ExpIntegralEi

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

Ei(z) is an analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function Ei(z) has an essential singularity at z==(OverTilde[∞]). At the same time, the point z==(OverTilde[∞]) is a branch point.

Branch points

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

Other series representations

Integral representations

On the real axis

Of the direct function

Contour integral 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

Related transformations

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

Involving one direct function and elementary functions

Involving functions of the direct function

Involving functions of the direct function and elementary functions

Definite integration

Involving the direct function

Integral transforms

Laplace transforms

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Involving hypergeometric HypergeometricU

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Exp

Classical cases for products of ExpIntegralEi

Through other functions

Representations through equivalent functions

With related functions

Theorems

The real part of the Heisenberg-Euler Lagrangian of quantum electrodynamics

The real part of the Heisenberg–Euler Lagrangian of quantum electrodynamics (Subscript[L, eff])(E,B) can be expressed as the following series.

Re((Subscript[L, eff])(E,B))==(Subscript[c, 1])-(α/(π^2))|(Subscript[c, 2])|(∑_(k=1)^∞)coth(π k (η/ξ))(Ci((k/ξ))cos((k/ξ))+Si((k/ξ))sin((k/ξ)))-coth(π k (ξ/η))(Ei(-(k/ξ))exp((k/ξ))+Ei((k/ξ))exp(-(k/ξ)))/;ξ==(e/π)(Sqrt[-(Subscript[c, 1])+(Sqrt[(Subscript[c, 1]^2)+(Subscript[c, 2]^2)])])∧η==(e/π)(Sqrt[(Subscript[c, 1])+(Sqrt[(Subscript[c, 1]^2)+(Subscript[c, 2]^2)])])∧(Subscript[c, 1])==(1/2)((E^2)-(B^2))∧(Subscript[c, 2])==E·B,

where E is the electric field, B is the magnetic induction and α is the fine structure constant.

The Nambu-Goldstone modes for weakly interacting fermions in a magnetic field

The Nambu-Goldstone modes for weakly interacting fermions in a magnetic field are described by (-Δ+(m^2)+V(r))ψ(r)=0 , where in 3+1–dimensional ladder quantum electrodynamics the potential  V(r) takes the form V(r)∝(e^((r^2)/(2(l^2))))Ei(-((r^2)/(2(l^2)))), where l is the magnetic length.

History

– A. M. Legendre (1811)
– O. Schlomilch (1846)
– F. Arndt (1847)
– J. W. L. Glaisher (1870) introduced the notation Ei

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