EulerE2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n

For fixed z

General characteristics

Domain and analyticity

(Subscript[E, n])(z) is a polynomial of z and as such an analytical function of z. (Subscript[E, n])(z) is defined in the whole complex z-plane and for n∈ℕ.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

The function (Subscript[E, n])(z) has a pole of order n at z==(OverTilde[∞]).

Branch points

With respect to z

The function (Subscript[E, n])(z) does not have branch points.

Branch cuts

With respect to z

The function (Subscript[E, n])(z) does not have branch cuts.

Series representations

Generalized power series

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

Expansions at z==0

Exponential Fourier series

Asymptotic series expansions

Other series representations

Integral representations

On the real axis

Of the direct function

Generating functions

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Multiple arguments

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Definite integration

For the direct function itself

Involving the direct function

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Summation

Finite summation

Infinite summation

Representations through more general functions

Through other functions

Involving Stirling numbers

Involving zeta functions

Representations through equivalent functions

With related functions

Inequalities

Other identities

Congruence properties

Theorems

The Boole summation formula

f(1)≈(1/2)(∑_(k=0)^n)(((Subscript[E, k])(1))/(k!))((f^((k)))(1)+(f^((k)))(0)).

History

– L. Euler
– H. F. Scherk (1825) suggested the name and calculated the first 30 numbers
– G. Chrystal (1889) introduced modern notations
– L. Saalschütz (1893) found the relation between Euler and Bernoulli numbers

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