Abstract
The Schottky–Klein prime function: a theoretical and computational tool for applications
Highlights
Special functions have always played a pivotal role in the applied sciences and, while the heyday of their study has arguably passed, they remain a powerful and ubiquitous tool
The S–K prime function for multiply connected circular domains In Sections 2 and 3 we have described the prime function associated with two of the simplest circular domains: the unit disc and a concentric annulus
We do not do that here and instead we present two methods each having more novel mathematical features: the method presented in Section 7 is based on a new transform approach to boundary value problems for analytic functions in circular domains recently formulated in Crowdy (2015a,b); the second method, described in Section 8, is closer to a more traditional boundary integral formulation but is unusual in relying on use of the so-called ‘generalized Neumann kernel’ (Nasser, 2009) rather than the usual Cauchy kernel
Summary
Special functions have always played a pivotal role in the applied sciences and, while the heyday of their study has arguably passed, they remain a powerful and ubiquitous tool. The most cited mathematical work of all time is Abramowitz and Stegun’s Handbook of Mathematical Functions (Abramowitz & Stegun, 1972), which in recent years has received around 2,000 citations per year. In a 10-year project, this classic monograph has been entirely revamped as the ‘Digital Library of Mathematical Functions’ (DLMF) which is both an online resource (http://dlmf.nist.gov) and a textbook reference called the NIST Handbook of Mathematical Functions (Olver et al, 2010).
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