Arbitrary-precision computation of the gamma function

We discuss the best methods available for computing the gamma function $\Gamma(z)$ in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small arguments; low or high precision; with or without precomputation. The methods also cover the log-gamma function $\log \Gamma(z)$, the digamma function $\psi(z)$, and derivatives $\Gamma^{(n)}(z)$ and $\psi^{(n)}(z)$. Besides attempting to summarize the existing state of the art, we present some new formulas, estimates, bounds and algorithmic improvements and discuss implementation results

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Relationships between algebra, differential equations and logic in England 1800-1860.

This thesis surveys the links between mathematics and algebraic logic in England in the first half of the 19th century. In particular, we show the impact that De Morgan's work on the calculus of functions in 1836 had on the shaping of his logic of relations in 1860. Similarly we study Boole’s background in D-operational methods and its impact on his calculus of logic in 1847. The starting point of the thesis is Lagrange’s algebraic calculus and Laplace’s analytical methods prominent in late 18th century French mathematics. Revival in mathematical research in early 19th century England was mainly effected through the diffusion of Lagrange’s calculus of operations as further developed by Arbogast, Servois and others in the 1800’s and of Laplace’s theory of attractions. .Lagrange’s algebraic calculus and Laplace’s methods in analysis – particularly on functional equations – were considerably developed by Herschel and Babbage during the period 1812-1820. Further research on the foundations of the calculus of operations and functions was provided by Murphy, De Morgan and Gregory in the late 1830’s. .Symbolic methods in analysis were further extended by Boole in 1844. Boole was followed by several analysts distinguished in their obsession in further vindicating these methods through applications on two differential equations which originally appeared in Laplace’s planetary physics. We record the main issues of De Morgan’s logic and their mathematical background. Special reference is given to his logic of relations and its connection with his foundational study of the calculus of functions. On similar lines we study Boole’s algebraic cast of logic drawing consequently a comparison between his two major works on logic. Moreover we emphasise his epistemological views and his evaluation of symbolical methods within logic and analysis