# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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ï^æ n J0

æ

(e) Show that ^ at ô1 (x) converges to f (x) in the mean if and only if

j-1 æ

/ r(x) f2(x) dx =Y\ a2.

J0 1

³=1

r 1

2

This result is known as Parseval’s equation.

In Problems 10 through 12 let ô1, ô2,.ôï, ... be the normalized eigenfunctions of the Sturm-Liouville problem (11), (12).

10. Show that if an is the nth Fourier coefficient of a square integrable function f, then lim an = 0.

ï^æ n

Hint: Use Bessel’s inequality, Problem 9(b).

11.6 Series of Orthogonal Functions: Mean Convergence

677

REFERENCES

11. Show that the series

ô^) + ô2^) +------------+ôï (x) +----

cannot be the eigenfunction series for any square integrable function.

Hint: See Problem 10.

12. Show that the series

ô-^ix) ôï (x)

^ix > + -TT + - + 7ÒÃ + •••

is not the eigenfunction series for any square integrable function.

Hint: Use Bessel’s inequality, Problem 9(b).

13. Show that Parseval’s equation in Problem 9(e) is obtained formally by squaring the series (10) corresponding to f, multiplying by the weight function r, and integrating term by term.

The following books were mentioned in the text in connection with certain theorems about Sturm-Liouville problems:

Birkhoff, G., and Rota, G.-C., Ordinary Differential Equations (4th ed.) (New York: Wiley, 1989). Sagan, H., Boundary and Eigenvalue Problems in Mathematical Physics (New York: Wiley, 1961;

New York: Dover, 1989).

Weinberger, H., A First Course in Partial Differential Equations (New York: Wiley, 1965; New York: Dover, 1995).

Yosida, K., Lectures on Differential and Integral Equations (New York: Wiley-Interscience, 1960).

The following books are convenient sources of numerical and graphical data about Bessel and Legendre functions:

Abramowitz, M., and Stegun, I. A. (eds.), Handbook of Mathematical Functions (New York: Dover, 1965); originally published by the National Bureau of Standards, Washington, DC, 1964.

Jahnke, E., and Emde, F., Tables of Functions with Formulae and Curves (Leipzig: Teubner, 1938; New York: Dover, 1945).

The following books also contain much information about Sturm-Liouville problems:

Cole, R. H., Theory of Ordinary Differential Equations (New York: Irvington, 1968).

Hochstadt,H., Differential Equations: A Modern Approach (New York: Holt, 1964;New York: Dover, 1975).

Miller, R. K., and Michel, A. N., Ordinary Differential Equations (New York: Academic Press, 1982). Tricomi, F. G., Differential Equations (New York: Hafner, 1961).

Answers to Problems

CHAPTER 1 Section 1.1, page 8

1. 7 ^ 3/2 as t ^æ 2. 7 diverges from 3/2 as t ^ æ

3. 7diverges from -3/2 as t ^æ 4. 7 1/2 as t ^æ

5. 7 diverges from -1/2 as t ^ æ 6. 7 diverges from -2 as t ^ æ

7. 7

3

1

2

II

X

8.

7

1

3

II

See SSM for 9. 1

7

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