module Math.JacobiTheta
(
jtheta1,
jtheta1',
jtheta2,
jtheta2',
jtheta3,
jtheta3',
jtheta4,
jtheta4',
jthetaAB,
jthetaAB',
jtheta1Dash0,
jtheta1Dash
)
where
import Data.Complex ( imagPart, magnitude, realPart, Complex(..) )
type Cplx = Complex Double
i_ :: Cplx
i_ :: Cplx
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 :: Cplx -> Cplx -> Bool
areClose Cplx
z1 Cplx
z2 = forall a. RealFloat a => Complex a -> a
magnitude (Cplx
z1 forall a. Num a => a -> a -> a
- Cplx
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 Cplx
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 Cplx
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 :: Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta4 Cplx
z Cplx
tau Int
passes Int
maxiter =
Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta3 (Cplx
z forall a. Num a => a -> a -> a
+ Cplx
0.5) Cplx
tau (Int
passesforall a. Num a => a -> a -> a
+Int
1) Int
maxiter
dologtheta3 :: Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta3 :: Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta3 Cplx
z Cplx
tau Int
passes Int
maxiterloc
| forall a. Complex a -> a
realPart Cplx
tau2 forall a. Ord a => a -> a -> Bool
> Double
0.6 = Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta4 Cplx
z (Cplx
tau2 forall a. Num a => a -> a -> a
- Cplx
1) (Int
passes forall a. Num a => a -> a -> a
+ Int
1) Int
maxiterloc
| forall a. Complex a -> a
realPart Cplx
tau2 forall a. Ord a => a -> a -> Bool
< -Double
0.6 = Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta4 Cplx
z (Cplx
tau2 forall a. Num a => a -> a -> a
+ Cplx
1) (Int
passes forall a. Num a => a -> a -> a
+ Int
1) Int
maxiterloc
| forall a. RealFloat a => Complex a -> a
magnitude Cplx
tau2 forall a. Ord a => a -> a -> Bool
< Double
0.98 Bool -> Bool -> Bool
&& forall a. Complex a -> a
imagPart Cplx
tau2 forall a. Ord a => a -> a -> Bool
< Double
0.98 =
Cplx
i_ forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Cplx
tauprime forall a. Num a => a -> a -> a
* Cplx
z forall a. Num a => a -> a -> a
* Cplx
z
forall a. Num a => a -> a -> a
+ Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta3 (Cplx
z forall a. Num a => a -> a -> a
* Cplx
tauprime) Cplx
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 Cplx
tau2 forall a. Fractional a => a -> a -> a
/ forall a. Floating a => a -> a
sqrt Cplx
i_)
| Bool
otherwise = Cplx -> Cplx -> Int -> Int -> Cplx
argtheta3 Cplx
z Cplx
tau2 Int
0 Int
maxiterloc
where
rPtau :: Double
rPtau = forall a. Complex a -> a
realPart Cplx
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 :: Cplx
tau2 = Double
rPtau2 forall a. a -> a -> Complex a
:+ forall a. Complex a -> a
imagPart Cplx
tau
tauprime :: Cplx
tauprime = -Cplx
1 forall a. Fractional a => a -> a -> a
/ Cplx
tau2
argtheta3 :: Cplx -> Cplx -> Int -> Int -> Cplx
argtheta3 :: Cplx -> Cplx -> Int -> Int -> Cplx
argtheta3 Cplx
z Cplx
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 = Cplx -> Cplx -> Int -> Int -> Cplx
argtheta3 (-Cplx
zuse) Cplx
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 =
-Cplx
2 forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Cplx
quotient forall a. Num a => a -> a -> a
* Cplx
i_ forall a. Num a => a -> a -> a
* Cplx
zmin
forall a. Num a => a -> a -> a
+ Cplx -> Cplx -> Int -> Int -> Cplx
argtheta3 Cplx
zmin Cplx
tau (Int
passes forall a. Num a => a -> a -> a
+ Int
1) Int
maxiterloc
forall a. Num a => a -> a -> a
- Cplx
i_ forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Cplx
tau forall a. Num a => a -> a -> a
* Cplx
quotient forall a. Num a => a -> a -> a
* Cplx
quotient
| Bool
otherwise = Cplx -> Cplx -> Cplx
calctheta3 Cplx
zuse Cplx
tau
where
iPz :: Double
iPz = forall a. Complex a -> a
imagPart Cplx
z
iPtau :: Double
iPtau = forall a. Complex a -> a
imagPart Cplx
tau
zuse :: Cplx
zuse = Double -> Int -> Double
modulo (forall a. Complex a -> a
realPart Cplx
z) Int
1 forall a. a -> a -> Complex a
:+ Double
iPz
quotient :: Cplx
quotient = Int -> Cplx
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 :: Cplx
zmin = Cplx
zuse forall a. Num a => a -> a -> a
- Cplx
tau forall a. Num a => a -> a -> a
* Cplx
quotient
fromInt :: Int -> Cplx
fromInt :: Int -> Cplx
fromInt = forall a b. (Integral a, Num b) => a -> b
fromIntegral
calctheta3 :: Cplx -> Cplx -> Cplx
calctheta3 :: Cplx -> Cplx -> Cplx
calctheta3 Cplx
z Cplx
tau =
Int -> Cplx -> Cplx
go Int
1 Cplx
1
where
qw :: Int -> Cplx
qw :: Int -> Cplx
qw Int
n = forall a. Floating a => a -> a
exp(Cplx
inpi forall a. Num a => a -> a -> a
* (Cplx
taun forall a. Num a => a -> a -> a
+ Cplx
2 forall a. Num a => a -> a -> a
* Cplx
z)) forall a. Num a => a -> a -> a
+ forall a. Floating a => a -> a
exp(Cplx
inpi forall a. Num a => a -> a -> a
* (Cplx
taun forall a. Num a => a -> a -> a
- Cplx
2 forall a. Num a => a -> a -> a
* Cplx
z))
where
n' :: Cplx
n' = forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
n
inpi :: Cplx
inpi = Cplx
i_ forall a. Num a => a -> a -> a
* Cplx
n' forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi
taun :: Cplx
taun = Cplx
n' forall a. Num a => a -> a -> a
* Cplx
tau
go :: Int -> Cplx -> Cplx
go Int
n Cplx
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."
| Int
n forall a. Ord a => a -> a -> Bool
>= Int
3 Bool -> Bool -> Bool
&& Cplx -> Cplx -> Bool
areClose Cplx
res Cplx
resnew = forall a. Floating a => a -> a
log Cplx
res
| Bool
otherwise = Int -> Cplx -> Cplx
go (Int
n forall a. Num a => a -> a -> a
+ Int
1) Cplx
resnew
where
modulus :: Double
modulus = forall a. RealFloat a => Complex a -> a
magnitude Cplx
res
resnew :: Cplx
resnew = Cplx
res forall a. Num a => a -> a -> a
+ Int -> Cplx
qw Int
n
tauFromQ :: Cplx -> Cplx
tauFromQ :: Cplx -> Cplx
tauFromQ Cplx
q = if forall a. Complex a -> a
imagPart Cplx
q forall a. Eq a => a -> a -> Bool
== Double
0 Bool -> Bool -> Bool
&& forall a. Complex a -> a
realPart Cplx
q forall a. Ord a => a -> a -> Bool
< Double
0
then Double
1 forall a. a -> a -> Complex a
:+ (-forall a. Floating a => a -> a
log(-(forall a. Complex a -> a
realPart Cplx
q)) forall a. Fractional a => a -> a -> a
/ forall a. Floating a => a
pi)
else -Cplx
i_ forall a. Num a => a -> a -> a
* forall a. Floating a => a -> a
log Cplx
q forall a. Fractional a => a -> a -> a
/ forall a. Floating a => a
pi
checkQ :: Cplx -> Cplx
checkQ :: Cplx -> Cplx
checkQ Cplx
q
| forall a. RealFloat a => Complex a -> a
magnitude Cplx
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."
| Cplx
q forall a. Eq a => a -> a -> Bool
== Cplx
0 =
forall a. HasCallStack => [Char] -> a
error [Char]
"The nome cannot be zero."
| Bool
otherwise = Cplx
q
getTauFromQ :: Cplx -> Cplx
getTauFromQ :: Cplx -> Cplx
getTauFromQ = Cplx -> Cplx
tauFromQ forall b c a. (b -> c) -> (a -> b) -> a -> c
. Cplx -> Cplx
checkQ
funM :: Cplx -> Cplx -> Cplx
funM :: Cplx -> Cplx -> Cplx
funM Cplx
z Cplx
tau = Cplx
i_ forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* (Cplx
z forall a. Num a => a -> a -> a
+ Cplx
tauforall a. Fractional a => a -> a -> a
/Cplx
4)
ljtheta1 :: Cplx -> Cplx -> Cplx
ljtheta1 :: Cplx -> Cplx -> Cplx
ljtheta1 Cplx
z Cplx
tau = Cplx -> Cplx -> Cplx
ljtheta2 (Cplx
z forall a. Num a => a -> a -> a
- Cplx
0.5) Cplx
tau
jtheta1 ::
Complex Double
-> Complex Double
-> Complex Double
jtheta1 :: Cplx -> Cplx -> Cplx
jtheta1 Cplx
z Cplx
q = forall a. Floating a => a -> a
exp(Cplx -> Cplx -> Cplx
ljtheta1 (Cplx
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Cplx
tau)
where
tau :: Cplx
tau = Cplx -> Cplx
getTauFromQ Cplx
q
jtheta1' ::
Complex Double
-> Complex Double
-> Complex Double
jtheta1' :: Cplx -> Cplx -> Cplx
jtheta1' Cplx
z Cplx
tau
| forall a. Complex a -> a
imagPart Cplx
tau forall a. Ord a => a -> a -> Bool
<= Double
0 = forall a. HasCallStack => [Char] -> a
error [Char]
"`tau` must have a nonnegative imaginary part."
| Bool
otherwise = forall a. Floating a => a -> a
exp(Cplx -> Cplx -> Cplx
ljtheta1 (Cplx
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Cplx
tau)
ljtheta2 :: Cplx -> Cplx -> Cplx
ljtheta2 :: Cplx -> Cplx -> Cplx
ljtheta2 Cplx
z Cplx
tau =
Cplx -> Cplx -> Cplx
funM Cplx
z Cplx
tau forall a. Num a => a -> a -> a
+ Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta3 (Cplx
z forall a. Num a => a -> a -> a
+ Cplx
0.5 forall a. Num a => a -> a -> a
* Cplx
tau) Cplx
tau Int
0 Int
1000
jtheta2 ::
Complex Double
-> Complex Double
-> Complex Double
jtheta2 :: Cplx -> Cplx -> Cplx
jtheta2 Cplx
z Cplx
q = forall a. Floating a => a -> a
exp(Cplx -> Cplx -> Cplx
ljtheta2 (Cplx
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Cplx
tau)
where
tau :: Cplx
tau = Cplx -> Cplx
getTauFromQ Cplx
q
jtheta2' ::
Complex Double
-> Complex Double
-> Complex Double
jtheta2' :: Cplx -> Cplx -> Cplx
jtheta2' Cplx
z Cplx
tau
| forall a. Complex a -> a
imagPart Cplx
tau forall a. Ord a => a -> a -> Bool
<= Double
0 = forall a. HasCallStack => [Char] -> a
error [Char]
"`tau` must have a nonnegative imaginary part."
| Bool
otherwise = forall a. Floating a => a -> a
exp(Cplx -> Cplx -> Cplx
ljtheta2 (Cplx
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Cplx
tau)
jtheta3 ::
Complex Double
-> Complex Double
-> Complex Double
jtheta3 :: Cplx -> Cplx -> Cplx
jtheta3 Cplx
z Cplx
q = forall a. Floating a => a -> a
exp(Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta3 (Cplx
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Cplx
tau Int
0 Int
1000)
where
tau :: Cplx
tau = Cplx -> Cplx
getTauFromQ Cplx
q
jtheta3' ::
Complex Double
-> Complex Double
-> Complex Double
jtheta3' :: Cplx -> Cplx -> Cplx
jtheta3' Cplx
z Cplx
tau
| forall a. Complex a -> a
imagPart Cplx
tau forall a. Ord a => a -> a -> Bool
<= Double
0 = forall a. HasCallStack => [Char] -> a
error [Char]
"`tau` must have a nonnegative imaginary part."
| Bool
otherwise = forall a. Floating a => a -> a
exp(Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta3 (Cplx
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Cplx
tau Int
0 Int
1000)
jtheta4 ::
Complex Double
-> Complex Double
-> Complex Double
jtheta4 :: Cplx -> Cplx -> Cplx
jtheta4 Cplx
z Cplx
q = forall a. Floating a => a -> a
exp(Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta4 (Cplx
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Cplx
tau Int
0 Int
1000)
where
tau :: Cplx
tau = Cplx -> Cplx
getTauFromQ Cplx
q
jtheta4' ::
Complex Double
-> Complex Double
-> Complex Double
jtheta4' :: Cplx -> Cplx -> Cplx
jtheta4' Cplx
z Cplx
tau
| forall a. Complex a -> a
imagPart Cplx
tau forall a. Ord a => a -> a -> Bool
<= Double
0 = forall a. HasCallStack => [Char] -> a
error [Char]
"`tau` must have a nonnegative imaginary part."
| Bool
otherwise = forall a. Floating a => a -> a
exp(Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta4 (Cplx
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi) Cplx
tau Int
0 Int
1000)
jthetaAB ::
Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
jthetaAB :: Cplx -> Cplx -> Cplx -> Cplx -> Cplx
jthetaAB Cplx
a Cplx
b Cplx
z Cplx
q = Cplx
c forall a. Num a => a -> a -> a
* Cplx -> Cplx -> Cplx
jtheta3 (Cplx
alpha forall a. Num a => a -> a -> a
+ Cplx
beta) Cplx
q
where
tau :: Cplx
tau = Cplx -> Cplx
getTauFromQ Cplx
q
alpha :: Cplx
alpha = forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Cplx
a forall a. Num a => a -> a -> a
* Cplx
tau
beta :: Cplx
beta = Cplx
z forall a. Num a => a -> a -> a
+ forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Cplx
b
c :: Cplx
c = forall a. Floating a => a -> a
exp(Cplx
i_ forall a. Num a => a -> a -> a
* Cplx
a forall a. Num a => a -> a -> a
* (Cplx
alpha forall a. Num a => a -> a -> a
+ Cplx
2forall a. Num a => a -> a -> a
*Cplx
beta))
jthetaAB' ::
Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
-> Complex Double
jthetaAB' :: Cplx -> Cplx -> Cplx -> Cplx -> Cplx
jthetaAB' Cplx
a Cplx
b Cplx
z Cplx
tau = if forall a. Complex a -> a
imagPart Cplx
tau forall a. Ord a => a -> a -> Bool
<= Double
0
then forall a. HasCallStack => [Char] -> a
error [Char]
"`tau` must have a nonnegative imaginary part."
else Cplx
c forall a. Num a => a -> a -> a
* forall a. Floating a => a -> a
exp(Cplx -> Cplx -> Int -> Int -> Cplx
dologtheta3 (Cplx
alphaforall a. Num a => a -> a -> a
+Cplx
beta) Cplx
tau Int
0 Int
1000)
where
alpha :: Cplx
alpha = Cplx
a forall a. Num a => a -> a -> a
* Cplx
tau
beta :: Cplx
beta = Cplx
zforall a. Fractional a => a -> a -> a
/forall a. Floating a => a
pi forall a. Num a => a -> a -> a
+ Cplx
b
c :: Cplx
c = forall a. Floating a => a -> a
exp(Cplx
i_ forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Num a => a -> a -> a
* Cplx
a forall a. Num a => a -> a -> a
* (Cplx
alpha forall a. Num a => a -> a -> a
+ Cplx
2forall a. Num a => a -> a -> a
*Cplx
beta))
jtheta1Dash0 ::
Complex Double
-> Complex Double
jtheta1Dash0 :: Cplx -> Cplx
jtheta1Dash0 Cplx
q =
-Cplx
2 forall a. Num a => a -> a -> a
* Cplx
i_ forall a. Num a => a -> a -> a
* Cplx
jab forall a. Num a => a -> a -> a
* Cplx
jab forall a. Num a => a -> a -> a
* Cplx
jab
where
tau :: Cplx
tau = Cplx -> Cplx
getTauFromQ Cplx
q
jab :: Cplx
jab = Cplx -> Cplx -> Cplx -> Cplx -> Cplx
jthetaAB' (Cplx
1forall a. Fractional a => a -> a -> a
/Cplx
6) Cplx
0.5 Cplx
0 (Cplx
3forall a. Num a => a -> a -> a
*Cplx
tau)
jtheta1Dash ::
Complex Double
-> Complex Double
-> Complex Double
jtheta1Dash :: Cplx -> Cplx -> Cplx
jtheta1Dash Cplx
z Cplx
q =
Int -> Cplx -> Cplx -> Cplx -> Cplx -> Cplx
go Int
0 (Double
0.0 forall a. a -> a -> Complex a
:+ Double
0.0) Cplx
1.0 (Cplx
1.0 forall a. Fractional a => a -> a -> a
/ Cplx
qsq) Cplx
1.0
where
q' :: Cplx
q' = Cplx -> Cplx
checkQ Cplx
q
qsq :: Cplx
qsq = Cplx
q' forall a. Num a => a -> a -> a
* Cplx
q'
go :: Int -> Cplx -> Cplx -> Cplx -> Cplx -> Cplx
go :: Int -> Cplx -> Cplx -> Cplx -> Cplx -> Cplx
go Int
n Cplx
out Cplx
alt Cplx
q_2n Cplx
q_n_np1
| Int
n forall a. Ord a => a -> a -> Bool
> Int
3000 = forall a. HasCallStack => [Char] -> a
error [Char]
"Reached 3000 iterations."
| Cplx -> Cplx -> Bool
areClose Cplx
out Cplx
outnew = Cplx
2.0 forall a. Num a => a -> a -> a
* forall a. Floating a => a -> a
sqrt (forall a. Floating a => a -> a
sqrt Cplx
q) forall a. Num a => a -> a -> a
* Cplx
out
| Bool
otherwise = Int -> Cplx -> Cplx -> Cplx -> Cplx -> Cplx
go (Int
n forall a. Num a => a -> a -> a
+ Int
1) Cplx
outnew (-Cplx
alt) Cplx
q_2np1 Cplx
q_np1_np2
where
q_2np1 :: Cplx
q_2np1 = Cplx
q_2n forall a. Num a => a -> a -> a
* Cplx
qsq
q_np1_np2 :: Cplx
q_np1_np2 = Cplx
q_n_np1 forall a. Num a => a -> a -> a
* Cplx
q_2np1
n' :: Cplx
n' = forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
n
k :: Cplx
k = Cplx
2.0 forall a. Num a => a -> a -> a
* Cplx
n' forall a. Num a => a -> a -> a
+ Cplx
1.0
outnew :: Cplx
outnew = Cplx
out forall a. Num a => a -> a -> a
+ Cplx
k forall a. Num a => a -> a -> a
* Cplx
alt forall a. Num a => a -> a -> a
* Cplx
q_np1_np2 forall a. Num a => a -> a -> a
* forall a. Floating a => a -> a
cos (Cplx
k forall a. Num a => a -> a -> a
* Cplx
z)