Zeta2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed s

For fixed a

Values at fixed points

Values at fixed points

General characteristics

Domain and analyticity

Domain and analyticity

ζ(s,a) is an analytical function of s,a which is defined in (ℂ^2). For fixed a, it is an entire function of s.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to a

For fixed s, the function ζ(s,a) does not have poles and essential singularities.

For positive integer s, the function (OverHat[ζ])(s,a) has an infinite set of singular points:
a) a==-n/;s==1∧n∈ℕ are the simple poles with residues 1;
b) a==-n/;s>1∧n∈ℕ are the poles of order s with residues 0;
c) a==(OverTilde[∞]) is the point of convergence of poles,  which is an essential singular point.

With respect to s

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

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

Branch points

With respect to a

For fixed s, the function ζ(s,a) does not have branch points.

With respect to s

For fixed a, the function ζ(s,a) does not have branch points.

For fixed a, the function (OverHat[ζ])(s,a) does not have branch points.

Branch cuts

With respect to a

For fixed s/;(s/2)∉ℤ, the function ζ(s,a) has infinitely many branch cuts parallel to the imaginary axis at Re(a)=-n/;n∈ℕ. In these cases the function ζ(s,a) is continuous from the left on the interval (-i ∞-n,-n)/;n∈ℕ and from the right on the interval (-n,-n+i ∞)/;n∈ℕ.

For integer (s/2), the function ζ(s,a) does not have branch cuts.

For fixed noninteger s, the function (OverHat[ζ])(s,a) is a single-valued function on the a-plane cut along the interval (-∞, 0), where  (OverHat[ζ])(s,a)  is continuous from above. This interval includes an infinite set of branch cut lines of power type along (-∞, -n)/;n∈ℕ.
 For integer s, the function (OverHat[ζ])(s,a) does not have branch cuts.

For fixed non integer s, the function (OverTilde[ζ])(s,a) is a single-valued function on the a-plane cut along the interval (-∞, 0), where  (OverTilde[ζ])(s,a)  is continuous from above. This interval includes an infinite set of branch cut lines of power type along (-∞, -n)/;n∈ℕ.

For fixed a, the function (OverTilde[ζ])(s,a) does not have branch cuts.

With respect to s

For fixed a, the function ζ(s,a) does not have branch cuts.

For fixed a, the function (OverHat[ζ])(s,a) does not have branch cuts.

For fixed a, the function (OverTilde[ζ])(s,a) does not have branch cuts.

Series representations

Generalized power series

Expansions at s==(Subscript[s, 0])/; (Subscript[s, 0])≠1

Expansions at s==1

Expansions at a==(Subscript[a, 0])/; (Subscript[a, 0])≠-n

Expansions at a==-n

Expansions on branch cuts

Asymptotic series expansions

For Zeta[±n,a] by a

For Derivative[1,0][Zeta][s,a] by a

Exponential Fourier series

Exponential Fourier series

Residue representations

Residue representations

Allan Cortzen

Allan Cortzen

Allan Cortzen

Other series representations

Other series representations

Allan Cortzen

Allan Cortzen

Integral representations

On the real axis

Of the direct function

Contour integral representations

Contour integral representations

Allan Cortzen

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple arguments

Argument involving numeric multiples of variable

Argument involving symbolic multiples of variable

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Major general cases

Pavlyk 0. (2006)

Differentiation

Low-order differentiation

With respect to s

With respect to a

Symbolic differentiation

With respect to s

With respect to a

Fractional integro-differentiation

With respect to s

With respect to a

Integration

Indefinite integration

Involving only one direct function with respect to s

Involving only one direct function with respect to a

Summation

Finite summation

Finite summation

Infinite summation

Infinite summation

Operations

Limit operation

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Through other functions

Through other functions

Representations through equivalent functions

Interrelations

Pavlyk 0. (2006)

With related functions

With related functions

History

– H. Mellin (1899)
– E. Lindelöf (1899)

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