Data.Complex

Complex numbers.

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.

The Foldable and Traversable instances traverse the real part first.

Constructors

!a :+ !a infix 6

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

Instances

Monad Complex

Since: 4.9.0.0

Methods

(>>=) :: Complex a -> (a -> Complex b) -> Complex b

(>>) :: Complex a -> Complex b -> Complex b

return :: a -> Complex a

fail :: String -> Complex a

Functor Complex

Methods

fmap :: (a -> b) -> Complex a -> Complex b

(<$) :: a -> Complex b -> Complex a

Applicative Complex

Since: 4.9.0.0

Methods

pure :: a -> Complex a

(<*>) :: Complex (a -> b) -> Complex a -> Complex b

liftA2 :: (a -> b -> c) -> Complex a -> Complex b -> Complex c

(*>) :: Complex a -> Complex b -> Complex b

(<*) :: Complex a -> Complex b -> Complex a

Foldable Complex

Methods

fold :: Monoid m => Complex m -> m

foldMap :: Monoid m => (a -> m) -> Complex a -> m

foldr :: (a -> b -> b) -> b -> Complex a -> b

foldr' :: (a -> b -> b) -> b -> Complex a -> b

foldl :: (b -> a -> b) -> b -> Complex a -> b

foldl' :: (b -> a -> b) -> b -> Complex a -> b

foldr1 :: (a -> a -> a) -> Complex a -> a

foldl1 :: (a -> a -> a) -> Complex a -> a

toList :: Complex a -> [a]

null :: Complex a -> Bool

length :: Complex a -> Int

elem :: Eq a => a -> Complex a -> Bool

maximum :: Ord a => Complex a -> a

minimum :: Ord a => Complex a -> a

sum :: Num a => Complex a -> a

product :: Num a => Complex a -> a

Traversable Complex

Methods

traverse :: Applicative f => (a -> f b) -> Complex a -> f (Complex b)

sequenceA :: Applicative f => Complex (f a) -> f (Complex a)

mapM :: Monad m => (a -> m b) -> Complex a -> m (Complex b)

sequence :: Monad m => Complex (m a) -> m (Complex a)

Eq a => Eq (Complex a)

Methods

(==) :: Complex a -> Complex a -> Bool

(/=) :: Complex a -> Complex a -> Bool

RealFloat a => Floating (Complex a)

Since: 2.1

Methods

pi :: Complex a

exp :: Complex a -> Complex a

log :: Complex a -> Complex a

sqrt :: Complex a -> Complex a

(**) :: Complex a -> Complex a -> Complex a

logBase :: Complex a -> Complex a -> Complex a

sin :: Complex a -> Complex a

cos :: Complex a -> Complex a

tan :: Complex a -> Complex a

asin :: Complex a -> Complex a

acos :: Complex a -> Complex a

atan :: Complex a -> Complex a

sinh :: Complex a -> Complex a

cosh :: Complex a -> Complex a

tanh :: Complex a -> Complex a

asinh :: Complex a -> Complex a

acosh :: Complex a -> Complex a

atanh :: Complex a -> Complex a

log1p :: Complex a -> Complex a

expm1 :: Complex a -> Complex a

log1pexp :: Complex a -> Complex a

log1mexp :: Complex a -> Complex a

RealFloat a => Fractional (Complex a)

Since: 2.1

Methods

(/) :: Complex a -> Complex a -> Complex a

recip :: Complex a -> Complex a

fromRational :: Rational -> Complex a

Data a => Data (Complex a)

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Complex a -> c (Complex a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Complex a)

toConstr :: Complex a -> Constr

dataTypeOf :: Complex a -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (Complex a))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Complex a))

gmapT :: (forall b. Data b => b -> b) -> Complex a -> Complex a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Complex a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Complex a -> r

gmapQ :: (forall d. Data d => d -> u) -> Complex a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Complex a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

RealFloat a => Num (Complex a)

Since: 2.1

Methods

(+) :: Complex a -> Complex a -> Complex a

(-) :: Complex a -> Complex a -> Complex a

(*) :: Complex a -> Complex a -> Complex a

negate :: Complex a -> Complex a

abs :: Complex a -> Complex a

signum :: Complex a -> Complex a

fromInteger :: Integer -> Complex a

Read a => Read (Complex a)

Methods

readsPrec :: Int -> ReadS (Complex a)

readList :: ReadS [Complex a]

readPrec :: ReadPrec (Complex a)

readListPrec :: ReadPrec [Complex a]

Show a => Show (Complex a)

Methods

showsPrec :: Int -> Complex a -> ShowS

show :: Complex a -> String

showList :: [Complex a] -> ShowS

Generic (Complex a)

Associated Types

type Rep (Complex a) :: * -> *

Methods

from :: Complex a -> Rep (Complex a) x

to :: Rep (Complex a) x -> Complex a

Storable a => Storable (Complex a)

Since: 4.8.0.0

Methods

sizeOf :: Complex a -> Int

alignment :: Complex a -> Int

peekElemOff :: Ptr (Complex a) -> Int -> IO (Complex a)

pokeElemOff :: Ptr (Complex a) -> Int -> Complex a -> IO ()

peekByteOff :: Ptr b -> Int -> IO (Complex a)

pokeByteOff :: Ptr b -> Int -> Complex a -> IO ()

peek :: Ptr (Complex a) -> IO (Complex a)

poke :: Ptr (Complex a) -> Complex a -> IO ()

Generic1 * Complex

Associated Types

type Rep1 Complex (f :: Complex -> *) :: k -> *

Methods

from1 :: f a -> Rep1 Complex f a

to1 :: Rep1 Complex f a -> f a

type Rep (Complex a)

type Rep (Complex a) = D1 * (MetaData "Complex" "Data.Complex" "base" False) (C1 * (MetaCons ":+" (InfixI NotAssociative 6) False) ((:*:) * (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 * a)) (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 * a))))

type Rep1 * Complex

type Rep1 * Complex = D1 * (MetaData "Complex" "Data.Complex" "base" False) (C1 * (MetaCons ":+" (InfixI NotAssociative 6) False) ((:*:) * (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1) (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1)))

realPart :: Complex a -> a

Extracts the real part of a complex number.

imagPart :: Complex a -> a

Extracts the imaginary part of a complex number.

Polar form

mkPolar :: Floating a => a -> a -> Complex a

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

cis :: Floating 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 :: Num a => Complex a -> Complex a

The conjugate of a complex number.