FresnelS

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

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

Symmetries and periodicities

Parity

S(z) is an odd function.

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

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

Branch points

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

Branch cuts

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

Series representations

Generalized power series

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

Expansions at z==0

Asymptotic series expansions

Residue representations

Integral representations

On the real axis

Of the direct function

Contour integral 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

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and a power function

Involving direct function and other elementary functions

Involving power of the direct function

Involving products of the direct function

Definite integration

For the direct function itself

Involving the direct function

Integral transforms

Laplace transforms

Mellin transforms

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 powers of Fresnel FresnelC,FresnelS

Generalized cases for the direct function itself

Generalized cases involving Cos,Sin and Fresnel FresnelC

Generalized cases involving powers of Fresnel FresnelC,FresnelS

Through other functions

Representations through equivalent functions

With related functions

Zeros

Theorems

Cornu spirals

Freeway exits will often have the shape of clothoids (Cornu spirals), which are curves with the parametrization {S(t),C(t)} .

History

– A.J. Fresnel (1798, 1818, 1826)
– K. W. Knochenhauer (1839) found a series representaion
– N. Nielsen (1906)

Copyright