Example 8.2 - Solve 
Example 8.3 - Analytical Solution to the Circular Fin ProblemMathematical Methods (10/24.539)
VIII. Special Functions
Introduction
If a particular differential equation (usually representing a linear variable coefficient system) and its power series solution occur frequently in applications, one gives them a name and introduces special symbols that define them. The properties of the functions are studied and tabulated and this information becomes a resource that can be exploited by the practicing engineer.
We have seen that linear constant coefficient systems have solutions that can be written in terms of elementary functions (sinusoids, exponentials, etc.). These functions are called elementary because they are treated in detail in introductory algebra, trigonometry, and calculus courses and they are used routinely in a variety of engineering applications. In short, since we are very familiar with these functions, they are easy to work with and we refer to them as elementary functions.
In contrast, functions that we are not as familiar with are more difficult to use in applications (at least initially) and sometimes these are referred to as non-elementary functions, special functions, or transcendental functions. We will use the
special function designation to emphasize their special significance in a variety of engineering applications. Also, once we gain a little experience with these special functions, we will no longer be imitated with their use and the non-elementary connotation will no longer be applicable (for example, using Bessel functions is as easy as using sinusoids, once you become comfortable with their use).The special feature of the so-called
special functions is a property called orthogonality. In this section of notes, we define this property, briefly identify several functions that share this special characteristic, and provide some additional details for two particular cases (for Legendre polynomials and Bessel functions). A generalization is made to include a full class of problems that have orthogonal functions as their solution - known as Sturm-Liouville Problems - in the next section.The current section on special functions is subdivided as follows:
Summary of Several Special Functions
Legendre’s Equation and Legendre Polynomials (in more detail)
Bessel’s Equation and Bessel Functions (in more detail)
Example 8.2 - Solve
Example 8.3 - Analytical Solution to the Circular Fin Problem|
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10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated October 1998).
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