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Properties of the Regular Sturm-Liouville Problem
1) There exists an infinite number of eigenvalues that can be arranged in increasing order
2) For each eigenvalue there is only one eigenfunction (except for non-zero constant multiples)
3) Eigenfunctions corresponding to different eigenvalues are linearly independent
4) The set of eigenfunctions corresponding to the set of eigenvalues is orthongonal with respect to the weight function p(x)
on the interval [a,b].
Regular Sterm-Liouville Problem
- where r, r', p, and q are continuous on the interval (a,b) and p(x)>0, r(x)>0 for all x in the interval (a,b)
- Subject to: A1y(a) + B1y'(a) = 0, A2y(b) + B2y'(b) = 0
- A12 + B12 > 0, A22 + B22 > 0
Convergence of a Fourier Series
and f '
be piecewise continuous on some interval [-p,p]. That is, let f
and f '
be continuous except at a finite number of points on the interval and have only finite discontinuities at these points. Then the Fourier series of f
on the interval converges to f(x)
at a point of continuity. At a point of discontinuity the Fourier series converges to the average
Orthogonal Functions Definition
Two functions f(x)
are orthogonal on an interval [a,b] if
Existence of Power Series Solutions Theorem
If x = x0
is an ordinary point of the differential equation
, we can always find two linearly independent solutions in the form of a power series,
centered at x0
. A series solution converges on some interval defined by |x-x0
| < R, where R is the distance from x0
to the closest singular point.
If x = x0
is a regular singular point of the differential equation, then there exists at least one solution of the form
where the number r is constant to be determined. The series will converge on some interval 0 < x-x0 < R
Definition of Gamma Function
, x > 0
Bessel Functions of the First Kind
- J-v = ...