Bessel's Equation and Rodrique Formula

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In summary, Bessel's equation is a second-order linear differential equation that is commonly used in physics and engineering to describe oscillatory phenomena. It is named after Friedrich Bessel, a German mathematician who introduced it in the early 19th century. The Rodrique formula is a special solution to Bessel's equation that simplifies its calculation and is particularly useful in solving problems involving cylindrical or spherical symmetry. It is named after Louis-Philippe Rodrique, a French mathematician who developed it in the mid-19th century. Together, these concepts are important tools in many areas of science and technology.
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Sudip Pradhan
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Is there any relation between Rodrique formula and Bessel's equation? Is it necessary to apply Rodrigue formula in the Bessel's Equation?
 
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I do not believe that there is a Rodrigues-type formula for the Bessel functions, since there is no representation of a Bessel function in terms of a polynomial of finite order. The Taylor series representations are infinite series. There is a slightly different type of presentation called a generating function that can be used to express the Bessel functions in terms of derivatives of an elementary function. See sect. 2.1 of http://www.math.psu.edu/papikian/Kreh.pdf for some details.
 

1. What is Bessel's Equation?

Bessel's Equation is a second-order linear differential equation that is named after mathematician Friedrich Bessel. It is used to solve problems in wave propagation, heat transfer, and other areas of science and engineering.

2. What is the Rodrique Formula?

The Rodrique Formula is a mathematical formula used to solve Bessel's Equation. It involves a power series expansion and can be used to find solutions to Bessel's Equation for different values of its parameters.

3. What are the applications of Bessel's Equation and the Rodrique Formula?

Bessel's Equation and the Rodrique Formula have a wide range of applications in physics, engineering, and mathematics. They are used in problems involving wave phenomena, heat transfer, and diffusion, as well as in solving differential equations in general.

4. What is the relationship between Bessel's Equation and the Rodrique Formula?

The Rodrique Formula is derived from Bessel's Equation and is used to solve it. The formula provides a systematic way to find solutions for different values of the parameters in Bessel's Equation.

5. Are there any practical limitations to using Bessel's Equation and the Rodrique Formula?

While Bessel's Equation and the Rodrique Formula are powerful tools in solving differential equations, they have limitations in certain scenarios. For example, they may not be applicable for non-linear problems or problems with irregular boundaries. It is important to carefully consider the limitations and assumptions of these methods when applying them to a problem.

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