# Integral of Bessel function, square root and gaussian

• vuellesse
In summary, the integral of a Bessel function is a mathematical function used to calculate the area under the curve of a Bessel function. It can be expressed in terms of the square root function and is commonly used in physics and engineering to solve problems involving oscillations and waves. It is also used in the calculation of Gaussian distributions and has a general formula known as Watson's integral. The integral of a Bessel function has many real-world applications in fields such as engineering, physics, and statistics.
vuellesse
Hi! Does anyone know how to solve the following integral analitically?

$\int^{1}_{0} dx \ e^{B x^{2}} J_{0}(i A \sqrt{1-x^{2}})$, where A and B are real numbers.

Thanks!

Not really sure, but maybe if I take a stab at the question, someone else will answer better.

would it help to replace x=cos(θ)? then x^2 = cos^2(θ), dx =-sin(θ)dθ, $\sqrt{1-x^2}=sin(\theta)$ and have your integral go from θ= ∏/2 to 0.

and you might try looking carefully at each of the "recurrence relations" for Bessel functions and see if they help.

Last edited:

## 1. What is the integral of a Bessel function?

The integral of a Bessel function is a mathematical function that calculates the area under the curve of a Bessel function. It is commonly used in physics and engineering to solve problems involving oscillations and waves.

## 2. What is the relationship between the integral of a Bessel function and the square root function?

The integral of a Bessel function can be expressed in terms of the square root function. This is because the Bessel function is closely related to the square root function through a mathematical transformation known as the Hankel transform.

## 3. How is the integral of a Bessel function used in Gaussian distributions?

The integral of a Bessel function is used in the calculation of Gaussian distributions, which are commonly used to model the probability distribution of continuous random variables. The Bessel function appears in the formula for calculating the probability density function of a Gaussian distribution.

## 4. Is there a general formula for the integral of a Bessel function?

Yes, there is a general formula for the integral of a Bessel function, which is called the Watson's integral. It involves a combination of Bessel functions and hypergeometric functions, and is often used in the study of special functions and mathematical physics.

## 5. Are there any applications of the integral of a Bessel function in real-world problems?

Yes, the integral of a Bessel function has many important applications in various fields such as engineering, physics, and statistics. Some examples include calculating the electric potential in a cylindrical conductor, solving diffusion equations in cylindrical coordinates, and modeling the spread of infectious diseases.

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