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Uniform approximations of the first symmetric elliptic integral in terms of elementary functions
(Springer, 2022)
info:eu-repo/semantics/article,
We consider the standard symmetric elliptic integral RF(x, y, z) for complex x, y, z. We derive convergent expansions of RF(x, y, z) in terms of elementary functions that hold uniformly for one of the three variables x, y ...
Asymptotic expansions for Moench's integral transform of hydrology
(MDPI, 2023)
Artículo / Artikulua,
Theis' theory (1935), later improved by Hantush & Jacob (1955) and Moench (1971), is a technique designed to study the water level in aquifers. The key formula in this theory is a certain integral transform H[g](r,t) of ...
A convergent and asymptotic Laplace method for integrals
(Elsevier, 2023)
Artículo / Artikulua,
Watson’s lemma and Laplace’s method provide asymptotic expansions of Laplace integrals F (z) := ∫ ∞
0
e
−zf (t)
g(t)dt for large values of the parameter z. They are useful
tools in the asymptotic approximation of ...
New analytic representations of the hypergeometric functions p+1Fp
(Springer, 2021)
info:eu-repo/semantics/article,
The power series expansions of the hypergeometric functions p+1Fp (a,b1,…,bp;c1,…,cp;z) converge either inside the unit disk |z|<1 or outside this disk |z|>1. Nørlund’s expansion in powers of z/(z−1) converges in the ...
Uniform convergent expansions of integral transforms
(American Mathematical Society, 2021)
info:eu-repo/semantics/article,
Several convergent expansions are available for most of the special functions of the mathematical physics, as well as some asymptotic expansions [NIST Handbook of Mathematical Functions, 2010]. Usually, both type of ...
The Pearcey integral in the highly oscillatory region
(Elsevier, 2016)
info:eu-repo/semantics/article,
We consider the Pearcey integral P(x, y) for large values of |y| and bounded values of |x|. The integrand of the Pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is complicated. ...
Series representations of the Volterra function and the Fransén–Robinson constant
(Elsevier, 2021)
info:eu-repo/semantics/article,
The Volterra function μ(t,β,α) was introduced by Vito Volterra in 1916 as the solution to certain integral equations with a logarithmic kernel. Despite the large number of applications of the Volterra function, the only ...
Convergent expansions of the Bessel functions in terms of elementary functions
(Springer US, 2018)
info:eu-repo/semantics/article,
We consider the Bessel functions Jν (z) and Yν (z) for ν > −1/2 and
z ≥ 0. We derive a convergent expansion of Jν (z) in terms of the derivatives of
(sin z)/z, and a convergent expansion of Yν (z) in terms of derivatives ...
Convergent expansions of the confluent hypergeometric functions in terms of elementary functions
(American Mathematical Society, 2018)
info:eu-repo/semantics/article,
We consider the confluent hypergeometric function M(a, b; z) for z ∈ C and Rb >Ra > 0, and the confluent hypergeometric function U(a, b; z) for b ∈ C, Ra > 0, and Rz > 0. We derive two convergent expansions of M(a, b; z); ...
Convergent expansions of the incomplete gamma functions in terms of elementary functions
(World Scientific Publishing, 2017)
info:eu-repo/semantics/article,
We consider the incomplete gamma function γ(a,z) for Ra>0 and z∈C. We derive several convergent expansions of z−aγ(a,z) in terms of exponentials and rational functions of z that hold uniformly in z with Rz bounded from ...