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Date: 28-7-2019
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Date: 25-4-2019
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Date: 24-3-2019
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A Fourier series-like expansion of a twice continuously differentiable function
(1) |
for , where is a zeroth order Bessel function of the first kind. The coefficients are then given by
(2) |
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(3) |
(Gradshteyn and Ryzhik 2000, p. 926), where and care should be taken to avoid the two typos of Iyanaga and Kawada (1980) and Itô (1986).
As an example, consider , which has and therefore
(4) |
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(5) |
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(6) |
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(7) |
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(8) |
so
(9) |
(Whittaker and Watson 1990, p. 378; Gradshteyn and Ryzhik 2000, p. 926). This is illustrated above with 1 (red), 2 (green), 3 (blue), and 4 terms (violet) included.
Similarly, for ,
(10) |
REFERENCES:
Gradshteyn, I. S. and Ryzhik, I. M. "The Series ." Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 926, 2000.
Itô, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 1803, 1986.
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1473, 1980.
Schlömilch. Z. für Math. Phys. 3, 137-165, 1857.
Whittaker, E. T. and Watson, G. N. "Schlömilch's Expansion of an Arbitrary Function in a Series of Bessel Coefficients of Order Zero." §17.82 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 377-378, 1990.
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