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</script></head><body><div id="package-header"><ul class="links" id="page-menu"><li><a href="index.html">Contents</a></li><li><a href="doc-index.html">Index</a></li></ul><p class="caption">haskell2010-1.1.0.1: Compatibility with Haskell 2010</p></div><div id="content"><div id="module-header"><table class="info"><tr><th>Safe Haskell</th><td>Safe</td></tr></table><p class="caption">Data.Complex</p></div><div id="table-of-contents"><p class="caption">Contents</p><ul><li><a href="#g:1">Rectangular form
</a></li><li><a href="#g:2">Polar form
</a></li><li><a href="#g:3">Conjugate
</a></li><li><a href="#g:4">Specification
</a></li></ul></div><div id="synopsis"><p id="control.syn" class="caption expander" onclick="toggleSection('syn')">Synopsis</p><ul id="section.syn" class="hide" onclick="toggleSection('syn')"><li class="src short"><span class="keyword">data</span> <a href="#t:Complex">Complex</a> a = !a <a href="#v::-43-">:+</a> !a</li><li class="src short"><a href="#v:realPart">realPart</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> a</li><li class="src short"><a href="#v:imagPart">imagPart</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> a</li><li class="src short"><a href="#v:mkPolar">mkPolar</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => a -> a -> <a href="Data-Complex.html#t:Complex">Complex</a> a</li><li class="src short"><a href="#v:cis">cis</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => a -> <a href="Data-Complex.html#t:Complex">Complex</a> a</li><li class="src short"><a href="#v:polar">polar</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> (a, a)</li><li class="src short"><a href="#v:magnitude">magnitude</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> a</li><li class="src short"><a href="#v:phase">phase</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> a</li><li class="src short"><a href="#v:conjugate">conjugate</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> <a href="Data-Complex.html#t:Complex">Complex</a> a</li></ul></div><div id="interface"><h1 id="g:1">Rectangular form
</h1><div class="top"><p class="src"><span class="keyword">data</span> <a name="t:Complex" class="def">Complex</a> a <a href="../base-4.5.1.0/src/Data-Complex.html#Complex" class="link">Source</a></p><div class="doc"><p>Complex numbers are an algebraic type.
</p><p>For a complex number <code>z</code>, <code><code><a href="Prelude.html#v:abs">abs</a></code> z</code> is a number with the magnitude of <code>z</code>,
but oriented in the positive real direction, whereas <code><code><a href="Prelude.html#v:signum">signum</a></code> z</code>
has the phase of <code>z</code>, but unit magnitude.
</p></div><div class="subs constructors"><p class="caption">Constructors</p><table><tr><td class="src">!a <a name="v::-43-" class="def">:+</a> !a</td><td class="doc"><p>forms a complex number from its real and imaginary
rectangular components.
</p></td></tr></table></div><div class="subs instances"><p id="control.i:Complex" class="caption collapser" onclick="toggleSection('i:Complex')">Instances</p><div id="section.i:Complex" class="show"><table><tr><td class="src"><a href="../base-4.5.1.0/Data-Typeable-Internal.html#t:Typeable1">Typeable1</a> <a href="Data-Complex.html#t:Complex">Complex</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:Eq">Eq</a> a => <a href="Prelude.html#t:Eq">Eq</a> (<a href="Data-Complex.html#t:Complex">Complex</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Prelude.html#t:Floating">Floating</a> (<a href="Data-Complex.html#t:Complex">Complex</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Prelude.html#t:Fractional">Fractional</a> (<a href="Data-Complex.html#t:Complex">Complex</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="../base-4.5.1.0/Data-Data.html#t:Data">Data</a> a => <a href="../base-4.5.1.0/Data-Data.html#t:Data">Data</a> (<a href="Data-Complex.html#t:Complex">Complex</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Prelude.html#t:Num">Num</a> (<a href="Data-Complex.html#t:Complex">Complex</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:Read">Read</a> a => <a href="Prelude.html#t:Read">Read</a> (<a href="Data-Complex.html#t:Complex">Complex</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:Show">Show</a> a => <a href="Prelude.html#t:Show">Show</a> (<a href="Data-Complex.html#t:Complex">Complex</a> a)</td><td class="doc empty"> </td></tr></table></div></div></div><div class="top"><p class="src"><a name="v:realPart" class="def">realPart</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> a<a href="../base-4.5.1.0/src/Data-Complex.html#realPart" class="link">Source</a></p><div class="doc"><p>Extracts the real part of a complex number.
</p></div></div><div class="top"><p class="src"><a name="v:imagPart" class="def">imagPart</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> a<a href="../base-4.5.1.0/src/Data-Complex.html#imagPart" class="link">Source</a></p><div class="doc"><p>Extracts the imaginary part of a complex number.
</p></div></div><h1 id="g:2">Polar form
</h1><div class="top"><p class="src"><a name="v:mkPolar" class="def">mkPolar</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => a -> a -> <a href="Data-Complex.html#t:Complex">Complex</a> a<a href="../base-4.5.1.0/src/Data-Complex.html#mkPolar" class="link">Source</a></p><div class="doc"><p>Form a complex number from polar components of magnitude and phase.
</p></div></div><div class="top"><p class="src"><a name="v:cis" class="def">cis</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => a -> <a href="Data-Complex.html#t:Complex">Complex</a> a<a href="../base-4.5.1.0/src/Data-Complex.html#cis" class="link">Source</a></p><div class="doc"><p><code><code><a href="Data-Complex.html#v:cis">cis</a></code> t</code> is a complex value with magnitude <code>1</code>
and phase <code>t</code> (modulo <code>2*<code><a href="Prelude.html#v:pi">pi</a></code></code>).
</p></div></div><div class="top"><p class="src"><a name="v:polar" class="def">polar</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> (a, a)<a href="../base-4.5.1.0/src/Data-Complex.html#polar" class="link">Source</a></p><div class="doc"><p>The function <code><a href="Data-Complex.html#v:polar">polar</a></code> takes a complex number and
returns a (magnitude, phase) pair in canonical form:
the magnitude is nonnegative, and the phase in the range <code>(-<code><a href="Prelude.html#v:pi">pi</a></code>, <code><a href="Prelude.html#v:pi">pi</a></code>]</code>;
if the magnitude is zero, then so is the phase.
</p></div></div><div class="top"><p class="src"><a name="v:magnitude" class="def">magnitude</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> a<a href="../base-4.5.1.0/src/Data-Complex.html#magnitude" class="link">Source</a></p><div class="doc"><p>The nonnegative magnitude of a complex number.
</p></div></div><div class="top"><p class="src"><a name="v:phase" class="def">phase</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> a<a href="../base-4.5.1.0/src/Data-Complex.html#phase" class="link">Source</a></p><div class="doc"><p>The phase of a complex number, in the range <code>(-<code><a href="Prelude.html#v:pi">pi</a></code>, <code><a href="Prelude.html#v:pi">pi</a></code>]</code>.
If the magnitude is zero, then so is the phase.
</p></div></div><h1 id="g:3">Conjugate
</h1><div class="top"><p class="src"><a name="v:conjugate" class="def">conjugate</a> :: <a href="Prelude.html#t:RealFloat">RealFloat</a> a => <a href="Data-Complex.html#t:Complex">Complex</a> a -> <a href="Data-Complex.html#t:Complex">Complex</a> a<a href="../base-4.5.1.0/src/Data-Complex.html#conjugate" class="link">Source</a></p><div class="doc"><p>The conjugate of a complex number.
</p></div></div><h1 id="g:4">Specification
</h1><div class="doc"><pre> 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))
</pre></div></div></div><div id="footer"><p>Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.11.0</p></div></body></html>
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