On the Logarithmic Integral

Yıl 2010, Cilt: 39 Sayı: 3, 393 - 401, 01.03.2010

Brian Fisher Biljana Jolevska-tuneska

Öz

The logarithmic integral li(x) and its associated functions li+(x) and li−(x) are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of these functions and other functions are then found.

Anahtar Kelimeler

Logarithmic integral, Distribution, Convolution, Neutrix, Neutrix convolution, 2000 AMS Classification: 33 B 10

Kaynakça

• Abramowitz, M. and Stegun, I. A. (Eds). Handbook of Mathematical Functions with Formu- las, Graphs and Mathematical Tables, 9th printing(New York: Dover, p. 879, 1972).
• van der Corput, J. G. Introduction to the neutrix calculus, J. Analyse Math. 7, 291–398, 1959-60.
• Fisher, B. Neutrices and the convolution of distributions, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 17, 119–135, 1987.
• Fisher, B., Jolevska-Tuneska, B. and Taka˘ci, A. On convolutions and neutrix convolutions involving the incomplete gamma function, Integral Transforms and Special Functions 15 (5), 404–415, 2004.
• Jolevska-Tuneska, B. and Taka˘ci, A. Results on the commutative neutrix convolution product of distributions, Hacettepe J. Math. Stat. 37 (2), 135–141, 2008.
• Gel’fand, I. M. and Shilov, G. E. Generalized functions, Vol. I (Academic Press Chap. 1, 1964).
• Jones, D. S. The convolution of generalized functions, Quart. J. Math. Oxford 24 (2), 145– 163, 1973.