# Complex Error Functions as Solutions to Gaussian Integrals

• Morberticus
In summary, the conversation discusses how a specific integral can be expressed analytically as a combination of error functions of t and can be obtained through numerical integration. The individual expresses difficulty with understanding how the expression becomes a combination of error functions and mentions a potential issue with the root t under the line. The solution is then provided, which involves changing t and using a specific identity.
Morberticus
I have been told that the following integral can be expressed analytically as a combination of error functions of t. And that the solution can be obtained from then by numerically integrating over t.

$$\int^a_b sin(k_1x)sin(k_2x')\int_0^\infty \frac{1}{\sqrt{t}}e^{-t(x-x')^2}dtdx dx'$$

While I don't have a problem with numerical integration, I can't see how the expression becomes a combination of error functions. The root t under the line is giving me trouble as well, as it makes the integral look divergent.

Thanks

-Strictly speaking, this is not course work, as the integral has come up in research. But if that forum is still more appropriate I can move it there.

First, change t = u²

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## What is a complex error function?

A complex error function is a mathematical function that is defined as the integral of a complex exponential function. It is also known as the Faddeeva function and is denoted by w(z).

## How is the complex error function related to Gaussian integrals?

The complex error function is closely related to Gaussian integrals as it can be used to express the Gaussian integral in terms of the error function. This allows for the evaluation of Gaussian integrals for complex values and has applications in statistics and physics.

## What are the properties of the complex error function?

The complex error function has several important properties, including being an entire function, having an essential singularity at infinity, and having a branch cut along the negative real axis. It also satisfies a functional equation that enables the computation of its values for negative arguments.

## What are some applications of complex error functions?

Complex error functions have various applications in mathematics, physics, and engineering. They are commonly used in the evaluation of complex integrals, solving partial differential equations, and modeling physical phenomena such as diffraction and heat transfer.

## How can complex error functions be approximated?

There are several methods for approximating complex error functions, including series expansions, continued fractions, and numerical methods such as the Taylor series and Padé approximants. These approximations can be useful when evaluating the complex error function for large or complex arguments.

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