Hypergeometric1F0

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Integral representations

Contour integral representations

Identities

Functional identities

Complex characteristics

Theorems

Gelfond-Schneider theorem

The number (α^β), where α, β are algebraic numbers, α≠0,1, β ∉ ℚ, is transcendental.

Mellin transformation and Parseval relation

(OverHat[f])(s)==(∫_0^∞)(x^(s-1))f(x)dx⇔ f(x)==(1/(2π i ))(∫_(γ-i ∞)^(γ+i ∞))(OverHat[f])(s)(x^(-s))ds;(∫_0^∞)(Subscript[f, 1])((x/t))(Subscript[f, 2])(t)((dt)/t)==(1/(2π i ))(∫_(γ-i ∞)^(γ+i ∞))(OverHat[(Subscript[f, 1])])(s)(OverHat[(Subscript[f, 2])])(s)(x^(-s))ds.

Hilbert transformation

(OverHat[f])(y)==(1/π)(∫_(-∞)^∞)((f(x))/(x-y))dx⇔f(x)==-(1/π)(∫_(-∞)^∞)(((OverHat[f])(y))/(y-x))dy.

The eigenfunctions of the Hilbert transformation are given by the following:

(1/π)𝒫(∫_(-∞)^∞)((f(x,n))/(x-y))dx==sgn(n) f(y,n)/;f(x,n)==((((1+i x))^n)/(((1-i x))^(n+1)))∧n∈ℤ

Zipf's law

The frequency f versus the rank r in a text has the form f=(r^(-β)), where β≈0.6 for natural languages.

The quantum mechanical density of states of a  d-dimensional lattice

The quantum mechanical density of states 𝒟(ε) of a  d-dimensional lattice exhibits van Hove singularities of the form 𝒟(ε)∝(ε^((d-1)/2)), where ε is the energy.

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