Complex components

Complex components#

Absolute value of a real or complex number#

ctx.abs(x)#

where ctx is math53, mathc53, ctxcpp or ctxflint.

Returns the absolute value of \(x\), \(|x|\).

An example in Python

>>> from xlcalcnet import XComplex
>>> XComplex.Abs(0.5)
XComplex('5.2359877559829887307E-1')
>>> XComplex.Abs('0.1')
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.Abs(0.5)
Gpc('5.2359877559829887307E-1')
>>> Gpc.Abs('0.1')
Gpc('5.3518479027559984754E-1')
ctx.fabs(x)#

where ctx is math53, mathc53 or ctxcpp. This is an alias of ctx.abs(x).

Sign of a real or complex number#

ctx.sign(x)#

where ctx is math53, mathc53 or ctxcpp.

Returns the sign of \(x\), defined as \(\mathrm{sign}(x) = x / |x|\) (with the special case \(\mathrm{sign}(0) = 0\)):

Note that the sign function is also defined for complex numbers, for which it gives the projection onto the unit circle:

An example in Python

>>> from xlcalcnet import XComplex
>>> XComplex.Sign(0.5)
XComplex('5.2359877559829887307E-1')
>>> XComplex.Sign('0.1')
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.Sign(0.5)
Gpc('5.2359877559829887307E-1')
>>> Gpc.Sign('0.1')
Gpc('5.3518479027559984754E-1')

Real part of a real or complex number#

ctx.real(x)#

where ctx is math53, mathc53 or ctxcpp.

Returns the real part of \(x\), \(\Re(x)\).

An example in Python

>>> from xlcalcnet import XComplex
>>> XComplex.Real(0.5)
XComplex('5.2359877559829887307E-1')
>>> XComplex.Real('0.1')
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.Real(0.5)
Gpc('5.2359877559829887307E-1')
>>> Gpc.Real('0.1')
Gpc('5.3518479027559984754E-1')

Imaginary part of a real or complex number#

ctx.imag(x)#

where ctx is math53, mathc53 or ctxcpp.

Returns the imaginary part of \(x\), \(\Im(x)\).

An example in Python

>>> from xlcalcnet import XComplex
>>> XComplex.Imaginary(0.5)
XComplex('5.2359877559829887307E-1')
>>> XComplex.Imaginary('0.1')
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.Imaginary(0.5)
Gpc('5.2359877559829887307E-1')
>>> Gpc.Imaginary('0.1')
Gpc('5.3518479027559984754E-1')

Phase (or argument) of a real or complex number#

ctx.phase(x)#

where ctx is math53, mathc53 or ctxcpp.

Computes the complex argument (phase) of \(x\), defined as the signed angle between the positive real axis and \(x\) in the complex plane: The angle is defined to satisfy \(-\pi < \arg(x) \le \pi\) and with the sign convention that a nonnegative imaginary part results in a nonnegative argument.

An example in Python

>>> from xlcalcnet import XComplex
>>> XComplex.Phase(0.5)
XComplex('5.2359877559829887307E-1')
>>> XComplex.Phase('0.1')
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.Phase(0.5)
Gpc('5.2359877559829887307E-1')
>>> Gpc.Phase('0.1')
Gpc('5.3518479027559984754E-1')

Conjugate of a real or complex number#

ctx.conj(x)#

where ctx is math53, mathc53 or ctxcpp.

Returns the complex conjugate of \(x\), \(\overline{x}\).

An example in Python

>>> from xlcalcnet import XComplex
>>> XComplex.Conj(0.5)
XComplex('5.2359877559829887307E-1')
>>> XComplex.Conj('0.1')
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.Conj(0.5)
Gpc('5.2359877559829887307E-1')
>>> Gpc.Conj('0.1')
Gpc('5.3518479027559984754E-1')

Polar representation of a real or complex number#

ctx.polar(z)#

where ctx is math53, mathc53 or ctxcpp.

Returns the polar representation of the complex number \(z\) as a pair \((r, \phi)\) such that \(z = r e^{i \phi}\):

See also Mpmath [709].

An example in Python

>>> from xlcalcnet import XComplex
>>> XComplex.Polar(0.5)
XComplex('5.2359877559829887307E-1')
>>> XComplex.Polar('0.1')
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.Polar(0.5)
Gpc('5.2359877559829887307E-1')
>>> Gpc.Polar('0.1')
Gpc('5.3518479027559984754E-1')

Rectangular coordinates calculated from the polar representation of a real or complex number#

ctx.rect(r, phi)#

where ctx is math53, mathc53 or ctxcpp.

Returns the complex number represented by polar coordinates \((r, \phi)\):

See also Mpmath [701].

An example in Python

>>> from xlcalcnet import XComplex
>>> XComplex.Rect(0.5)
XComplex('5.2359877559829887307E-1')
>>> XComplex.Rect('0.1')
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.Rect(0.5)
Gpc('5.2359877559829887307E-1')
>>> Gpc.Rect('0.1')
Gpc('5.3518479027559984754E-1')
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