# Jacobi Theta Function modularity translation quick q

• binbagsss
In summary, the conversation discusses the Jacobi theta series and how to show that ##\theta^{m}(\tau + 1) = \theta^{m}(\tau)##. The attempt at a solution involves an extra factor of ##e^{2\pi i n}## and defining a new Fourier coefficient, ##r'{m}##. However, it is noted that if n is an integer, ##e^{i 2n\pi} = 1##.
binbagsss

## Homework Statement

I have the Jacobi theta series: ##\theta^{m}(\tau) = \sum\limits^{\infty}_{n=0} r_{m}(\tau) q^{n} ##,

where ##q^{n} = e^{2\pi i n \tau} ## and I want to show that ##\theta^{m}(\tau + 1) = \theta^{m}(\tau) ##

(dont think its needed but) where ##r_{m} = ## number of ways of writing ##m## as the sum of ##n## squares.

the above

## The Attempt at a Solution

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so i get an extra ##e^{2\pi i n} ## factor, ## \theta^{m}(\tau) = \sum\limits^{\infty}_{n=0}r_{m}(\tau) q^{n} e^{2\pi i n} ##

I think it should be obvious what to do now, but I don't know what to do next?

Something like defining a new Fourier coefficient, ##r'{m}= r_{m} e^{2\pi i n} ## and then since the sum is to ? but that doesn't seem proper enough?

Many thanks.

Hmm. You know that if $n$ is an integer, $e^{i 2n\pi} = 1$?

stevendaryl said:
Hmm. You know that if $n$ is an integer, $e^{i 2n\pi} = 1$?
oh my ! thank you ha :)

## What is the Jacobi Theta Function modularity translation?

The Jacobi Theta Function is a mathematical function that is used to express the relationship between two quantities that are related by a modular transformation. In other words, it is a function that describes how a quantity changes when its underlying structure is transformed.

## What is the significance of modularity in the Jacobi Theta Function?

Modularity refers to the property of the Jacobi Theta Function that allows it to remain unchanged under a specific transformation known as a modular transformation. This property is important because it allows for the simplification of complex mathematical expressions involving the Jacobi Theta Function.

## What is meant by "translation" in the context of Jacobi Theta Function modularity translation?

In this context, "translation" refers to the shifting or moving of the Jacobi Theta Function in order to express it in a different form. This translation can involve changing the variables, coefficients, or other aspects of the function.

## What is the role of "quick q" in Jacobi Theta Function modularity translation?

"Quick q" is a term used to describe a specific type of transformation that is often used in the context of Jacobi Theta Function modularity translation. This transformation involves changing the value of the variable q, which is a key component of the Jacobi Theta Function.

## How is Jacobi Theta Function modularity translation used in scientific research?

Jacobi Theta Function modularity translation is used in a variety of fields in scientific research, including mathematics, physics, and engineering. It is particularly useful in solving complex mathematical problems and describing the behavior of physical systems. It can also be used in data analysis and signal processing applications.

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