Composition of a theta function with function

In summary, the conversation discusses integrals involving Heaviside theta functions with non-trivial arguments and the difficulty in implementing them in Mathematica. The possibility of using a formula similar to the one for Dirac delta functions is also mentioned, along with a suggested resource for finding relevant formulas.
  • #1
muppet
608
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Hi all,

I have some integrals involving Heaviside theta functions of non-trivial arguments, in particular [itex]\theta(1-x^2)[/itex] and [itex]\theta(x^2-1)[/itex]. As x is a radial coordinate it's easy enough for me to break these up by hand, but it's cumbersome for me to implement in Mathematica and I can't work out how I'd go about attacking a more general problem. Is there an expression for [itex]\theta(f(x))[/itex] analagous to [tex]\delta(f(x))=\sum_i{\frac{\delta(x-x_i)}{|g'(x_i)|}}[/tex]?
Thanks in advance.
 
Mathematics news on Phys.org

1. What is a theta function?

A theta function is a special type of mathematical function that is widely used in number theory and analysis. It is defined as a complex-valued function of several variables and has many important properties, such as periodicity and symmetry.

2. How is a theta function composed with another function?

Composition of a theta function with another function is simply the process of plugging in the theta function as an argument into another function. For example, if we have a theta function f(x,y) and another function g(x), their composition would be g(f(x,y)).

3. What are the properties of a theta function composition?

The properties of a theta function composition depend on the properties of the individual functions being composed. In general, the composition of theta functions is also a theta function, and it inherits some of the properties of the original functions, such as periodicity and symmetry.

4. How is a theta function composition used in mathematics?

Theta function composition is used in various areas of mathematics, including number theory, algebra, and analysis. It is particularly useful in studying the behavior of complex functions, as well as in solving problems related to modular forms and elliptic curves.

5. Can any function be composed with a theta function?

No, not all functions can be composed with a theta function. The theta function must have compatible arguments with the other function in order for their composition to be defined. Additionally, some functions may not have the necessary properties to be composed with a theta function.

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