close Warning: Can't synchronize with repository "(default)" ("(default)" is not readable or not a Git repository.). Look in the Trac log for more information.

# Elliptic Theta Functions

## Introduction

All of the elliptic theta functions are functions of two parameters: the argument z and a second argument. For these routines the second argument is the nome q, `|q| < 1` . Some references use the the lattice parameter tau, with the imaginary part of tau strictly positive.

The nome and lattice parameter are related by `q = exp(%i*%pi*tau)`.

### Theta 1

`(elliptic-theta-1 z q)`

First theta function defined by

```  theta1(z, q) = 2*q^(1/4)*sum((-1)^n*q^(n*(n+1))*sin((2*n+1)*z), n, 0, inf)
```

### Theta 2

`(elliptic-theta-2 z q)`

Second theta function defined by

```  theta2(z, q) = 2*q^(1/4)*sum(q^(n*(n+1))*cos((2*n+1)*z), n, 0, inf)
```

### Theta 3

`(elliptic-theta-3 z q)`

Third theta function defined by

```  theta3(z, q) = 1 + 2 * sum(q^(n^2)*cos(2*n*z), n, 1, inf)
```

### Theta 4

`(elliptic-theta-4 z q)`

Fourth theta function defined by

```  theta4(z, q) = 1 + 2*sum((-1)^n*q^(n^2)*cos(2*n*z), n, 1, inf)
```

## Miscellaneous

`(elliptic-theta n z q)`
Convenience function where n is 1, 2, 3, or 4 for one of the four theta functions
`(elliptic-nome m)`
Compute the nome q from the parameter m.