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User:Dnessett/Sturm-Liouville/Orthogonality proof

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This article proves that solutions to the Sturm–Liouville equation corresponding to distinct eigenvalues are orthogonal. For background see Sturm–Liouville theory.

Theorem

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, where and are solutions to the Sturm–Liouville equation corresponding to distinct eigenvalues and is the "weight" or "density" function.

Proof

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Let and be solutions of the Sturm-Liouville equation [1] corresponding to eigenvalues and respectively. Multiply the equation for by (the complex conjugate of ) to get:

(Only , , , and may be complex; all other quantities are real.) Complex conjugate this equation, exchange and , and subtract the new equation from the original:

Integrate this between the limits and

The right side of this equation vanishes because of the boundary conditions, which are either:

periodic boundary conditions, i.e., that , , and their first derivatives (as well as ) have the same values at as at , or
that independently at and at either:
the condition cited in equation [2] or [3] holds or:

So: .

If we set , so that the integral surely is non-zero, then it follows that ; that is, the eigenvalues are real, making the differential operator in the Sturm-Liouville equation self-adjoint (hermitian); so:

It follows that, if and have distinct eigenvalues, then they are orthogonal. QED.

See also

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References

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1. Ruel V. Churchill, "Fourier Series and Boundary Value Problems", pp. 70–72, (1963) McGraw–Hill, ISBN 0-07-010841-2.