2 module Complex(Complex((:+)), realPart, imagPart, conjugate, mkPolar,
3 cis, polar, magnitude, phase) where
7 data (RealFloat a) => Complex a = !a :+ !a deriving (Eq,Read,Show)
10 realPart, imagPart :: (RealFloat a) => Complex a -> a
14 conjugate :: (RealFloat a) => Complex a -> Complex a
15 conjugate (x:+y) = x :+ (-y)
17 mkPolar :: (RealFloat a) => a -> a -> Complex a
18 mkPolar r theta = r * cos theta :+ r * sin theta
20 cis :: (RealFloat a) => a -> Complex a
21 cis theta = cos theta :+ sin theta
23 polar :: (RealFloat a) => Complex a -> (a,a)
24 polar z = (magnitude z, phase z)
26 magnitude, phase :: (RealFloat a) => Complex a -> a
27 magnitude (x:+y) = scaleFloat k
28 (sqrt ((scaleFloat mk x)^2 + (scaleFloat mk y)^2))
29 where k = max (exponent x) (exponent y)
32 phase (x:+y) = atan2 y x
35 instance (RealFloat a) => Num (Complex a) where
36 (x:+y) + (x':+y') = (x+x') :+ (y+y')
37 (x:+y) - (x':+y') = (x-x') :+ (y-y')
38 (x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')
39 negate (x:+y) = negate x :+ negate y
40 abs z = magnitude z :+ 0
42 signum z@(x:+y) = x/r :+ y/r where r = magnitude z
43 fromInteger n = fromInteger n :+ 0
45 instance (RealFloat a) => Fractional (Complex a) where
46 (x:+y) / (x':+y') = (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
47 where x'' = scaleFloat k x'
49 k = - max (exponent x') (exponent y')
52 fromRational a = fromRational a :+ 0
54 instance (RealFloat a) => Floating (Complex a) where
56 exp (x:+y) = expx * cos y :+ expx * sin y
58 log z = log (magnitude z) :+ phase z
61 sqrt z@(x:+y) = u :+ (if y < 0 then -v else v)
62 where (u,v) = if x < 0 then (v',u') else (u',v')
64 u' = sqrt ((magnitude z + abs x) / 2)
66 sin (x:+y) = sin x * cosh y :+ cos x * sinh y
67 cos (x:+y) = cos x * cosh y :+ (- sin x * sinh y)
68 tan (x:+y) = (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
74 sinh (x:+y) = cos y * sinh x :+ sin y * cosh x
75 cosh (x:+y) = cos y * cosh x :+ sin y * sinh x
76 tanh (x:+y) = (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
82 asin z@(x:+y) = y':+(-x')
83 where (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
84 acos z@(x:+y) = y'':+(-x'')
85 where (x'':+y'') = log (z + ((-y'):+x'))
86 (x':+y') = sqrt (1 - z*z)
87 atan z@(x:+y) = y':+(-x')
88 where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))
90 asinh z = log (z + sqrt (1+z*z))
91 acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))
92 atanh z = log ((1+z) / sqrt (1-z*z))