Mathematical Methods (10/24.539)
VIII. Special Functions
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
where n is a positive integer. A particular function might be 
. 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 mth function as 
, etc..
The orthogonality property can be stated mathematically as
where
and 
is the Kronecker delta function that takes on the value of unity if m = n and a value of zero if 
. If 
, then gm(x) is said to be an orthonormal function.
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
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. (8.4) by the mth function, gm(x), and integrate over the domain of interest. Doing this gives
or
where the summation symbol is eliminated in the last equality in eqn. (8.5) 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. (8.2)]. Thus we will see that this is a very important characteristic.
The Generalized Fourier Series given in eqn. (8.4) 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
. Although of theoretical interest, a rigorous proof of completeness is quite often unnecessary because the series is almost always truncated to low order in practical problems.
Finally, we note that, in many cases, the basis functions may be orthogonal with respect to a weight function, p(x). This means that
where the normalization is given by
For this case the basic series expansion relationship is unchanged [i.e. eqn. (8.4) is the same], but the expression for the expansion coefficients is modified accordingly to give
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10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated October 1998).
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