Integrating a function involved error functions and a Gaussian kernel

In summary, the speaker is facing difficulty integrating a complicated expression with the boundaries of 0 and infinity. They have searched through handbooks but only found a similar integration with different boundaries. They are seeking guidance for solving this problem and wondering if there is a systematic approach for finding the solution.
  • #1
bjteoh
3
0
I am currently facing a problem of integrating
exp(-ax^2)*erf(bx+c)*erf(dx+f) with the integral boundaries 0 and
infinity.

I have gone through some handbooks but what I could locate is the integration of exp(-ax^2)*erf(bx)*erf(cx) from 0 to infinity which yields arctan(b*c/sqrt(a*(b^2+c^2+a)))/sqrt(a*pi).

I would really appreciate if anyone could guide me through solving this problem. Thanks.
 
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  • #2
I was trying perform sort of educated guess based on arctan(b*c/sqrt(a*(b^2+c^2+a)))/sqrt(a*pi) - solution for the integration of exp(-ax^2)*erf(bx)*erf(cx) from 0 to infinity but no luck, since the one I need is a translated version for the above.

I am wondering is that any systematic back tracking procedure for that?
 
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What is a Gaussian kernel?

A Gaussian kernel is a mathematical function that is used in many areas of science and engineering. It is a type of probability distribution that is often used to model random variables. The shape of a Gaussian kernel is bell-shaped, with a peak at the mean and symmetrical tails on either side.

What is the purpose of integrating a function with error functions and a Gaussian kernel?

The purpose of integrating a function with error functions and a Gaussian kernel is to accurately calculate the area under the curve of the function. This is useful in various areas of science and engineering, such as signal processing and statistics.

How do error functions and a Gaussian kernel work together in integration?

Error functions, also known as Gauss error functions, are used in conjunction with the Gaussian kernel to describe the shape and spread of a distribution. When integrating a function, the error function is used to adjust the height and width of the Gaussian kernel to accurately represent the function being integrated.

What are the benefits of using a Gaussian kernel in integration?

Gaussian kernels are useful for integration because they are smooth and continuous, making them easier to work with mathematically. They also have useful properties, such as being symmetric and having a known integral, which makes them ideal for approximating the area under a curve.

Are there any limitations to using a Gaussian kernel in integration?

One limitation of using a Gaussian kernel in integration is that it assumes a normal distribution, which may not always be the case in real-world applications. Additionally, the accuracy of the integration may be affected by the width and height of the kernel, which must be carefully chosen to accurately represent the function being integrated.

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