{-|
Module      : Math.JacobiTheta
Description : Jacobi theta functions.
Copyright   : (c) Stéphane Laurent, 2023
License     : BSD3
Maintainer  : laurent_step@outlook.fr

Provides the four usual Jacobi theta functions, the Jacobi theta function 
with characteristics, the derivative of the first Jacobi theta function, 
as well as a function for the derivative at @0@ only of the first Jacobi 
theta function.
-}
module Math.JacobiTheta
  (
    jtheta1,
    jtheta2,
    jtheta3,
    jtheta4,
    jthetaAB,
    jtheta1Dash0,
    jtheta1Dash 
  )
  where
import Data.Complex ( imagPart, magnitude, realPart, Complex(..) )

type Cplx = Complex Double

i_ :: Cplx
i_ :: Complex Double
i_ = Double
0.0 forall a. a -> a -> Complex a
:+ Double
1.0

machinePrecision :: Double
machinePrecision :: Double
machinePrecision = Double
2forall a. Floating a => a -> a -> a
**(-Double
52)

areClose :: Cplx -> Cplx -> Bool
areClose :: Complex Double -> Complex Double -> Bool
areClose Complex Double
z1 Complex Double
z2 = forall a. RealFloat a => Complex a -> a
magnitude (Complex Double
z1 forall a. Num a => a -> a -> a
- Complex Double
z2) forall a. Ord a => a -> a -> Bool
< Double
epsilon forall a. Num a => a -> a -> a
* Double
h
  where
    epsilon :: Double
epsilon = Double
2.0 forall a. Num a => a -> a -> a
* Double
machinePrecision
    magn2 :: Double
magn2 = forall a. RealFloat a => Complex a -> a
magnitude Complex Double
z2
    h :: Double
h = if Double
magn2 forall a. Ord a => a -> a -> Bool
< Double
epsilon then Double
1.0 else forall a. Ord a => a -> a -> a
max (forall a. RealFloat a => Complex a -> a
magnitude Complex Double
z1) Double
magn2

modulo :: Double -> Int -> Double
modulo :: Double -> Int -> Double
modulo Double
a Int
p = 
  let p' :: Double
p' = forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
p
  in
  if Double
a forall a. Ord a => a -> a -> Bool
> Double
0 
    then Double
a forall a. Num a => a -> a -> a
- forall a b. (Integral a, Num b) => a -> b
fromIntegral(Int
p forall a. Num a => a -> a -> a
* forall a b. (RealFrac a, Integral b) => a -> b
floor(Double
aforall a. Fractional a => a -> a -> a
/Double
p'))  
    else Double
a forall a. Num a => a -> a -> a
- forall a b. (Integral a, Num b) => a -> b
fromIntegral(Int
p forall a. Num a => a -> a -> a
* forall a b. (RealFrac a, Integral b) => a -> b
ceiling(Double
aforall a. Fractional a => a -> a -> a
/Double
p'))

dologtheta4 :: Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta4 :: Complex Double -> Complex Double -> Int -> Int -> Complex Double
dologtheta4 Complex Double
z Complex Double
tau Int
passes Int
maxiter = 
  Complex Double -> Complex Double -> Int -> Int -> Complex Double
dologtheta3 (Complex Double
z forall a. Num a => a -> a -> a
+ Complex Double
0.5) Complex Double
tau (Int
passesforall a. Num a => a -> a -> a
+Int
1) Int
maxiter

dologtheta3 :: Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta3 :: Complex Double -> Complex Double -> Int -> Int -> Complex Double
dologtheta3 Complex Double
z Complex Double
tau Int
passes Int
maxiterloc
  | forall a. Complex a -> a
realPart Complex Double
tau2 forall a. Ord a => a -> a -> Bool
> Double
0.6  = Complex Double -> Complex Double -> Int -> Int -> Complex Double
dologtheta4 Complex Double
z (Complex Double
tau2 forall a. Num a => a -> a -> a
- Complex Double
1) (Int
passes forall a. Num a => a -> a -> a
+ Int
1) Int
maxiterloc
  | forall a. Complex a -> a
realPart Complex Double
tau2 forall a. Ord a => a -> a -> Bool
< -Double
0.6 = Complex Double -> Complex Double -> Int -> Int -> Complex Double
dologtheta4 Complex Double
z (Complex Double
tau2 forall a. Num a => a -> a -> a
+ Complex Double
1) (Int
passes forall a. Num a => a -> a -> a
+ Int
1) Int
maxiterloc
  | forall a. RealFloat a => Complex a -> a
magnitude Complex Double
tau2 forall a. Ord a => a -> a -> Bool
< Double
0.98 Bool -> Bool -> Bool
&& forall a. Complex a -> a
imagPart Complex Double
tau2 forall a. Ord a => a -> a -> Bool
< Double
0.98 = 
      Complex Double
i_ forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Complex Double
tauprime forall a. Num a => a -> a -> a
* Complex Double
z forall a. Num a => a -> a -> a
* Complex Double
z 
      forall a. Num a => a -> a -> a
+ Complex Double -> Complex Double -> Int -> Int -> Complex Double
dologtheta3 (Complex Double
z forall a. Num a => a -> a -> a
* Complex Double
tauprime) Complex Double
tauprime (Int
passes forall a. Num a => a -> a -> a
+ Int
1) Int
maxiterloc
      forall a. Num a => a -> a -> a
- forall a. Floating a => a -> a
log(forall a. Floating a => a -> a
sqrt Complex Double
tau2 forall a. Fractional a => a -> a -> a
/ forall a. Floating a => a -> a
sqrt Complex Double
i_) 
  | Bool
otherwise = Complex Double -> Complex Double -> Int -> Int -> Complex Double
argtheta3 Complex Double
z Complex Double
tau2 Int
0 Int
maxiterloc
    where
      rPtau :: Double
rPtau = forall a. Complex a -> a
realPart Complex Double
tau
      rPtau2 :: Double
rPtau2 = if Double
rPtau forall a. Ord a => a -> a -> Bool
> Double
0
        then Double -> Int -> Double
modulo (Double
rPtau forall a. Num a => a -> a -> a
+ Double
1) Int
2 forall a. Num a => a -> a -> a
- Double
1
        else Double -> Int -> Double
modulo (Double
rPtau forall a. Num a => a -> a -> a
- Double
1) Int
2 forall a. Num a => a -> a -> a
+ Double
1
      tau2 :: Complex Double
tau2 = Double
rPtau2 forall a. a -> a -> Complex a
:+ forall a. Complex a -> a
imagPart Complex Double
tau
      tauprime :: Complex Double
tauprime = -Complex Double
1 forall a. Fractional a => a -> a -> a
/ Complex Double
tau2

argtheta3 :: Cplx -> Cplx -> Int -> Int -> Cplx
argtheta3 :: Complex Double -> Complex Double -> Int -> Int -> Complex Double
argtheta3 Complex Double
z Complex Double
tau Int
passes Int
maxiterloc
  | Int
passes forall a. Ord a => a -> a -> Bool
> Int
maxiterloc = forall a. HasCallStack => [Char] -> a
error [Char]
"Reached maximal iteration."
  | Double
iPz forall a. Ord a => a -> a -> Bool
< -Double
iPtau forall a. Fractional a => a -> a -> a
/ Double
2 = Complex Double -> Complex Double -> Int -> Int -> Complex Double
argtheta3 (-Complex Double
zuse) Complex Double
tau (Int
passes forall a. Num a => a -> a -> a
+ Int
1) Int
maxiterloc
  | Double
iPz forall a. Ord a => a -> a -> Bool
>= Double
iPtau forall a. Fractional a => a -> a -> a
/ Double
2 = 
      -Complex Double
2 forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Complex Double
quotient forall a. Num a => a -> a -> a
* Complex Double
i_ forall a. Num a => a -> a -> a
* Complex Double
zmin 
      forall a. Num a => a -> a -> a
+ Complex Double -> Complex Double -> Int -> Int -> Complex Double
argtheta3 Complex Double
zmin Complex Double
tau (Int
passes forall a. Num a => a -> a -> a
+ Int
1) Int
maxiterloc
      forall a. Num a => a -> a -> a
- Complex Double
i_ forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Complex Double
tau forall a. Num a => a -> a -> a
* Complex Double
quotient forall a. Num a => a -> a -> a
* Complex Double
quotient
  | Bool
otherwise = Complex Double -> Complex Double -> Complex Double
calctheta3 Complex Double
zuse Complex Double
tau
    where
      iPz :: Double
iPz = forall a. Complex a -> a
imagPart Complex Double
z
      iPtau :: Double
iPtau = forall a. Complex a -> a
imagPart Complex Double
tau
      zuse :: Complex Double
zuse = Double -> Int -> Double
modulo (forall a. Complex a -> a
realPart Complex Double
z) Int
1 forall a. a -> a -> Complex a
:+ Double
iPz
      quotient :: Complex Double
quotient = Int -> Complex Double
fromInt forall a b. (a -> b) -> a -> b
$ forall a b. (RealFrac a, Integral b) => a -> b
floor(Double
iPz forall a. Fractional a => a -> a -> a
/ Double
iPtau forall a. Num a => a -> a -> a
+ Double
0.5)
      zmin :: Complex Double
zmin = Complex Double
zuse forall a. Num a => a -> a -> a
- Complex Double
tau forall a. Num a => a -> a -> a
* Complex Double
quotient
      fromInt :: Int -> Cplx
      fromInt :: Int -> Complex Double
fromInt = forall a b. (Integral a, Num b) => a -> b
fromIntegral

calctheta3 :: Cplx -> Cplx -> Cplx
calctheta3 :: Complex Double -> Complex Double -> Complex Double
calctheta3 Complex Double
z Complex Double
tau = 
    Int -> Complex Double -> Complex Double
go Int
1 Complex Double
1
    where
      qw :: Int -> Cplx
      qw :: Int -> Complex Double
qw Int
n = forall a. Floating a => a -> a
exp(Complex Double
inpi forall a. Num a => a -> a -> a
* (Complex Double
taun forall a. Num a => a -> a -> a
+ Complex Double
2 forall a. Num a => a -> a -> a
* Complex Double
z)) forall a. Num a => a -> a -> a
+ forall a. Floating a => a -> a
exp(Complex Double
inpi forall a. Num a => a -> a -> a
* (Complex Double
taun forall a. Num a => a -> a -> a
- Complex Double
2 forall a. Num a => a -> a -> a
* Complex Double
z))
        where
          n' :: Complex Double
n' = forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
n 
          inpi :: Complex Double
inpi = Complex Double
i_ forall a. Num a => a -> a -> a
* Complex Double
n' forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi 
          taun :: Complex Double
taun = Complex Double
n' forall a. Num a => a -> a -> a
* Complex Double
tau     
      go :: Int -> Complex Double -> Complex Double
go Int
n Complex Double
res
        | forall a. RealFloat a => a -> Bool
isNaN Double
modulus = forall a. HasCallStack => [Char] -> a
error [Char]
"NaN has occured in the summation."
        | forall a. RealFloat a => a -> Bool
isInfinite Double
modulus = forall a. HasCallStack => [Char] -> a
error [Char]
"Infinity reached in the summation."
--        | modulus == 0 = error "Zero has occured in the summation."

        | Int
n forall a. Ord a => a -> a -> Bool
>= Int
3 Bool -> Bool -> Bool
&& Complex Double -> Complex Double -> Bool
areClose Complex Double
res Complex Double
resnew = forall a. Floating a => a -> a
log Complex Double
res
        | Bool
otherwise = Int -> Complex Double -> Complex Double
go (Int
n forall a. Num a => a -> a -> a
+ Int
1) Complex Double
resnew
          where
            modulus :: Double
modulus = forall a. RealFloat a => Complex a -> a
magnitude Complex Double
res
            resnew :: Complex Double
resnew = Complex Double
res forall a. Num a => a -> a -> a
+ Int -> Complex Double
qw Int
n

-------------------------------------------------------------------------------

tauFromQ :: Cplx -> Cplx
tauFromQ :: Complex Double -> Complex Double
tauFromQ Complex Double
q = -Complex Double
i_ forall a. Num a => a -> a -> a
* forall a. Floating a => a -> a
log Complex Double
q forall a. Fractional a => a -> a -> a
/ forall a. Floating a => a
pi

checkQ :: Cplx -> Cplx
checkQ :: Complex Double -> Complex Double
checkQ Complex Double
q
  | forall a. RealFloat a => Complex a -> a
magnitude Complex Double
q forall a. Ord a => a -> a -> Bool
>= Double
1 = 
    forall a. HasCallStack => [Char] -> a
error [Char]
"The modulus of the nome must be smaller than one."
  | forall a. Complex a -> a
imagPart Complex Double
q forall a. Eq a => a -> a -> Bool
== Double
0 Bool -> Bool -> Bool
&& forall a. Complex a -> a
realPart Complex Double
q forall a. Ord a => a -> a -> Bool
<= Double
0 = 
    forall a. HasCallStack => [Char] -> a
error [Char]
"If the nome is real, it must be positive."
  | Bool
otherwise = Complex Double
q

getTauFromQ :: Cplx -> Cplx
getTauFromQ :: Complex Double -> Complex Double
getTauFromQ = Complex Double -> Complex Double
tauFromQ forall b c a. (b -> c) -> (a -> b) -> a -> c
. Complex Double -> Complex Double
checkQ

funM :: Cplx -> Cplx -> Cplx
funM :: Complex Double -> Complex Double -> Complex Double
funM Complex Double
z Complex Double
tau = Complex Double
i_ forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* (Complex Double
z forall a. Num a => a -> a -> a
+ Complex Double
tauforall a. Fractional a => a -> a -> a
/Complex Double
4)

ljtheta1 :: Cplx -> Cplx -> Cplx
ljtheta1 :: Complex Double -> Complex Double -> Complex Double
ljtheta1 Complex Double
z Complex Double
tau = Complex Double -> Complex Double -> Complex Double
ljtheta2 (Complex Double
z forall a. Num a => a -> a -> a
- Complex Double
0.5) Complex Double
tau

-- | First Jacobi theta function.

jtheta1 ::
     Complex Double -- ^ z

  -> Complex Double -- ^ q, the nome

  -> Complex Double
jtheta1 :: Complex Double -> Complex Double -> Complex Double
jtheta1 Complex Double
z Complex Double
q = forall a. Floating a => a -> a
exp(Complex Double -> Complex Double -> Complex Double
ljtheta1 (Complex Double
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Complex Double
tau)
  where
    tau :: Complex Double
tau = Complex Double -> Complex Double
getTauFromQ Complex Double
q

ljtheta2 :: Cplx -> Cplx -> Cplx
ljtheta2 :: Complex Double -> Complex Double -> Complex Double
ljtheta2 Complex Double
z Complex Double
tau = 
  Complex Double -> Complex Double -> Complex Double
funM Complex Double
z Complex Double
tau forall a. Num a => a -> a -> a
+ Complex Double -> Complex Double -> Int -> Int -> Complex Double
dologtheta3 (Complex Double
z forall a. Num a => a -> a -> a
+ Complex Double
0.5 forall a. Num a => a -> a -> a
* Complex Double
tau) Complex Double
tau Int
0 Int
1000

-- | Second Jacobi theta function.

jtheta2 ::
     Complex Double -- ^ z

  -> Complex Double -- ^ q, the nome

  -> Complex Double
jtheta2 :: Complex Double -> Complex Double -> Complex Double
jtheta2 Complex Double
z Complex Double
q = forall a. Floating a => a -> a
exp(Complex Double -> Complex Double -> Complex Double
ljtheta2 (Complex Double
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Complex Double
tau)
  where
    tau :: Complex Double
tau = Complex Double -> Complex Double
getTauFromQ Complex Double
q

-- | Third Jacobi theta function.

jtheta3 ::
     Complex Double -- ^ z

  -> Complex Double -- ^ q, the nome

  -> Complex Double
jtheta3 :: Complex Double -> Complex Double -> Complex Double
jtheta3 Complex Double
z Complex Double
q = forall a. Floating a => a -> a
exp(Complex Double -> Complex Double -> Int -> Int -> Complex Double
dologtheta3 (Complex Double
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Complex Double
tau Int
0 Int
1000)
  where
    tau :: Complex Double
tau = Complex Double -> Complex Double
getTauFromQ Complex Double
q

-- | Fourth Jacobi theta function.

jtheta4 ::
     Complex Double -- ^ z

  -> Complex Double -- ^ q, the nome

  -> Complex Double
jtheta4 :: Complex Double -> Complex Double -> Complex Double
jtheta4 Complex Double
z Complex Double
q = forall a. Floating a => a -> a
exp(Complex Double -> Complex Double -> Int -> Int -> Complex Double
dologtheta4 (Complex Double
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Complex Double
tau Int
0 Int
1000)
  where
    tau :: Complex Double
tau = Complex Double -> Complex Double
getTauFromQ Complex Double
q

jthetaAB' ::
     Complex Double -- ^ characteristic a

  -> Complex Double -- ^ characteristic b

  -> Complex Double -- ^ z

  -> Complex Double -- ^ tau

  -> Complex Double
jthetaAB' :: Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
jthetaAB' Complex Double
a Complex Double
b Complex Double
z Complex Double
tau = Complex Double
c forall a. Num a => a -> a -> a
* forall a. Floating a => a -> a
exp(Complex Double -> Complex Double -> Int -> Int -> Complex Double
dologtheta3 (Complex Double
alphaforall a. Num a => a -> a -> a
+Complex Double
beta) Complex Double
tau Int
0 Int
1000)
  where
    alpha :: Complex Double
alpha = Complex Double
a forall a. Num a => a -> a -> a
* Complex Double
tau 
    beta :: Complex Double
beta  = Complex Double
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi forall a. Num a => a -> a -> a
+ Complex Double
b
    c :: Complex Double
c     = forall a. Floating a => a -> a
exp(Complex Double
i_ forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Complex Double
a forall a. Num a => a -> a -> a
* (Complex Double
alpha forall a. Num a => a -> a -> a
+ Complex Double
2forall a. Num a => a -> a -> a
*Complex Double
beta)) 


-- | Jacobi theta function with characteristics. This is a family of functions, 

--  containing the first Jacobi theta function (@a=b=0.5@), the second Jacobi 

--  theta function (@a=0.5, b=0@), the third Jacobi theta function (@a=b=0@)

--  and the fourth Jacobi theta function (@a=0, b=0.5@). The examples given 

--  below show the periodicity-like properties of these functions:

--  

-- >>> import Data.Complex

-- >>> a = 2 :+ 0.3

-- >>> b = 1 :+ (-0.6)

-- >>> z = 0.1 :+ 0.4

-- >>> tau = 0.2 :+ 0.3

-- >>> im = 0 :+ 1 

-- >>> q = exp(im * pi * tau)

-- >>> jab = jthetaAB a b z q

-- >>> jthetaAB a b (z + pi) q

-- (-5.285746223832433e-3) :+ 0.1674462628348814

-- 

-- >>> jab * exp(2 * im * pi * a)

-- (-5.285746223831987e-3) :+ 0.16744626283488154

-- 

-- >>> jtheta_ab a b (z + pi*tau) q

-- 0.10389127606987271 :+ 0.10155646232306936

-- 

-- >>> jab * exp(-im * (pi*tau + 2*z + 2*pi*b))

-- 0.10389127606987278 :+ 0.10155646232306961

jthetaAB ::
     Complex Double -- ^ characteristic a

  -> Complex Double -- ^ characteristic b

  -> Complex Double -- ^ z

  -> Complex Double -- ^ q, the nome

  -> Complex Double
jthetaAB :: Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
jthetaAB Complex Double
a Complex Double
b Complex Double
z Complex Double
q = Complex Double
c forall a. Num a => a -> a -> a
* Complex Double -> Complex Double -> Complex Double
jtheta3 (Complex Double
alpha forall a. Num a => a -> a -> a
+ Complex Double
beta) Complex Double
q
  where
    tau :: Complex Double
tau = Complex Double -> Complex Double
getTauFromQ Complex Double
q
    alpha :: Complex Double
alpha = forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Complex Double
a forall a. Num a => a -> a -> a
* Complex Double
tau 
    beta :: Complex Double
beta  = Complex Double
z forall a. Num a => a -> a -> a
+ forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Complex Double
b
    c :: Complex Double
c     = forall a. Floating a => a -> a
exp(Complex Double
i_ forall a. Num a => a -> a -> a
* Complex Double
a forall a. Num a => a -> a -> a
* (Complex Double
alpha forall a. Num a => a -> a -> a
+ Complex Double
2forall a. Num a => a -> a -> a
*Complex Double
beta)) 
    -- c     = q**(a*a) * exp(2 * i_ * a * beta)


-- | Derivative at 0 of the first Jacobi theta function. This is much more 

--  efficient than evaluating @jtheta1Dash@ at @0@.

jtheta1Dash0 :: 
     Complex Double -- ^ q, the nome

  -> Complex Double
jtheta1Dash0 :: Complex Double -> Complex Double
jtheta1Dash0 Complex Double
q = 
  -Complex Double
2 forall a. Num a => a -> a -> a
* Complex Double
i_ forall a. Num a => a -> a -> a
* Complex Double
jab forall a. Num a => a -> a -> a
* Complex Double
jab forall a. Num a => a -> a -> a
* Complex Double
jab
  where
    tau :: Complex Double
tau = Complex Double -> Complex Double
getTauFromQ Complex Double
q
    jab :: Complex Double
jab = Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
jthetaAB' (Complex Double
1forall a. Fractional a => a -> a -> a
/Complex Double
6) Complex Double
0.5 Complex Double
0 (Complex Double
3forall a. Num a => a -> a -> a
*Complex Double
tau)

-- | Derivative of the first Jacobi theta function.

jtheta1Dash :: 
     Complex Double -- ^ z

  -> Complex Double -- ^ q, the nome

  -> Complex Double
jtheta1Dash :: Complex Double -> Complex Double -> Complex Double
jtheta1Dash Complex Double
z Complex Double
q = 
  Int
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
go Int
0 (Double
0.0 forall a. a -> a -> Complex a
:+ Double
0.0) Complex Double
1.0 (Complex Double
1.0 forall a. Fractional a => a -> a -> a
/ Complex Double
qsq) Complex Double
1.0
  where 
    q' :: Complex Double
q' = Complex Double -> Complex Double
checkQ Complex Double
q
    qsq :: Complex Double
qsq = Complex Double
q' forall a. Num a => a -> a -> a
* Complex Double
q'
    go :: Int -> Cplx -> Cplx -> Cplx -> Cplx -> Cplx
    go :: Int
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
go Int
n Complex Double
out Complex Double
alt Complex Double
q_2n Complex Double
q_n_np1 
      | Int
n forall a. Ord a => a -> a -> Bool
> Int
3000 = forall a. HasCallStack => [Char] -> a
error [Char]
"Reached 3000 iterations."
      | Complex Double -> Complex Double -> Bool
areClose Complex Double
out Complex Double
outnew = Complex Double
2.0 forall a. Num a => a -> a -> a
* forall a. Floating a => a -> a
sqrt (forall a. Floating a => a -> a
sqrt Complex Double
q) forall a. Num a => a -> a -> a
* Complex Double
out
      | Bool
otherwise = Int
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
go (Int
n forall a. Num a => a -> a -> a
+ Int
1) Complex Double
outnew (-Complex Double
alt) Complex Double
q_2np1 Complex Double
q_np1_np2
        where
          q_2np1 :: Complex Double
q_2np1 = Complex Double
q_2n forall a. Num a => a -> a -> a
* Complex Double
qsq
          q_np1_np2 :: Complex Double
q_np1_np2 = Complex Double
q_n_np1 forall a. Num a => a -> a -> a
* Complex Double
q_2np1
          n' :: Complex Double
n' = forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
n 
          k :: Complex Double
k = Complex Double
2.0 forall a. Num a => a -> a -> a
* Complex Double
n' forall a. Num a => a -> a -> a
+ Complex Double
1.0
          outnew :: Complex Double
outnew = Complex Double
out forall a. Num a => a -> a -> a
+ Complex Double
k forall a. Num a => a -> a -> a
* Complex Double
alt forall a. Num a => a -> a -> a
* Complex Double
q_np1_np2 forall a. Num a => a -> a -> a
* forall a. Floating a => a -> a
cos (Complex Double
k forall a. Num a => a -> a -> a
* Complex Double
z)