CosIntegral

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

Ci(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 Ci(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

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

Involving the direct function

Involving related functions

Integral transforms

Laplace 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,SinIntegral

Generalized cases for the direct function itself

Generalized cases for powers of CosIntegral and SinIntegral

Generalized cases involving Cos,Sin,SinIntegral

Through other functions

Representations through equivalent functions

With related functions

Theorems

The averaged radiated intensity from a thin linear antenna

The averaged radiated intensity Π from a thin linear antenna of length l driven with a time-dependent current i=(Subscript[i, 0])cos(ω t+ n π (x/l)), n∈(SuperPlus[ℕ]), where x is the coordinate along the antenna, -l<x<l, is given by Π∝(Subscript[i, 0]^(2 ))(γ+log(2π n)-Ci(2π n)).

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 the notations Si and Ci

Copyright