Iterated Integral (Fourier Integral)

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In summary, there are certain conditions that allow for the order of iterated integrals to be changed, specifically for improper integrals. However, there are cases where this approach may not work, as shown in the example provided. In these cases, other methods must be used, as the commonly referenced theorem on Wikipedia is not sufficient.
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MHD93
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What conditions are there that allow to change the order of iterated integrals (improper ones).
For example, the following doesn't seem to work:
[tex]\int_{0}^\infty\left(\int_{0}^\infty f(t) \cos(\alpha t) dt\right)\cos(\alpha x)d\alpha = \int_{0}^\infty f(t)\left(\int_{0}^\infty \cos(\alpha t) \cos(\alpha x) d\alpha\right)dt [/tex]
The integral in parentheses on the RHS obviously fails to converge.
 
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See theorem II on wikipedia http://en.wikipedia.org/wiki/Order_of_integration_(calculus). That is the commonly referenced theorem. The trouble with it is it is very weak in the sense that many integrals that can be reversed do not meet the conditions. In those cases other methods are used, but they are not general.
 

What is an iterated integral?

An iterated integral, also known as a multiple integral, is an extension of the concept of a single integral to integrate a multi-dimensional function over a certain region in space. It involves integrating a function with respect to multiple variables, one after another.

What is the purpose of an iterated integral?

The purpose of an iterated integral is to calculate the total value of a multi-dimensional function over a specific region in space. This can be useful in many fields of science, such as physics, engineering, and economics.

What is the difference between a single and an iterated integral?

A single integral involves integrating a function with respect to one variable, while an iterated integral involves integrating a function with respect to multiple variables. In other words, a single integral is a special case of an iterated integral, with one of the variables being constant.

What is the Fourier integral in an iterated integral?

The Fourier integral is a type of iterated integral that involves integrating a complex-valued function over the entire real line. It is used to decompose a periodic function into its individual frequency components, which can be useful in signal processing and analysis.

What are some applications of iterated integrals?

Iterated integrals have a wide range of applications in various fields, such as calculating areas and volumes, finding the center of mass of an object, determining the work done by a force, and solving differential equations. They are also used in probability and statistics to calculate probabilities and expected values.

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