What is the result of Fourier transformation of functional theta function?

In summary, a functional theta function is a type of mathematical function that is used in number theory and mathematical physics. It is a more general form of a regular theta function, as it is a function of multiple variables and can take on values in more general number fields. Functional theta functions have various applications in mathematics and physics, including the study of elliptic curves, modular forms, and number theory. They are also closely related to regular theta functions, with a regular theta function being expressible as a functional theta function with certain parameters. There are several open problems related to functional theta functions, including the Riemann hypothesis and the conjecture that there exists a functional theta function for every modular form.
  • #1
jarowit
4
0
what is the result of Fourier transformation of functional theta [Heaviside] function?
 
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  • #2
Is it not the case -

[tex] \int_{-\infty}^\infty}\theta (a)e^{ikx}dx = \int_{a}^{\infty}e^{ikx}dx[/tex]
 
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  • #3
what you have written is Fourier transforamtion of the "usual" theta function.
i would like to know how to transform theta function which in arguments have other function s [i.e. functional theta function].
 
  • #4
True. Did you know there is a math area in the PF ? Just scroll the main page down.
 

1. What is a functional theta function?

A functional theta function is a type of mathematical function that is used in number theory and mathematical physics. It is a special type of modular form, which is a function that is invariant under certain transformations.

2. How is a functional theta function different from a regular theta function?

A functional theta function is a more general form of a regular theta function. While a regular theta function is a complex-valued function of a single variable, a functional theta function is a function of multiple variables and can take on values in more general number fields.

3. What are some applications of functional theta functions?

Functional theta functions have various applications in mathematics and physics. They are used in the study of elliptic curves, modular forms, and number theory. They also have applications in quantum field theory, string theory, and mathematical finance.

4. How are functional theta functions related to modular forms?

Functional theta functions are a special type of modular form. They are closely related to regular theta functions, which are also modular forms. In fact, a regular theta function can be expressed as a functional theta function with certain parameters.

5. Are there any known open problems related to functional theta functions?

Yes, there are several open problems related to functional theta functions. One major open problem is the Riemann hypothesis, which states that all non-trivial zeros of the Riemann zeta function lie on the critical line. Another open problem is the conjecture that there exists a functional theta function that corresponds to every modular form.

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