Page 525 - Advanced engineering mathematics

EIGENFUNCTION EXPANSIONS LEGENDRE
POLYNOMIALS BESSEL FUNCTIONS
CHAPTER 15

Special

Functions and

Eigenfunction

Expansions

15.1 Eigenfunction Expansions

Functions may be designated as special functions when they arise in important applications and
contexts, often as solutions of differential equations. The most familiar special functions are
cos(kx) and sin(kx), which are solutions of
2
y + k y = 0.

Other special functions include Legendre polynomials and Bessel functions, which are solutions
of Legendre’s and Bessel’s differential equations. We will develop these functions shortly.
Fourier series are used to solve many problems involving partial differential equations mod-
eling diffusion processes and wave motion. We will see, however, that some problems require
series expansions in terms of special functions. This chapter develops a framework in which to
make such expansions.
Begin with the ordinary differential equation

y + R(x)y + (Q(x) + λP(x))y = 0
on some interval (a,b) or [a,b], with λ a constant to be determined along with y. Assume that
the coefficient functions are continuous on the interval. First manipulate the differential equation
into a special form. Multiply it by

r(x) = e R(x)dx
to obtain

y e R(x)dx + R(x)y e R(x)dx + (Q(x) + λP(x))e R(x)dx y = 0.

This is

R(x)dx R(x)dx R(x)dx

y e + Q(x)e + λP(x)e y = 0,
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October 14, 2010 15:20 THM/NEIL Page-505 27410_15_ch15_p505-562