

# Defining theta functions

Theta functions are a type of mathematical function that are used in various branches of mathematics and physics. They are defined using the Greek letter theta (θ) and are related to the trigonometric functions. Theta functions are used to describe periodic functions and have applications in number theory, geometry, and quantum mechanics.

In this section, we will define theta functions and their properties. We will start by introducing the basic definition of theta functions and then explore some of their important properties.

Theta functions are defined as follows:

$$\theta_k(x) = \sum_{n=-\infty}^{\infty} e^{-kx^2 + 2\pi i nx}$$

Here, $k$ is a positive constant, and $x$ is the variable. The theta function is an infinite sum of complex exponentials. The variable $x$ is often referred to as the "argument" of the theta function.

Let's consider a simple example of a theta function:

$$\theta_1(x) = \sum_{n=-\infty}^{\infty} e^{-x^2 + 2\pi i nx}$$

This is a theta function with $k=1$. Notice that the exponent of $x$ in the exponential is $x^2$.

Theta functions are periodic functions. This means that their values repeat themselves at regular intervals. For example, the theta function $\theta_1(x)$ has a period of 1, which means that its values repeat every 1 unit of $x$.

## Exercise
Find the period of the theta function $\theta_2(x)$.

### Solution
The period of the theta function $\theta_2(x)$ is $\sqrt{1/2}$.

# Properties of theta functions



One of the most important properties of theta functions is their symmetry. The theta function is an even function, which means that for any value of $x$, the corresponding value of the theta function is the same as the value of the theta function at $-x$. In mathematical notation, this property is expressed as:

$$\theta_k(x) = \theta_k(-x)$$

Theta functions are also analytic functions. This means that they have a power series expansion that converges to the function for all values of the argument $x$. The power series expansion of the theta function is given by:

$$\theta_k(x) = \sum_{n=0}^{\infty} (-1)^n q^{n(n+1)/2}$$

Here, $q = e^{-kx^2}$.

Theta functions are also related to the Jacobi elliptic functions. The Jacobi elliptic functions are a set of three functions that are defined using the theta function. These functions have applications in number theory and geometry.

# Applications of theta functions in mathematics

Theta functions have a wide range of applications in various branches of mathematics. Some of the most important applications of theta functions include:

- Number theory: Theta functions are used in the study of modular forms and in the theory of the Riemann zeta function.

- Geometry: Theta functions are used in the study of elliptic curves and in the construction of the Poincaré half-plane model of hyperbolic geometry.

- Quantum mechanics: Theta functions are used in the study of quantum systems with periodic potentials, such as the quantum Hall effect.

- Statistical mechanics: Theta functions are used in the study of statistical systems with long-range interactions, such as the Ising model.

# Implementing theta functions in python



To implement the theta function $\theta_1(x)$ in python, we can use the following code:

```python
import math

def theta_1(x):
    result = 0
    for n in range(math.inf, -math.inf, -1):
        result += math.exp(-x**2 + 2j * math.pi * n * x)
    return result
```

This code defines a function `theta_1(x)` that computes the theta function $\theta_1(x)$. The function uses a for loop to compute the infinite sum of the complex exponentials.

We can also use the `numpy` library to compute the theta function more efficiently. The `numpy` library provides a function `exp` that can be used to compute the exponential of an array of values.

```python
import numpy as np

def theta_1(x):
    n = np.arange(-np.inf, np.inf, -1)
    result = np.sum(np.exp(-x**2 + 2j * np.pi * n * x))
    return result
```

This code uses the `numpy` library to compute the theta function $\theta_1(x)$ more efficiently.

# Examples and case studies



One of the most famous examples of a theta function is the Riemann zeta function. The Riemann zeta function is defined as:

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$

The Riemann zeta function is related to the theta function $\theta_0(x)$ by the identity:

$$\zeta(s) = \frac{1}{2} + \frac{1}{2}\theta_0(0)s + \frac{1}{2}\theta_0(0)s^2 + \frac{1}{2}\theta_0(0)s^3 + \cdots$$

Another example of a theta function is the theta function of the Jacobi elliptic functions. The Jacobi elliptic functions are defined as:

$$\vartheta_k(x) = \theta_k(x) - \frac{1}{2}x^2 - \frac{1}{2}k$$

The Jacobi elliptic functions have many important properties and applications in number theory and geometry.

# Solving mathematical problems using theta functions in python



For example, we can use the theta function $\theta_1(x)$ to compute the value of the Riemann zeta function at a given point $s$. We can do this by using the identity:

$$\zeta(s) = \frac{1}{2} + \frac{1}{2}\theta_0(0)s + \frac{1}{2}\theta_0(0)s^2 + \frac{1}{2}\theta_0(0)s^3 + \cdots$$

We can implement this in python as follows:

```python
def zeta(s):
    theta_0 = theta_1(0)
    result = 0.5 + 0.5 * theta_0 * s
    for n in range(2, math.inf):
        result += 0.5 * theta_0 * s**n
    return result
```

This code defines a function `zeta(s)` that computes the value of the Riemann zeta function at a given point $s$.

# Advanced topics in theta functions and python



- The relationship between theta functions and the Riemann zeta function.

- The use of theta functions in the study of quantum systems with periodic potentials.

- The application of theta functions in the construction of numerical algorithms for solving mathematical problems.

We will also learn how to use the `sympy` library in python to compute theta functions and perform symbolic computations.

The `sympy` library provides a powerful set of tools for symbolic computation in python. It can be used to compute theta functions and perform a wide range of symbolic computations.

# Conclusion

In this textbook, we have learned about theta functions, their properties, applications, and how to implement them in python. We have also explored some advanced topics in theta functions and python.

Theta functions are powerful mathematical tools that have applications in various branches of mathematics and physics. By studying theta functions, we can gain a deeper understanding of the connections between different areas of mathematics and develop new techniques for solving mathematical problems.

In conclusion, theta functions are an important topic in the study of mathematics and physics. They provide a valuable tool for understanding the behavior of periodic functions and their applications in number theory, geometry, and quantum mechanics.

In this textbook, we have covered the basics of theta functions and their properties, as well as how to implement them in python. We have also explored some advanced topics in theta functions and python. This textbook is a valuable resource for anyone interested in learning more about theta functions and their applications in mathematics and physics.