Additional elementary complex functions

Additional elementary complex functions#

Auxiliary function \(\sqrt{1-z^2}\)#

math53.sqrt1mz2(z)#

Returns the square root of \(1-z^2\). For very large \(z\) (when \(z^2\) would overflow) we have \(\sqrt{1-z^2} = \pm iz\), where the sign is chosen to make \(\Re \sqrt{1-z^2} \ge 0\); otherwise the result is computed from the definition.

See also Wikipedia [1234], MathWorld [843], BoostMath [97].

An example in Python

>>> from xlcalcnet import ecplx
>>> ecplx.Cuberoot(0.5)
ecplx('5.2359877559829887307E-1')
>>> ecplx.Cuberoot('0.1')
ecplx('5.3518479027559984754E-1')

Cube root, \(\mathrm{cuberoot}(x) = \sqrt[3]{x} = y\), with \(\mathrm{arg}(y)\) closest to \(\mathrm{arg}(x)\)#

math53.cuberoot(z)#

Returns the cube root of \(x\), \(x^{1/3}\) in a way which gives a negative number for negative input. See also Wikipedia [1234], MathWorld [843], BoostMath [97].

An example in Python

>>> from xlcalcnet import ecplx
>>> ecplx.Cuberoot(0.5)
ecplx('5.2359877559829887307E-1')
>>> ecplx.Cuberoot('0.1')
ecplx('5.3518479027559984754E-1')

Nth root, \(\mathrm{surd}(x, n) = \sqrt[n]{x} = y\), with \(\mathrm{arg}(y)\) closest to \(\mathrm{arg}(x)\)#

mathc53.surd(x, n)#

Returns the complex nth root \(w = z^{1/n}\) with arg(\(w\)) closest to arg(\(z\)), e.g. surd(-8, 3) = -2 or surd(\(i\), 5) = \(i\), compared to the cnroot results \(\sqrt[3]{-8} = 1+i \sqrt{3}\) and \(\sqrt[5]{i} = \cos(\pi/10) + i\sin(\pi/10)\). See Ehrhardt [297] (4.2.60).

An example in Python

>>> from xlcalcnet import ecplx
>>> ecplx.Surd(0.5)
ecplx('5.2359877559829887307E-1')
>>> ecplx.Surd('0.1')
ecplx('5.3518479027559984754E-1')

Continuous inverse cotangent, \(\mathrm{acotc}(x)\)#

ctx.acotc(x)#

where ctx is math53, mathc53, ctxcpp or ctxflint.

Returns the continuous inverse cotangent of \(x\), \(\mathrm{acotc}(x) = \pi/2 - \mathrm{atan}(x)\). See also Wikipedia [1261], MathWorld [870], NIST [488], Mpmath [570].

An example in Python

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

Continuous inverse hyperbolic cotangent, \(\mathrm{acothc}(x)\)#

ctx.acothc(x)#

where ctx is math53, mathc53, ctxcpp or ctxflint.

Returns the continuous inverse hyperbolic cotangent of \(x\), \(\mathrm{acothc}(x) = \pi/2 - \mathrm{atan}(x)\). See also Wikipedia [1261], MathWorld [870], NIST [488], Mpmath [570].

An example in Python

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