Number.Complex

Complex numbers.

Cartesian form

data T a

Complex numbers are an algebraic type.

Instances

Functor T

Defined in Number.Complex

Methods

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

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

C T

Defined in Number.Complex

Methods

zero :: C a => T a

(<+>) :: C a => T a -> T a -> T a

(*>) :: C a => a -> T a -> T a

C a b => C a (T b)

The '(*>)' method can't replace scale because it requires the Algebra.Module constraint

Defined in Number.Complex

Methods

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

C a b => C a (T b)

Defined in Number.Complex

(Show v, C v, C v, C a v) => C a (T v)

Defined in Number.Complex

Methods

toScalar :: T v -> a

toMaybeScalar :: T v -> Maybe a

fromScalar :: a -> T v

(C a, C a v) => C a (T v)

Defined in Number.Complex

Methods

norm :: T v -> a

(Ord a, C a v) => C a (T v)

Defined in Number.Complex

Methods

norm :: T v -> a

(C a, Sqr a b) => C a (T b)

Defined in Number.Complex

Methods

norm :: T b -> a

Sqr a b => Sqr a (T b)

Defined in Number.Complex

Methods

normSqr :: T b -> a

Eq a => Eq (T a)

Defined in Number.Complex

Methods

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

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

(Floating a, Eq a) => Fractional (T a)

Defined in Number.Complex

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational :: Rational -> T a

(Floating a, Eq a) => Num (T a)

Defined in Number.Complex

Methods

(+) :: T a -> T a -> T a

(-) :: T a -> T a -> T a

(*) :: T a -> T a -> T a

negate :: T a -> T a

abs :: T a -> T a

signum :: T a -> T a

fromInteger :: Integer -> T a

Read a => Read (T a)

Defined in Number.Complex

Methods

readsPrec :: Int -> ReadS (T a)

readList :: ReadS [T a]

readPrec :: ReadPrec (T a)

readListPrec :: ReadPrec [T a]

Show a => Show (T a)

Defined in Number.Complex

Methods

showsPrec :: Int -> T a -> ShowS

show :: T a -> String

showList :: [T a] -> ShowS

Arbitrary a => Arbitrary (T a)

Defined in Number.Complex

Methods

arbitrary :: Gen (T a)

shrink :: T a -> [T a]

Storable a => Storable (T a)

Defined in Number.Complex

Methods

sizeOf :: T a -> Int

alignment :: T a -> Int

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

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

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

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

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

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

C a => C (T a)

Defined in Number.Complex

Methods

compare :: T a -> T a -> Ordering

C a => C (T a)

Defined in Number.Complex

Methods

zero :: T a

(+) :: T a -> T a -> T a

(-) :: T a -> T a -> T a

negate :: T a -> T a

C a => C (T a)

Defined in Number.Complex

Methods

isZero :: T a -> Bool

C a => C (T a)

Defined in Number.Complex

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Defined in Number.Complex

Methods

div :: T a -> T a -> T a

mod :: T a -> T a -> T a

divMod :: T a -> T a -> (T a, T a)

(Ord a, C a) => C (T a)

Defined in Number.Complex

Methods

isUnit :: T a -> Bool

stdAssociate :: T a -> T a

stdUnit :: T a -> T a

stdUnitInv :: T a -> T a

(Ord a, C a, C a) => C (T a)

Defined in Number.Complex

Methods

extendedGCD :: T a -> T a -> (T a, (T a, T a))

gcd :: T a -> T a -> T a

lcm :: T a -> T a -> T a

(C a, C a, C a) => C (T a)

Defined in Number.Complex

Methods

abs :: T a -> T a

signum :: T a -> T a

C a => C (T a)

Defined in Number.Complex

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

(^-) :: T a -> Integer -> T a

(C a, C a, Power a) => C (T a)

Defined in Number.Complex

Methods

sqrt :: T a -> T a

root :: Integer -> T a -> T a

(^/) :: T a -> Rational -> T a

(C a, C a, C a, Power a) => C (T a)

Defined in Number.Complex

Methods

pi :: T a

exp :: T a -> T a

log :: T a -> T a

logBase :: T a -> T a -> T a

(**) :: T a -> T a -> T a

sin :: T a -> T a

cos :: T a -> T a

tan :: T a -> T a

asin :: T a -> T a

acos :: T a -> T a

atan :: T a -> T a

sinh :: T a -> T a

cosh :: T a -> T a

tanh :: T a -> T a

asinh :: T a -> T a

acosh :: T a -> T a

atanh :: T a -> T a

imaginaryUnit :: C a => T a

fromReal :: C a => a -> T a

(+:) :: a -> a -> T a infix 6

Construct a complex number from real and imaginary part.

(-:) :: C a => a -> a -> T a

Construct a complex number with negated imaginary part.

scale :: C a => a -> T a -> T a

Scale a complex number by a real number.

exp :: C a => T a -> T a

Exponential of a complex number with minimal type class constraints.

quarterLeft :: C a => T a -> T a

Turn the point one quarter to the right.

quarterRight :: C a => T a -> T a

Turn the point one quarter to the right.

Polar form

fromPolar :: C a => a -> a -> T a

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

cis :: C a => a -> T a

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

signum :: (C a, C a) => T a -> T a

Scale a complex number to magnitude 1.

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.

signumNorm :: (C a, C a a, C a) => T a -> T a

toPolar :: (C a, C a) => T a -> (a, a)

The function toPolar 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 :: C a => T a -> a

magnitudeSqr :: C a => T a -> a

phase :: (C a, C a) => T 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 :: C a => T a -> T a

The conjugate of a complex number.

Properties

propPolar :: (C a, C a) => T a -> Bool

Auxiliary classes

class C a => Power a where

We like to build the Complex Algebraic instance on top of the Algebraic instance of the scalar type. This poses no problem to sqrt. However, root requires computing the complex argument which is a transcendent operation. In order to keep the type class dependencies clean for more sophisticated algebraic number types, we introduce a type class which actually performs the radix operation.

Minimal complete definition

power

Methods

power :: Rational -> T a -> T a

Instances

Power Double

Defined in Number.Complex

Methods

power :: Rational -> T Double -> T Double

Power Float

Defined in Number.Complex

Methods

power :: Rational -> T Float -> T Float

Power T

Defined in Number.Positional.Check

Methods

power :: Rational -> T0 T -> T0 T

defltPow :: (C a, C a) => Rational -> T a -> T a