×

A link between the Fourier and Heaviside representations of the square wave. (English) Zbl 0773.42002

Summary: By Laplace transforming the Fourier expansion of the odd periodic extension of the square wave and then taking inverse transforms on the results, we arrive at a representation of this periodic function as an infinite combination of delayed step functions. This helps to clarify the connection between solutions obtained for constant coefficient linear differential equations with periodic step forcing functions, when using, on the one hand, the Fourier series approach and on the other, the Laplace transform.

MSC:

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
44A10 Laplace transform
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A30 Linear ordinary differential equations and systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Spencer A. J. M., Engineering Mathematics 1 (1978)
[2] Gower N. W., Fourier Series (1974)
[3] Chawla M. M., Proc. Amer. Math. Soc 99 pp 199– (1987)
[4] DOI: 10.1214/aoms/1177697386 · Zbl 0198.23802 · doi:10.1214/aoms/1177697386
[5] DOI: 10.1214/aoms/1177729552 · Zbl 0044.14701 · doi:10.1214/aoms/1177729552
[6] Seaman, J. W. and Odell, P. L. 1988.The Encyclopedia of Statistical Sciences, Edited by: Kotz, S., Johnson, N. L. and Read, C. B. Vol. 9, 480–84. New York: John Wiley & Sons.
[7] DOI: 10.2307/1267286 · Zbl 0237.62038 · doi:10.2307/1267286
[8] Van Dantzig, D. 1951.Koninklijke Nederlandse Akademie van Wetenschappen Proceedings1–8. Series A, 54,
[9] DOI: 10.2307/1427126 · Zbl 0569.60014 · doi:10.2307/1427126
[10] DOI: 10.1080/0020739870180609 · doi:10.1080/0020739870180609
[11] DOI: 10.2307/2324416 · doi:10.2307/2324416
[12] Wasyliw, B. 15 (November) 1986.Mathematics in School15 (November), 39
[13] Glaister P., Int. J. Math. Educ. Sci. Technol. 21 pp 1002– (1990)
[14] Chester W., Mechanics (1979)
[15] Nayfeh A. H., Perturbation Methods (1973) · Zbl 0265.35002
[16] Rudin W., rinciples of Mathematical Analysis,, 3. ed. (1976)
[17] DOI: 10.1090/S0002-9939-1985-0770538-8 · doi:10.1090/S0002-9939-1985-0770538-8
[18] Brand, L. 1957.Vector and Tensor Analysis, sixth printing, 106–231-2. New York: John Wiley and Sons.
[19] Struik, D. J. 1961.Lectures on Classical Differential Geometry,, second edition, 33New York: Addison-Wesley. · Zbl 0105.14707
[20] Weatherburn, C. E. 1964.Differential Geometry of Three Dimensions, 20London: Cambridge University Press. · JFM 53.0658.08
[21] DOI: 10.1215/S0012-7094-66-03341-2 · Zbl 0148.10902 · doi:10.1215/S0012-7094-66-03341-2
[22] DOI: 10.1080/0020739850160601 · Zbl 0561.44002 · doi:10.1080/0020739850160601
[23] DOI: 10.1016/0022-247X(91)90060-D · Zbl 0715.44004 · doi:10.1016/0022-247X(91)90060-D
[24] Widder D. V., An Introduction to Transform Theory (1971) · Zbl 0219.44001
[25] DOI: 10.1093/imamat/42.3.241 · Zbl 0685.44002 · doi:10.1093/imamat/42.3.241
[26] DOI: 10.1155/S0161171291000704 · Zbl 0735.44004 · doi:10.1155/S0161171291000704
[27] Goldstein S., Proc. London Math. Soc. pp 34– (1932)
[28] Erdelyi A., Tables of Integral Transforms (1954)
[29] Spanier J., An Atlas of Functions (1987) · Zbl 0618.65007
[30] DOI: 10.1093/biomet/56.3.690 · Zbl 0183.48701 · doi:10.1093/biomet/56.3.690
[31] DOI: 10.1080/0020739890200101 · Zbl 0684.34005 · doi:10.1080/0020739890200101
[32] Boyce W. E., Elementary Differential Equations and Boundary Value Problems (1977) · Zbl 0353.34001
[33] Halmos P. R., Finite-Dimensional Vector Spaces (1974) · Zbl 0288.15002 · doi:10.1007/978-1-4612-6387-6
[34] Buchberger, B., Collins, G. E. and Loos, R. 1983.Computer Algebra, 4–7. New York: Springer-Verlag.
[35] Oman P., IEEE Software 7 pp 93– (1990)
[36] DOI: 10.1080/0020739870180504 · doi:10.1080/0020739870180504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.