AppellF1

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at ((Subscript[z, 1]),(Subscript[z, 2]))==(0,0)

Values at (Subscript[z, 2])==0

Values at (Subscript[z, 2])==1

Values at (Subscript[z, 2])==±(Subscript[z, 1])

Values at fixed points

For fixed a,(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[z, 1]),(Subscript[z, 2])

For fixed (Subscript[z, 1]),(Subscript[z, 2])

General characteristics

Some abbreviations

Domain and analyticity

(Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) is an analytical function of a, (Subscript[b, 1]), (Subscript[b, 2]), c, (Subscript[z, 1]), (Subscript[z, 2]), which is defined in (ℂ^6). For fixed a, (Subscript[b, 1]), c, (Subscript[z, 1]), (Subscript[z, 2]), it is an entire function of (Subscript[b, 2]). For fixed a, (Subscript[b, 2]), c, (Subscript[z, 1]), (Subscript[z, 2]), it is an entire function of (Subscript[b, 1]). For fixed (Subscript[b, 1]), (Subscript[b, 2]), c, (Subscript[z, 1]), (Subscript[z, 2]), it is an entire function of a. For negative integer a, (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) degenerates to a polynomial in (Subscript[z, 1]), (Subscript[z, 2]). For negative integer (Subscript[b, k]), (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) degenerates to a polynomial in (Subscript[z, k])of order -(Subscript[b, k]),1≤k≤2.

Symmetries and periodicities

Mirror symmetry

Permutation symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to (Subscript[z, k])

With respect to c

For fixed a, (Subscript[b, 1]), (Subscript[b, 2]), (Subscript[z, 1]), (Subscript[z, 2]), the function (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) has an infinite set of singular points with respect to c:
a) the points c==- k/;k∈ℕ are the simple poles with residues ((((-1))^k)/(k!)) (Subscript[, 2])(Subscript[(OverTilde[F]), 1])(a,b;-k;z);
b) c== (OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

With respect to (Subscript[b, k])

The function (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) does not have poles and essential singularities with respect to (Subscript[b, k]),1≤k≤2.

With respect to a

The function (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) does not have poles and essential singularities with respect to a.

Branch points

With respect to (Subscript[z, k])

With respect to c

The function (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) does not have branch points with respect to c.

With respect to (Subscript[b, k])

The function (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) does not have branch points with respect to (Subscript[b, k]),1≤k≤2.

With respect to a

The function (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) does not have branch points with respect to a.

Branch cuts

With respect to (Subscript[z, 1])

With respect to (Subscript[z, 2])

With respect to c

The function (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) does not have branch cuts with respect to c.

With respect to (Subscript[b, k])

The function (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) does not have branch cuts with respect to (Subscript[b, k]),1≤k≤2.

With respect to a

The function (Subscript[F, 1])(a;(Subscript[b, 1]),(Subscript[b, 2]);c;(Subscript[z, 1]),(Subscript[z, 2])) does not have branch cuts with respect to a.

Series representations

Generalized power series

Expansions at ((Subscript[z, 1]),(Subscript[z, 2]))==(0,0)

Expansions at (Subscript[z, 1])==0

Expansions at (Subscript[z, 2])==0

Expansions at (Subscript[z, 1])==1

Expansions at (Subscript[z, 2])==1

Expansions at (Subscript[z, 1])==∞

Expansions at (Subscript[z, 2])==∞

Residue representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Multiple integral representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Identities

Functional identities

Major general cases

Differentiation

Low-order differentiation

With respect to (Subscript[z, 1])

With respect to (Subscript[z, 2])

Symbolic differentiation

With respect to (Subscript[z, 1])

With respect to (Subscript[z, 2])

Fractional integro-differentiation

With respect to (Subscript[z, 1])

With respect to (Subscript[z, 2])

Integration

Indefinite integration

Involving only one direct function with respect to (Subscript[z, 1])

Involving only one direct function with respect to (Subscript[z, 2])

Representations through more general functions

Through hypergeometric functions of two variables

Through Meijer G

Classical cases for the direct function itself

History

– L. Pochhammer (1870)
– P. E. Appell (1880)
– J. Horn (1889)
– P.E. Appell and J. Kampé de Fériet (1926)

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