Bessel's Integrals with Cosine or Sine?

In summary, the conversation discusses the definition of Bessel's integrals and the differences between the Wikipedia and professor's notation. The integral is defined as the product of two exponential functions integrated over a specific interval, and the question asks for justification of the two expressions being exactly the same without any modifications.
  • #1
FQVBSina
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TL;DR Summary
Bessel's integral form: is it e to the power of a cosine or sine?
Hello all,

This is knowledge needed to solve my take-home final exam but I just want to ask about the definition of Bessel's integrals. This is not a problem on the exam. Wikipedia says the integral is defined as:

$$J_n(x) = \frac {1} {2\pi} \int_{-\pi}^{\pi} e^{i(xsin(\theta) - n\theta)} \, d\theta$$

My professor wrote it as:

$$J_m(Z) = \frac {1} {2\pi} \int_{-\pi}^{\pi} e^{i(Zcos(\theta))} e^{(- im\theta)} \, d\theta$$

Ignoring notation differences and I understand that cosine and sine form an orthogonal basis and are essentially the same as they can be easily expressed in terms of each other, but how do I justify that these two expressions are EXACTLY the same without any modifications with negative signs and such?

Thanks!

Jesse
 
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  • #2
I don't think they have the same constants. To see this, try defining a new variable equal to theta+pi/2, and evaluating the integral.
 
Last edited:

1. What are Bessel's integrals with cosine or sine?

Bessel's integrals with cosine or sine are mathematical functions that are used to solve problems involving vibrations, waves, and oscillations. They were discovered by the German mathematician Friedrich Bessel in the 19th century.

2. What is the difference between Bessel's integral with cosine and sine?

The main difference between Bessel's integral with cosine and sine is the type of oscillation they represent. Bessel's integral with cosine is used to represent oscillations with a fixed amplitude, while Bessel's integral with sine is used to represent oscillations with a decreasing amplitude.

3. How are Bessel's integrals with cosine or sine used in science?

Bessel's integrals with cosine or sine are used in a variety of scientific fields, including physics, engineering, and astronomy. They are particularly useful in solving problems involving vibrations, such as those found in musical instruments, electronic circuits, and celestial bodies.

4. What is the formula for Bessel's integral with cosine or sine?

The formula for Bessel's integral with cosine or sine is: ∫cos(nx)dx = (1/n)sin(nx) + C or ∫sin(nx)dx = -(1/n)cos(nx) + C, where n is a constant and C is the integration constant.

5. Are there any real-life applications of Bessel's integrals with cosine or sine?

Yes, there are many real-life applications of Bessel's integrals with cosine or sine. They are used in the design and analysis of structures and systems that involve vibrations, such as bridges, buildings, and aircraft. They are also used in signal processing to filter out unwanted noise and in image processing to enhance image quality.

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