How do you do a gaussian integral when it contains a heaviside function?

• nirovanton
In summary, the conversation discusses the difficulty of finding information on how to handle a gaussian integral with a heaviside function. The integral is equivalent to the error integral and must be solved numerically or with a table.

nirovanton

How do you do a gaussian integral when it contains a heaviside function!?

Very few textbooks cover gaussian integrals effectively. This isn't a big deal as they are easy to locate in integral tables, but something I cannot find anywhere is how to handle a gaussian with a heaviside

heaviside = theta

$$\int_0^\infty \theta(v-b)e^{-av^2}dv$$

where b is an arbitrary value of v where the heaviside 'turns on'

If anyone can help shed some light on this for me it would be greatly appreciated.

Last edited:

That integral is the error integral $$(\sqrt{\pi}/2)erfc(b)$$.
It must be done numerically or by using a table of erfc or erf.

What is a Heaviside function?

A Heaviside function, also known as the unit step function, is a mathematical function that is defined as 0 for negative inputs and 1 for positive inputs. It is commonly used in signal processing and control theory.

Why does a Gaussian integral contain a Heaviside function?

A Gaussian integral is a type of definite integral that involves the Gaussian function, which is a bell-shaped curve. The Heaviside function is often used to define the limits of the integral, as it allows for the integral to be split into two parts for easier computation.

How do you evaluate a Gaussian integral with a Heaviside function?

To evaluate a Gaussian integral with a Heaviside function, the integral is typically split into two parts: one where the Heaviside function is 0 and the other where it is 1. The limits of integration are adjusted accordingly and the integral is then solved using standard techniques.

Can a Gaussian integral with a Heaviside function be solved analytically?

Yes, a Gaussian integral with a Heaviside function can be solved analytically using methods such as integration by parts or substitution. However, in some cases, numerical methods may be needed to obtain an accurate solution.

What are the applications of Gaussian integrals with Heaviside functions?

Gaussian integrals with Heaviside functions have various applications in physics, engineering, and mathematics. They are commonly used in solving differential equations, calculating probabilities in statistics, and modeling physical phenomena such as diffusion and heat transfer.

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