Lemniscate functions

Lemniscate functions#

Lemniscate sine function, \(\mathrm{sinlemn}(x)\)#

math53.sin_lemniscate(x)#

Returns the lemniscate sine function \(\displaystyle \mathrm{sinlemn}(x) = \frac{\sqrt{2}}{2} \mathrm{sd}\left(x \sqrt{2}, \frac{\sqrt{2}}{2}\right)\), where \(\mathrm{sd}(\cdot)\) is a Jacobi elliptic function.

See also: Wikipedia [1435], MathWorld [1042], Ehrhardt [309] (3.2.16.1).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.SinLemniscate(1.5, 0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.SinLemniscate(1.5, '0.51')
ereal('5.3518479027559984754E-1')

Lemniscate cosine function, \(\mathrm{coslemn}(x)\)#

math53.cos_lemniscate(x)#

Returns the lemniscate cosine function \(\displaystyle \mathrm{coslemn}(x) = \mathrm{cn}\left(x \sqrt{2}, \frac{\sqrt{2}}{2}\right)\), where \(\mathrm{cn}(\cdot)\) is a Jacobi elliptic function.

See also: Wikipedia [1435], MathWorld [1042], Ehrhardt [309] (3.2.16.2).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.CosLemniscate(1.5, 0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.CosLemniscate(1.5, '0.51')
ereal('5.3518479027559984754E-1')

Inverse lemniscate cosine function, \(\mathrm{arcsl}(x)\)#

math53.acos_lemniscate(x)#

Returns the inverse lemniscate cosine function \(\displaystyle \mathrm{arcsl}(x) = \frac{1}{\sqrt{2}} \mathrm{arcsd}\left(x \sqrt{2}, \frac{\sqrt{2}}{2}\right), |x| \le 1\), where \(\mathrm{arcsd}(\cdot)\) is an inverse Jacobi elliptic function.

See also: Wikipedia [1435], MathWorld [1042], Ehrhardt [309] (3.2.16.3).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.AcosLemniscate(1.5, 0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.AcosLemniscate(1.5, '0.51')
ereal('5.3518479027559984754E-1')

Inverse lemniscate sine function, \(\mathrm{arccl}(x)\)#

math53.asin_lemniscate(x)#

Returns the inverse lemniscate sine function \(\displaystyle \mathrm{arccl}(x) = \frac{1}{\sqrt{2}} \mathrm{arccn}\left(x \sqrt{2}, \frac{\sqrt{2}}{2}\right), |x| \le 1\), where \(\mathrm{arccn}(\cdot)\) is an inverse Jacobi elliptic function.

See also: Wikipedia [1435], MathWorld [1042], Ehrhardt [309] (3.2.16.4).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.AsinLemniscate(1.5, 0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.AsinLemniscate(1.5, '0.51')
ereal('5.3518479027559984754E-1')
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