Abstract
In the first part of these notes new expressions—simpler than any previously obtained—are presented in integral form for the derivatives of the α 0 n -functions (required for an interpretation of the observed light changes of eclipsing variables) with respect to the fractional radiir 1, 2 and projected separation δ of their centres in terms of the modified Bessel functionsK 0, 1 (x) of the second kind; and utilized for establishing new asymptotic formulae for the computation of ‘boundary integrals’ of the formJ 0−1 ,n(μ).
In the second part of this paper, by a resort to bi-polar coordinates, we shall establish a new type of expansions for the α 0 n -functions valid for any type of eclipses, and converging faster than the expansions of the cross-correlation integral of the form (1) for α 0 n that have so far been established.
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References
Erdélyi, A. (ed.): 1953,Higher Trascendental Functions Vol. 1, McGraw-Hill, New York.
Gradsteyn, I. S. and Ryzhik, I. M.: 1980,Table of Integrals, Series and Products, Academic Press, London and New York.
Kopal, Z.: 1947,Harvard Obs. Circ., No. 450.
Kopal, Z.: 1959,Close Binary Systems, Chapman-Hall and John Wiley, London and New York.
Kopal, Z.: 1977,Astrophys. Space Sci. 50, 225.
Kopal, Z.: 1979,Language of the Stars, D. Reidal Publ. Co., Dordrecht, Holland.
Magnus, W., Oberhettinger, F., and Soni, R. P.: 1966,Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York.
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Kopal, Z. Notes on the associated alpha-functions and related integrals. Astrophys Space Sci 90, 445–454 (1983). https://doi.org/10.1007/BF00650075
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DOI: https://doi.org/10.1007/BF00650075