Mathematical Methods (10/24.539) VIII. Special …

1 Mathematical Methods (10 ) VIII. Special functions and orthogonality 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.)

2 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.

3 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 ).

4 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 and the subject of orthogonality is subdivided as follows: Orthogonal functions Summary of Several Special functions Legendre Polynomials Associated Legendre functions Hermite Polynomials Laguerre Polynomials Bessel functions The Gamma Function Math Methods -- Section VIII: Special functions and orthogonality 2 Legendre s Equation and Legendre Polynomials (in more detail) Solution via the Power Series Method Standard Form for Legendre Polynomials Some Low-Order Legendre Polynomials Some Important Relationships The Matlab legendre Function Application Notes Bessel s Equation and Bessel functions (in more detail)

5 Bessel s Equation One Solution via the Power Series Method Linear Independence Ordinary Bessel functions of the Second Kind Summary Expressions for Various Bessel functions Additional Properties and Relationships Some Plots and Limiting Values Equations Solvable in Terms of Bessel functions Some Analytical Examples using Bessel functions Example - Solve y''y0+= Example - Solve ()()2482xy''x4x3y'4x5x3y0+ + += Example - Analytical Solution to the Circular Fin Problem Lecture Notes for Math Methods by Dr. John R. White, UMass-Lowell (updated Nov.)

6 2003) Math Methods -- Section VIII: Special functions and orthogonality 3 Orthogonal functions Two functions are said to be orthogonal if, when multiplied together and integrated over the domain of interest, the integral becomes zero. The property of orthogonality is usually applied to a class of functions that differ by one or more variables (and usually represent the basis solutions to a homogeneous eigenvalue problem with an infinite number of eigenfunction solutions). For example, we can represent a class of sinusoids as ( ) n(x)sin nxfor n1, 2, 3, =="where n is a positive integer.

7 A particular function might be f(2x)(x)sin2x= =. For an arbitrary function belonging to this set, we simply refer to the discrete index n, where the nth function is denoted as , or the mn(x)th function as m(x) , The orthogonality property can be stated mathematically as b2mnmnmmn2am0mggg(x)g (x)dxggm nn == = = ( ) where 2mmggnormof thefunction == ( ) and is the Kronecker delta function that takes on the value of unity if m = n and a value of zero if m. If mn n mg1=, then gm(x) is said to be an orthonormal function.

8 The orthogonality property is important because functions with this characteristic are often used to expand arbitrary functions with an infinite series expansion in terms of the given basis functions . For example, the function f(x) can be written in terms of a Generalized Fourier Series (implies completeness), or ( ) nnn1f(x)a g (x) == where the an are the expansion coefficients. The orthogonality property comes into play when one tries to determine an expression for the an coefficients. To see this, we multiply eqn. ( ) by the mth function, gm(x), and integrate over the domain of interest.

9 Doing this gives 2mnmnnmmnn1n1g(x)f (x)ag(x)g (x)agag ==== = 2mm ( ) or mmmg(x)f (x)ag2= ( ) Lecture Notes for Math Methods by Dr. John R. White, UMass-Lowell (updated Nov. 2003) Math Methods -- Section VIII: Special functions and orthogonality 4where the summation symbol is eliminated in the last equality in eqn. ( ) because orthogonality forces all the terms in the infinite sum to zero except for the single term where n = m. This simplification is essential in many practical applications, and it would not be possible without the orthogonality property [as defined in eqn.]

10 ( )]. Thus we will see that this is a very important characteristic. The Generalized Fourier Series given in eqn. ( ) is an eigenfunction expansion in terms of a complete set of orthogonal basis functions . The choice of the basis functions is usually determined by the domain of interest and the boundary conditions imposed upon f(x). The basis functions are usually obtained from a Sturm-Liouville Problem which results in a set of orthogonal eigenfunctions (see the next section for further details). The term completeness implies that the Generalized Fourier Series converges as.