Hypergeometric1F1

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed a,b

For fixed a,z

For fixed b,z

For fixed z and with symbolical integers in parameters

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

For fixed z

For fixed z and a=-(11/2)

For fixed z and a=-(9/2)

For fixed z and a=-(7/2)

For fixed z and a=-(5/2)

For fixed z and a=-(3/2)

For fixed z and a=-(1/2)

For fixed z and a=(1/2)

For fixed z and a=1

For fixed z and a=(3/2)

For fixed z and a=2

For fixed z and a=(5/2)

For fixed z and a=3

For fixed z and a=(7/2)

For fixed z and a=4

For fixed z and a=(9/2)

For fixed z and a=5

For fixed z and a=(11/2)

For fixed z and a=6

General characteristics

Domain and analyticity

(Subscript[, 1])(Subscript[F, 1])(a;b;z) is an analytical function of a,  b and z which is defined in (ℂ^3). For fixed a, b, it is an entire function of z. For fixed b, z, it is an entire function of a. For negative integer a, (Subscript[, 1])(Subscript[F, 1])(a;b;z) degenerates to a polynomial in z of order -a.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

With respect to b

For fixed a, z, the function (Subscript[, 1])(Subscript[F, 1])(a;b;z) has an infinite set of singular points:
a) b==-k/;k∈ℕ, are the simple poles with residues ((((-1))^k)/(k!))(Subscript[, 1])(Subscript[(OverTilde[F]), 1])(a;-k;z);
b) b==(OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

With respect to a

For fixed b, z, the function (Subscript[, 1])(Subscript[F, 1])(a;b;z) has only one singular point at a= (OverTilde[∞]).  It is an essential singular point.

Branch points

With respect to z

The function (Subscript[, 1])(Subscript[F, 1])(a;b;z) does not have branch points with respect to z.

With respect to b

The function (Subscript[, 1])(Subscript[F, 1])(a;b;z) does not have branch points with respect to b.

With respect to a

The function (Subscript[, 1])(Subscript[F, 1])(a;b;z) does not have branch points with respect to a.

Branch cuts

With respect to z

The function (Subscript[, 1])(Subscript[F, 1])(a;b;z) does not have branch cuts with respect to z.

With respect to b

The function (Subscript[, 1])(Subscript[F, 1])(a;b;z) does not have branch cuts with respect to b.

With respect to a

The function (Subscript[, 1])(Subscript[F, 1])(a;b;z) does not have branch cuts with respect to a.

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Expansions at z==∞ for polynomial cases

Asymptotic series expansions

Residue representations

Limit representations

Continued fraction representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Products, sums, and powers of the direct function

Products of the direct function

Sums of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Relations of special kind

Division on even and odd parts and generalization

Differentiation

Low-order differentiation

With respect to a

With respect to b

With respect to element of parameters |||  With respect to element of parameters

With respect to z

Symbolic differentiation

With respect to a

With respect to b

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

For the direct function itself

Involving the direct function

Integral transforms

Laplace transforms

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Involving Hypergeometric1F1Regularized

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Exp

Classical cases involving  Exp and Cosh

Classical cases involving  Exp and Sinh

Classical cases for products of Hypergeometric1F1

Classical cases involving  Exp and  products of Hypergeometric1F1

Classical cases involving Hypergeometric1F1Regularized

Classical cases involving  Exp and Hypergeometric1F1Regularized

Classical cases involving hypergeometric HypergeometricU

Classical cases involving  Exp and hypergeometric HypergeometricU

Classical cases involving Laguerre LaguerreL

Classical cases involving Exp and Laguerre LaguerreL

Generalized cases for the direct function itself

Generalized cases for products of Hypergeometric1F1

Generalized cases involving  Exp and  products of Hypergeometric1F1

Generalized cases involving Hypergeometric1F1Regularized

Generalized cases involving  Exp and Hypergeometric1F1Regularized

Generalized cases involving hypergeometric HypergeometricU

Generalized cases involving  Exp and hypergeometric HypergeometricU

Generalized cases involving Laguerre LaguerreL

Generalized cases involving Exp and Laguerre LaguerreL

Representations through equivalent functions

With related functions

Theorems

The solution to the two dimensional time-independent Schrödinger equation

The solution to the two dimensional time–independent Schrödinger equation with the harmonic oscillator  potential

-(((∂^2)(Subscript[ψ, n,m])(x,y))/(∂(x^2)))-(((∂^2)(Subscript[ψ, n,m])(x,y))/(∂(y^2)))+((ω^2)/4)((x^2)+(y^2))(Subscript[ψ, n,m])(x,y)==(Subscript[ε, n,m])(Subscript[ψ, n,m])(x,y)

in polar coordinates is given by

(Subscript[ε, n,m])==ω(|m|+2n+1)/;m∈ℤ,n∈ℕ,
(Subscript[ψ, n,m])(r,ϕ)==(e^(i m ϕ))  (r^(|m|))(e^(-(ω/4)(r^2)))(Subscript[, 1])(Subscript[F, 1])(-n;|m|+1;(ω/2)(r^2))/;m∈ℤ,n∈ℕ.

One-parameter family of solutions of Burgers' equation

The function

(Subscript[w, α])(x,y)==(Γ(α+1) (Γ(((α+1)/2))x   1(Subscript[F, 1])(((α+1)/2);(3/2);((x^2)/(4 y)))- Γ((α/2))(Sqrt[y])  1(Subscript[F, 1])((α/2);(1/2);((x^2)/(4 y)))))/(Γ((α/2)+1) Γ(α)x (Sqrt[y]) 1(Subscript[F, 1])((α/2)+1;(3/2);((x^2)/(4 y)))-Γ(α) Γ(((α+1)/2))y   1(Subscript[F, 1])(((α+1)/2);(1/2);((x^2)/(4 y))))

is a one-parameter family of solutions of Burgers' equation

((∂w(x,y))/(∂y))+((∂w(x,y))/(∂x)) w(x,y)==(((∂^2)w(x,y))/(∂(x^2))).

Padé approximation to the exponential function

The function (Subscript[, 1])(Subscript[F, 1])(-p;-p-q;z)/(Subscript[, 1])(Subscript[F, 1])(-q;-p-q;-z) is the [p,q] –Padé approximation to exp(z).

Effective potential of the hydrogen atom in a strong magnetic field

The function

(Subscript[V, m])(x)==(∫_0^∞)(((r^(2 m)) (e^(-(r^2))))/(Sqrt[(r^2)+(x^2)]))dr==(1/2) (((Γ(-m) Γ(m+1/2))/(Sqrt[π])) (Subscript[, 1])(Subscript[F, 1])(m+(1/2);m+1;(x^2)) (x^(2 m))+Γ(m) (Subscript[, 1])(Subscript[F, 1])((1/2);1-m;(x^2)))

is the effective potential of the hydrogen atom in a strong magnetic field where m is the angular momentum and x is the distance from the nucleus.

Dirac equation with d-dimensional Coulomb potential

The d-dimensional Dirac equation i(∑_(μ=0)^d)(γ^μ)((∂_μ)+i e (Subscript[A, μ]))ψ(x,t)=m ψ(x,t) where the d+1 matrices (γ^μ) obey the commutation relations (γ^μ).(γ^ν)+(γ^ν).(γ^μ)=2(η^(μ ν))1 and (η^(μ ν))=(Subscript[δ, μ 0])(Subscript[δ, μ ν])-(1-(Subscript[δ, μ 0]))(Subscript[δ, μ ν]) and e (Subscript[A, 0])=V(r)=-Z α (r^(-1)), Z α>0, (Subscript[A, 1])=(Subscript[A, 2])=…=(Subscript[A, d])=0 yields after separting the angular part for the radial part the coupled equations

+((∂ G(r))/(∂r))+(k/r)G(r)=(ε-V(r)-m)F(r)
-((∂ F(r))/(∂r))+(k/r)F(r)=(ε-V(r)+m)G(r)

where k=±(l+(OverTilde[d])) ((OverTilde[d])=d/2 for even d and (OverTilde[d])=(d-1)/2 for odd d and l is the highest weight angular momentum) and ε is the energy of the stationary states. The bound state solutions of these two coupled differential equations are

({{(Subscript[F, ε])(r)}, {(Subscript[G, ε])(r)}})}=((((m^2)-(ε^2))^(1/4))/(Γ(2λ+1)))(Sqrt[(((m±ε)ε Γ(ν+2λ+1))/(2(m^2)τ (k+τ m/ε)ν!))])(ρ^λ)(e^(-ρ/2))((k+τ m/ε)(Subscript[, 1])(Subscript[F, 1])(-ν;2λ+1;ρ)∓ν (Subscript[, 1])(Subscript[F, 1])(1-ν;2λ+1;ρ))

where ρ=2r ((((m^2)-(ε^2)))^(1/2)), κ=((((k^2)-(ξ^2)))^(1/2)), τ=ε ξ ((((m^2)-(ε^2)))^(-1/2)), ν=τ-λ∈ℕ, and 0<ε<m. 
This yields the energy eigenvalues ε=m (((1+(((ξ^2)(((((k^2)-(ξ^2)))^(1/2))+ν))^(-2))))^(-1/2)).

History

– E. E. Kummer (1836)

Copyright