Data.Complex

Rectangular form

data Complex a :: * -> *

Complex numbers are an algebraic type.

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.

Constructors

!a :+ !a infix 6

forms a complex number from its real and imaginary rectangular components.

Instances

Eq a => Eq (Complex a)

RealFloat a => Floating (Complex a)

RealFloat a => Fractional (Complex a)

Data a => Data (Complex a)

RealFloat a => Num (Complex a)

Read a => Read (Complex a)

Show a => Show (Complex a)

Typeable (* -> *) Complex

realPart :: RealFloat a => Complex a -> a

Extracts the real part of a complex number.

imagPart :: RealFloat a => Complex a -> a

Extracts the imaginary part of a complex number.

Polar form

mkPolar :: RealFloat a => a -> a -> Complex a

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

cis :: RealFloat a => a -> Complex a

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

polar :: RealFloat a => Complex a -> (a, a)

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

magnitude :: RealFloat a => Complex a -> a

The nonnegative magnitude of a complex number.

phase :: RealFloat a => Complex a -> a

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

Conjugate

conjugate :: RealFloat a => Complex a -> Complex a

The conjugate of a complex number.

Specification

module Data.Complex(Complex((:+)), realPart, imagPart, conjugate, mkPolar,
                    cis, polar, magnitude, phase)  where

infix  6  :+

data  (RealFloat a)     => Complex a = !a :+ !a  deriving (Eq,Read,Show)


realPart, imagPart :: (RealFloat a) => Complex a -> a
realPart (x:+y)        =  x
imagPart (x:+y)        =  y

conjugate      :: (RealFloat a) => Complex a -> Complex a
conjugate (x:+y) =  x :+ (-y)

mkPolar                :: (RealFloat a) => a -> a -> Complex a
mkPolar r theta        =  r * cos theta :+ r * sin theta

cis            :: (RealFloat a) => a -> Complex a
cis theta      =  cos theta :+ sin theta

polar          :: (RealFloat a) => Complex a -> (a,a)
polar z                =  (magnitude z, phase z)

magnitude :: (RealFloat a) => Complex a -> a
magnitude (x:+y) =  scaleFloat k
                   (sqrt ((scaleFloat mk x)^2 + (scaleFloat mk y)^2))
                  where k  = max (exponent x) (exponent y)
                        mk = - k

phase :: (RealFloat a) => Complex a -> a
phase (0 :+ 0) = 0
phase (x :+ y) = atan2 y x


instance  (RealFloat a) => Num (Complex a)  where
    (x:+y) + (x':+y') =  (x+x') :+ (y+y')
    (x:+y) - (x':+y') =  (x-x') :+ (y-y')
    (x:+y) * (x':+y') =  (x*x'-y*y') :+ (x*y'+y*x')
    negate (x:+y)     =  negate x :+ negate y
    abs z             =  magnitude z :+ 0
    signum 0          =  0
    signum z@(x:+y)   =  x/r :+ y/r  where r = magnitude z
    fromInteger n     =  fromInteger n :+ 0

instance  (RealFloat a) => Fractional (Complex a)  where
    (x:+y) / (x':+y') =  (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
                         where x'' = scaleFloat k x'
                               y'' = scaleFloat k y'
                               k   = - max (exponent x') (exponent y')
                               d   = x'*x'' + y'*y''

    fromRational a    =  fromRational a :+ 0

instance  (RealFloat a) => Floating (Complex a)       where
    pi             =  pi :+ 0
    exp (x:+y)     =  expx * cos y :+ expx * sin y
                      where expx = exp x
    log z          =  log (magnitude z) :+ phase z

    sqrt 0         =  0
    sqrt z@(x:+y)  =  u :+ (if y < 0 then -v else v)
                      where (u,v) = if x < 0 then (v',u') else (u',v')
                            v'    = abs y / (u'*2)
                            u'    = sqrt ((magnitude z + abs x) / 2)

    sin (x:+y)     =  sin x * cosh y :+ cos x * sinh y
    cos (x:+y)     =  cos x * cosh y :+ (- sin x * sinh y)
    tan (x:+y)     =  (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
                      where sinx  = sin x
                            cosx  = cos x
                            sinhy = sinh y
                            coshy = cosh y

    sinh (x:+y)    =  cos y * sinh x :+ sin  y * cosh x
    cosh (x:+y)    =  cos y * cosh x :+ sin y * sinh x
    tanh (x:+y)    =  (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
                      where siny  = sin y
                            cosy  = cos y
                            sinhx = sinh x
                            coshx = cosh x

    asin z@(x:+y)  =  y':+(-x')
                      where  (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
    acos z@(x:+y)  =  y'':+(-x'')
                      where (x'':+y'') = log (z + ((-y'):+x'))
                            (x':+y')   = sqrt (1 - z*z)
    atan z@(x:+y)  =  y':+(-x')
                      where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))

    asinh z        =  log (z + sqrt (1+z*z))
    acosh z        =  log (z + (z+1) * sqrt ((z-1)/(z+1)))
    atanh z        =  log ((1+z) / sqrt (1-z*z))