Copyright	(c) Stéphane Laurent 2023
License	BSD3
Maintainer	laurent_step@outlook.fr
Safe Haskell	Safe-Inferred
Language	Haskell2010

Math.JacobiTheta

Description

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.

Synopsis

jtheta1 :: Complex Double -> Complex Double -> Complex Double
jtheta1' :: Complex Double -> Complex Double -> Complex Double
jtheta2 :: Complex Double -> Complex Double -> Complex Double
jtheta2' :: Complex Double -> Complex Double -> Complex Double
jtheta3 :: Complex Double -> Complex Double -> Complex Double
jtheta3' :: Complex Double -> Complex Double -> Complex Double
jtheta4 :: Complex Double -> Complex Double -> Complex Double
jtheta4' :: Complex Double -> Complex Double -> Complex Double
jthetaAB :: Complex Double -> Complex Double -> Complex Double -> Complex Double -> Complex Double
jthetaAB' :: Complex Double -> Complex Double -> Complex Double -> Complex Double -> Complex Double
jtheta1Dash0 :: Complex Double -> Complex Double
jtheta1Dash :: Complex Double -> Complex Double -> Complex Double

Documentation

jtheta1 Source #

Arguments

:: Complex Double	z
-> Complex Double	q, the nome
-> Complex Double

First Jacobi theta function in function of the nome.

jtheta1' Source #

Arguments

:: Complex Double	z
-> Complex Double	tau
-> Complex Double

First Jacobi theta function in function of tau.

jtheta2 Source #

Arguments

:: Complex Double	z
-> Complex Double	q, the nome
-> Complex Double

Second Jacobi theta function in function of the nome.

jtheta2' Source #

Arguments

:: Complex Double	z
-> Complex Double	tau
-> Complex Double

Second Jacobi theta function in function of tau.

jtheta3 Source #

Arguments

:: Complex Double	z
-> Complex Double	q, the nome
-> Complex Double

Third Jacobi theta function in function of the nome.

jtheta3' Source #

Arguments

:: Complex Double	z
-> Complex Double	tau
-> Complex Double

Third Jacobi theta function in function of tau.

jtheta4 Source #

Arguments

:: Complex Double	z
-> Complex Double	q, the nome
-> Complex Double

Fourth Jacobi theta function in function of the nome.

jtheta4' Source #

Arguments

:: Complex Double	z
-> Complex Double	tau
-> Complex Double

Fourth Jacobi theta function in function of tau.

jthetaAB Source #

Arguments

:: Complex Double	characteristic a
-> Complex Double	characteristic b
-> Complex Double	z
-> Complex Double	q, the nome
-> Complex Double

Jacobi theta function with characteristics. This is a family of functions, containing the opposite of 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 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' Source #

Arguments

:: Complex Double	characteristic a
-> Complex Double	characteristic b
-> Complex Double	z
-> Complex Double	tau
-> Complex Double

Jacobi theta function with characteristics in function of tau.

jtheta1Dash0 Source #

Arguments

:: Complex Double	q, the nome
-> Complex Double

Derivative at 0 of the first Jacobi theta function. This is much more efficient than evaluating jtheta1Dash at 0.

jtheta1Dash Source #

Arguments

:: Complex Double	z
-> Complex Double	q, the nome
-> Complex Double

Derivative of the first Jacobi theta function.