On Volterra functions and Ramanujan integrals
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Roberto Garrappa
Abstract
Volterra functions were introduced at the beginning of the twentieth century as solutions of some integral equations of convolution type with logarithmic kernel. Since then, few authors have studied this family of functions and faced with the problem of providing a clear understanding of their asymptotic behavior for small and large arguments. This paper reviews some of the most important results on Volterra functions and in particular collects, into a quite general framework, several results on their asymptotic expansions; these results turn out to be useful not only for the full understanding of the behavior of the Volterra functions but also for their numerical computation. The connections with integrals of Ramanujan type, which have several important applications, are also discussed.
The authors appreciate constructive remarks and suggestions of the referees that helped to improve the manuscript. Furthermore, F. Mainardi likes to thank Prof. Alexander Apelblat for providing him with the copies of his books and for keeping a correspondence via e-mail on the matters related to Volterra functions. Without this correspondence, this survey paper would not have been conceived. For several years, Apelblat has devoted his attention to Volterra functions as a chemist engineer, not as a mathematician. His current position is Emeritus Professor at the Chemical Engineering Department, Ben-Gurion University of the Negev, Beer Sheva, Israel.
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Weighted mixed spherical means and singular ultrahyperbolic equation
- Fractional approximation of solutions of evolution equations
- Forecasting of random sequences and Prony decomposition of finance data
- On Volterra functions and Ramanujan integrals
- An inverse problem for a multidimensional fractional diffusion equation
- Analysis of fractional diffusion equations of distributed order: Maximum principles and their applications
Articles in the same Issue
- Frontmatter
- Weighted mixed spherical means and singular ultrahyperbolic equation
- Fractional approximation of solutions of evolution equations
- Forecasting of random sequences and Prony decomposition of finance data
- On Volterra functions and Ramanujan integrals
- An inverse problem for a multidimensional fractional diffusion equation
- Analysis of fractional diffusion equations of distributed order: Maximum principles and their applications
