HypergeometricU

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

Brychkov Yu.A. (2006)

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)

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

U(a,b,z) is an analytical function of a,  b and z which is defined in (ℂ^3).  For negative integer a, U(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 U(a,b,z) has only one singular point at b= (OverTilde[∞]).  It is an essential singular point.

With respect to a

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

Branch points

With respect to z

With respect to b

For fixed a, z, the function U(a,b,z) does not have branch points.

With respect to a

For fixed b, z, the function U(a,b,z) does not have branch points.

Branch cuts

With respect to z

With respect to b

For fixed a, z, the function U(a,b,z) does not have branch cuts.

With respect to a

For fixed b, z, the function U(a,b,z) does not have branch cuts.

Series representations

Generalized power series

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

Expansions on branch cuts

Expansions at z==0

Asymptotic series expansions

Residue representations

Limit 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

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

Brychkov Yu.A. (2007)

Brychkov Yu.A. (2007)

With respect to b

Mixed derivatives by parameters

Brychkov Yu.A. (2007)

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

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric1F1

Involving Hypergeometric1F1Regularized

Involving HypergeometricPFQ

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 hypergeometric HypergeometricU

Classical cases involving  Exp and  products of hypergeometric HypergeometricU

Classical cases involving Hypergeometric1F1

Classical cases involving Hypergeometric1F1Regularized

Classical cases involving  Exp and Hypergeometric1F1

Classical cases involving  Exp and Hypergeometric1F1Regularized

Classical cases involving LaguerreL LaguerreL

Classical cases involving  Exp and LaguerreL LaguerreL

Generalized cases for the direct function itself

Generalized cases for products of hypergeometric HypergeometricU

Generalized cases involving  Exp and  products of hypergeometric HypergeometricU

Generalized cases involving Hypergeometric1F1

Generalized cases involving  Exp and Hypergeometric1F1

Generalized cases involving  Exp and Hypergeometric1F1Regularized

Generalized cases involving LaguerreL LaguerreL

Generalized cases involving  Exp and LaguerreL LaguerreL

Representations through equivalent functions

With related functions

Theorems

The Green's function of the time-independent Schrödinger equation for the Coulomb problem in 3 dimensions

The Green's function G(x',x;ε) of the time–independent Schrödinger equation for the Coulomb problem in 3 dimensions

-Δ G(x',x;ε)+(α/(|x|))G(x',x;ε)-ε G(x',x;E)==-i δ(x'-x)

is given by

G(x',x;ε)==(i/(8 π r (SubPlus[r])))((e^( i z ((SubMinus[r])+(SubPlus[r]))/2)) Γ(1-i ν) (2  1(Subscript[F, 1])(1-i ν;2;-i z (SubMinus[r])) (i (i+ν) U(-i ν,-1,-i z (SubPlus[r]))+U(-i ν,0,-i z (SubPlus[r]))) (SubMinus[r])-U(-i ν,0,-i z (SubPlus[r])) (2  1(Subscript[F, 1])(1-i ν;2;-i z (SubMinus[r]))-z (i+ν)  1(Subscript[F, 1])(2-i ν;3;-i z (SubMinus[r])) (SubMinus[r])) (SubPlus[r])))/;(SubPlus[r])==|x|+|x'| +|x-x'|∧(SubMinus[r])==|x|+|x'| -|x-x'|∧r==|x-x'|∧z==(Sqrt[s])∧ν==α /(4(Sqrt[s])).

History

– F. Tricomi (1927)

Copyright