Data.Complex

Synopsis

Rectangular form

data RealFloat a => Complex a Source

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

Instances

 Typeable1 Complex RealFloat a => Eq (Complex a) RealFloat a => Floating (Complex a) RealFloat a => Fractional (Complex a) (Data a, RealFloat a) => Data (Complex a) RealFloat a => Num (Complex a) (Read a, RealFloat a) => Read (Complex a) RealFloat a => Show (Complex a)

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

Extracts the real part of a complex number.

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

Extracts the imaginary part of a complex number.

Polar form

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

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

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

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

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

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 -> aSource

The nonnegative magnitude of a complex number.

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

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 aSource

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))

```