Data.Complex.Compat

cis t is a complex value with magnitude 1 and phase t (modulo 2*pi).

cis = mkPolar 1

Examples

>>> cis 01.0 :+ 0.0

The following examples are not perfectly zero due to IEEE 754

>>> cis pi(-1.0) :+ 1.2246467991473532e-16

>>> cis (4 * pi) - cis (2 * pi)0.0 :+ (-2.4492935982947064e-16)

The conjugate of a complex number.

prop> conjugate (conjugate x) = x

Examples

>>> conjugate (3.0 :+ 3.0)3.0 :+ (-3.0)

>>> conjugate ((3.0 :+ 3.0) * (2.0 :+ 2.0))0.0 :+ (-12.0)

Extracts the imaginary part of a complex number.

Examples

>>> imagPart (5.0 :+ 3.0)3.0

>>> imagPart ((5.0 :+ 3.0) * (2.0 :+ 3.0))21.0

The non-negative magnitude of a complex number.

Examples

>>> magnitude (1.0 :+ 1.0)1.4142135623730951

>>> magnitude (1.0 + 0.0)1.0

>>> magnitude (0.0 :+ (-5.0))5.0

Form a complex number from polar components of magnitude and phase.

Examples

>>> mkPolar 1 (pi / 4)0.7071067811865476 :+ 0.7071067811865475

>>> mkPolar 1 01.0 :+ 0.0

The phase of a complex number, in the range (-pi, pi]. If the magnitude is zero, then so is the phase.

Examples

>>> phase (0.5 :+ 0.5) / pi0.25

>>> phase (0 :+ 4) / pi0.5

The function polar takes a complex number and returns a (magnitude, phase) pair in canonical form: the magnitude is non-negative, and the phase in the range (-pi, pi]; if the magnitude is zero, then so is the phase.

polar z = (magnitude z, phase z)

Examples

>>> polar (1.0 :+ 1.0)(1.4142135623730951,0.7853981633974483)

>>> polar ((-1.0) :+ 0.0)(1.0,3.141592653589793)

>>> polar (0.0 :+ 0.0)(0.0,0.0)

Extracts the real part of a complex number.

Examples

>>> realPart (5.0 :+ 3.0)5.0

>>> realPart ((5.0 :+ 3.0) * (2.0 :+ 3.0))1.0

A data type representing complex numbers.

You can read about complex numbers on wikipedia.

In haskell, complex numbers are represented as a :+ b which can be thought of as representing a + bi. For a complex number z, abs z is a number with the magnitude of z, but oriented in the positive real direction, whereas signum z has the phase of z, but unit magnitude. Apart from the loss of precision due to IEEE754 floating point numbers, it holds that z == abs z * signum z.

Note that Complex's instances inherit the deficiencies from the type parameter's. For example, Complex Float's Ord instance has similar problems to Float's.

As can be seen in the examples, the Foldable and Traversable instances traverse the real part first.

Examples

>>> (5.0 :+ 2.5) + 6.511.5 :+ 2.5

>>> abs (1.0 :+ 1.0) - sqrt 2.00.0 :+ 0.0

>>> abs (signum (4.0 :+ 3.0))1.0 :+ 0.0

>>> foldr (:) [] (1 :+ 2)[1,2]

>>> mapM print (1 :+ 2)12