Bessel Functions and Shifted Integral Limits: How Are They Related?

In summary, the nth order Bessel function of the first kind is defined by an integral with limits of -pi to pi, while a similar expression with limits of -pi/2 to 3pi/2 also relates to Bessel functions and is equal to the first expression. This has been proven through mathematical calculations and a MathCAD file. The Bessel function used in the calculations is Jn(m, x), which is equivalent to Jm(x). The proof is a classical calculus and does not require citation of an author.
  • #1
dfrenette
3
0
A nth order bessel function of the first kind is defined as:

Jn(B)=(1/2pi)*integral(exp(jBsin(x)-jnx))dx

where the integral limits are -pi to pi

I have an expression that is the exact same as above, but the limits are shifted by 90 degrees; from -pi/2 to 3pi/2

My question is how does this new expression relate to Bessel functions? My first thought was that the two function are equal since the integral limits are over one period. But I am not sure.
 
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  • #2
I think they are different. I can not offer a mathematical proof, but using MathCAD function for Jn and its numerical integration, I obtained different numbers.
 
  • #3
My question is how does this new expression relate to Bessel functions? My first thought was that the two function are equal since the integral limits are over one period. But I am not sure.
You are right. The two functions are equal.
A more tedious proof is given in the joint page :
 

Attachments

  • Bessel.JPG
    Bessel.JPG
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  • #4
Super,
Thanks for the proof!

I did set up a little mathcad file (attached) that shows the two implementations, proving to myself that indeed they are equal, but this proof will come in handy.

Where did you get it from so I can cite it?

Thanks again,

Darren
 

Attachments

  • BesselFunction.zip
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  • #5
JJacquelin said:
You are right. The two functions are equal.
A more tedious proof is given in the joint page :
Nice proof. I stand corrected.

The Bessel function I tried in MathCAD is: Jn(m, x) which returns Jm(x). Is this the same thing?
 
  • #6
Where did you get it from so I can cite it?
It's written by myself. You can copy it. No need to cite an author : the calculus is rather classical.
 
  • #7
Perfect. Thanks again.
 

Related to Bessel Functions and Shifted Integral Limits: How Are They Related?

1. What is a Bessel function?

A Bessel function is a mathematical function that arises in many areas of physics and engineering, particularly in problems involving wave propagation and oscillation. It was first introduced by the mathematician Daniel Bernoulli in 1732.

2. How is a Bessel function different from other mathematical functions?

Bessel functions are different from other mathematical functions because they have a complex argument and are often used to solve differential equations that describe physical phenomena, such as heat transfer and wave propagation. They also have a unique form that includes both exponential and trigonometric functions.

3. How are Bessel functions used in scientific research?

Bessel functions are used in a variety of scientific research, particularly in the fields of physics, engineering, and mathematics. They are commonly used to describe and analyze physical phenomena, such as the behavior of electromagnetic waves, vibrations in mechanical systems, and heat transfer in materials.

4. What are some real-world applications of Bessel functions?

Bessel functions have a wide range of real-world applications, including in signal processing, acoustics, optics, and astrophysics. They are also used in the design and analysis of electronic circuits, antennas, and other engineering systems.

5. How are Bessel functions related to other special functions in mathematics?

Bessel functions are part of a larger group of special functions in mathematics, including hypergeometric functions, Legendre polynomials, and Hermite polynomials. These functions are all related and have various applications in different fields of science and engineering.

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