Inverse Jacobi elliptic functions

Inverse Jacobi elliptic functions#

The inverse Jacobi elliptic functions can be defined like the inverse trigonometric functions: e.g. if \(\mathrm{sn}(y, k) = x\) then \(y = \mathrm{arcsn}(x, k)\); they are multivalued and their principal values are returned. The functions can be represented as elliptic integrals [30, §22.15(ii)] and they are computed with the incomplete elliptic integral \(F(., k)\) using the table from Abramowitz and Stegun [1, p.596].

Inverse Jacobi elliptic function \(\mathrm{arcsn}(x, k)\)#

math53.jacobi_arcsn(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arcsn}(x, k)\) for \(|x| \le 1\) and \(|kx| \le 1\). See also NIST [501], Ehrhardt [309] (3.2.12.1).

\[\mathrm{arcsn}(x, k) = F(\mathrm{arcsin}(x), k).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arccn}(x, k)\)#

math53.jacobi_arccn(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arccn}(x, k)\) for \(|x| < 1\) if \(k \le 1\), and \(x^2 > 1 - 1/k^2\) if \(k > 1\). See also NIST [501], Ehrhardt [309] (3.2.12.2).

\[\mathrm{arccn}(x, k) = F(\mathrm{arccos}(x), k).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arcdn}(x, k)\)#

math53.jacobi_arcdn(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arcdn}(x, k)\) for \(0 \le x \le 1\) and \(k^2 + x^2 > 1\) if \(|k| < 1\); and \(|x| \le 1\) if \(|k| > 1\). See also NIST [501], Ehrhardt [309] (3.2.12.3).

\[\mathrm{arcdn}(x, k) = F\left(\mathrm{arcsin}\left(\sqrt{\frac{1-x^2}{k^2}} \right), k \right).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arccd}(x, k)\)#

math53.jacobi_arccd(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arccd}(x, k)\) for \(|x| \le 1\) if \(|k| < 1\); and \(|x| \ge 1\) if \(|k| > 1\). See also NIST [501], Ehrhardt [309] (3.2.12.4).

\[\mathrm{arccd}(x, k) = F\left(\mathrm{arcsin}\left(\sqrt{\frac{1-x^2}{1-k^2 x^2}} \right), k \right).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arcsd}(x, k)\)#

math53.jacobi_arcsd(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arcsd}(x, k)\) for \(x \in \mathbb{R}\) if \(|k| \ge 1\) and \(|x| < 1/\sqrt{1 - k^2}\) otherwise. See also NIST [501], Ehrhardt [309] (3.2.12.5).

\[\mathrm{arcsd}(x, k) = F\left(\mathrm{arcsin}\left(\sqrt{\frac{x^2}{1+k^2 x^2}} \right), k \right).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arcnd}(x, k)\)#

math53.jacobi_arcnd(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arcnd}(x, k)\) for \(x \ge 1\) and \(x^2 \le k^2/(1 - k^2)\) if \(|k| < 1\). See also NIST [501], Ehrhardt [309] (3.2.12.6).

\[\mathrm{arcnd}(x, k) = F\left(\mathrm{arcsin}\left(\sqrt{\frac{x^2-1}{k^2 x^2}} \right), k \right).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arcdc}(x, k)\)#

math53.jacobi_arcdc(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arcdc}(x, k)\) for \(|x| \ge 1\) if \(|k| < 1\); and \(|x| \le 1\) if \(|k| > 1\). See also NIST [501], Ehrhardt [309] (3.2.12.7).

\[\mathrm{arcdc}(x, k) = F\left(\mathrm{arcsin}\left(\sqrt{\frac{1-x^2-1}{k^2 - x^2}} \right), k \right).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arcnc}(x, k)\)#

math53.jacobi_arcnc(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arcnc}(x, k)\) for \(x \ge 1\), and \(x^2 \le k^2/(k^2 - 1)\) for \(|k| > 1\). See also NIST [501], Ehrhardt [309] (3.2.12.8).

\[\mathrm{arcnc}(x, k) = F(\mathrm{arcsec}(x), k).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arcsc}(x, k)\)#

math53.jacobi_arcsc(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arcsc}(x, k)\) for \(x \in \mathbb{R}\) if \(|k| \le 1\) and \(|x| \le 1/\sqrt{ k^2 - 1}\) if \(|k| > 1\). See also NIST [501], Ehrhardt [309] (3.2.12.9).

\[\mathrm{arcsc}(x, k) = F(\mathrm{arctan}(x), k).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arcns}(x, k)\)#

math53.jacobi_arcns(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arcns}(x, k)\) for \(|x| \ge 1\) if \(|k| < 1\) and \(|x| \ge |k|\) if \(|k| > 1\). See also NIST [501], Ehrhardt [309] (3.2.12.10).

\[\mathrm{arcns}(x, k) = F(\mathrm{arccsc}(x), k).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arcds}(x, k)\)#

math53.jacobi_arcds(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arcds}(x, k)\) for \(x \in \mathbb{R}\) if \(|k| > 1\) and \(|x| \ge \sqrt{ 1 - k^2}\) if \(|k| < 1\). See also NIST [501], Ehrhardt [309] (3.2.12.11).

\[\mathrm{arcds}(x, k) = F\left(\mathrm{arcsin}\left(\sqrt{\frac{1}{k^2 + x^2}} \right), k \right).\]

An example in Python

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

An example in Visual Basic

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

Inverse Jacobi elliptic function \(\mathrm{arccs}(x, k)\)#

math53.jacobiArcCS(x, k)#

Returns the inverse Jacobi elliptic function \(\mathrm{arccs}(x, k)\) for \(x \in \mathbb{R}\) if \(|k| < 1\) and \(|x| \ge \sqrt{k^2 - 1}\) if \(|k| > 1\). See also NIST [501], Ehrhardt [309] (3.2.12.12).

\[\mathrm{arccs}(x, k) = F(\mathrm{arccot}(x), k).\]

An example in Python

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

An example in Visual Basic

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