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# Complex

Complex numbers are an algebraic type. The constructor (:+) forms a complex number from its real and imaginary rectangular components. This constructor is strict: if either the real part or the imaginary part of the number is _|_, the entire number is _|_. A complex number may also be formed from polar components of magnitude and phase by the function mkPolar. The function cis produces a complex number from an angle t. Put another way, cis t is a complex value with magnitude 1 and phase t (modulo 2p).

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 (- p, p]; if the magnitude is zero, then so is the phase.

The functions realPart and imagPart extract the rectangular components of a complex number and the functions magnitude and phase extract the polar components of a complex number. The function conjugate computes the conjugate of a complex number in the usual way.

The magnitude and sign of a complex number are defined as follows:

```abs z   =  magnitude z :+ 0
signum 0   =  0
signum z@(x:+y)   =  x/r :+ y/r  where r = magnitude z
```

That is, 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.

```module 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))

```