3)
xreal('5.2359877559829887307E-1')
>>> xreal.El3(2, '0.51', 3)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.El3(2, 0.5, 3)
Gpr('5.2359877559829887307E-1')
>>> Gpr.El3(2, '0.51', 3)
Gpr('5.3518479027559984754E-1')
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