16,129 research outputs found
Numerical computation of real or complex elliptic integrals
Algorithms for numerical computation of symmetric elliptic integrals of all
three kinds are improved in several ways and extended to complex values of the
variables (with some restrictions in the case of the integral of the third
kind). Numerical check values, consistency checks, and relations to Legendre's
integrals and Bulirsch's integrals are included
Analytical expressions and numerical evaluation of the luminosity distance in a flat cosmology
Accurate and efficient methods to evaluate cosmological distances are an
important tool in modern precision cosmology. In a flat CDM cosmology,
the luminosity distance can be expressed in terms of elliptic integrals. We
derive an alternative and simple expression for the luminosity distance in a
flat CDM based on hypergeometric functions. Using a timing experiment
we compare the computation time for the numerical evaluation of the various
exact formulae, as well as for two approximate fitting formulae available in
the literature. We find that our novel expression is the most efficient exact
expression in the redshift range . Ideally, it can be combined with
the expression based on Carlson's elliptic integrals in the range
for high precision cosmology distance calculations over the entire redshift
range. On the other hand, for practical work where relative errors of about
0.1% are acceptable, the analytical approximation proposed by Adachi & Kasai
(2012) is a suitable alternative.Comment: 4 pages, 1 figure, accepted for publication in MNRA
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
We consider the calculation of the master integrals of the three-loop massive
banana graph. In the case of equal internal masses, the graph is reduced to
three master integrals which satisfy an irreducible system of three coupled
linear differential equations. The solution of the system requires finding a matrix of homogeneous solutions. We show how the maximal cut can be
used to determine all entries of this matrix in terms of products of elliptic
integrals of first and second kind of suitable arguments. All independent
solutions are found by performing the integration which defines the maximal cut
on different contours. Once the homogeneous solution is known, the
inhomogeneous solution can be obtained by use of Euler's variation of
constants.Comment: 39 pages, 3 figures; Fixed a typo in eq. (6.16
Fast and stable contour integration for high order divided differences via elliptic functions
In this paper, we will present a new method for evaluating high order divided
differences for certain classes of analytic, possibly, operator valued
functions. This is a classical problem in numerical mathematics but also
arises in new applications such as, e.g., the use of generalized convolution
quadrature to solve retarded potential integral equations. The functions which
we will consider are allowed to grow exponentially to the left complex half
plane, polynomially to the right half plane and have an oscillatory behaviour
with increasing imaginary part. The interpolation points are scattered in a
large real interval. Our approach is based on the representation of divided
differences as contour integral and we will employ a subtle parameterization
of the contour in combination with a quadrature approximation by the
trapezoidal rule
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