SinIntegral

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

Si(z) is an entire function of z which is defined in the whole complex z-plane.

Symmetries and periodicities

Parity

Si(z) is an odd function.

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function Si(z) has only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

Branch points

The function Si(z) does not have branch points.

Branch cuts

The function Si(z) does not have branch cuts.

Series representations

Generalized power series

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

Expansions at z==0

Andrea Piccolroaz

Asymptotic series expansions

Residue representations

Other series representations

Integral representations

On the real axis

Of the direct function

Contour integral 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

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

Involving direct function and Gamma-, Beta-, Erf-type functions

Definite integration

For the direct function itself

Involving the direct function

Involving related functions

Integral transforms

Laplace transforms

Mellin transforms

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Classical cases for powers of CosIntegral and SinIntegral

Classical cases involving Cos,Sin,CosIntegral

Generalized cases for the direct function itself

Generalized cases for powers of CosIntegral and SinIntegral

Generalized cases involving Cos,Sin,CosIntegral

Through other functions

Representations through equivalent functions

With related functions

Zeros

Theorems

The Hartree and exchange energy of a metal surface within the semi-infinite jellium model

The Hartree and exchange energy of a metal surface within the semi-infinite jellium model can be expressed using the sine integral.

History

– L. Mascheroni (1790, 1819)
– F. W. Bessel (1812)
–  C. A. Bretschneider (1843)
– O. Schlomilch (1846)
– F. Arndt (1847)
– J. W. L. Glaisher (1870) introduced the notations Si and Ci
– N. Nielsen (1904) used notations Si and Ci

Copyright