Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Khinchin






Mathematica Notation

Traditional Notation









Constants > Khinchin > Introduction to the classical constants





The best-known properties and formulas for classical constants and the imaginary unit


The imaginary unit satisfies the following relation:

For evaluation of the eight classical constants , , , , , , , and , Mathematica uses procedures that are based on the following formulas or methods:

The formula for is called Chudnovsky's formula.

The eight classical constants , , , , , , , and are positive real numbers. The constant is a quadratic irrational number. The constants , , and are irrational and transcendental over . Whether and are irrational is not known. The imaginary unit is an algebraic number.

The five classical constants , , , , and have numerous series representations, for example, the following:

The four classical constants , , , and can be represented by the following formulas:

The five classical constants , (and ), , , , and have numerous integral representations, for example:

The following integral is called the Gaussian probability density integral: .

The following integrals are called the Fresnel integrals: .

The six classical constants , , , , , and have numerous limit representations, for example:

The four classical constants , , , and have numerous closed‐form continued fraction representations, for example:

The golden ratio satisfies the following special functional identities:

The eight classical constants (, , , , , , , and ) and the imaginary unit have the following complex characteristics:

Derivatives of the eight classical constants (, , , , , , , and ) and imaginary unit constant satisfy the following relations:

Simple indefinite integrals of the eight classical constants (, , , , , , , and ) and imaginary unit constant have the following values:

All Fourier integral transforms and Laplace direct and inverse integral transforms of the eight classical constants (, , , , , , , and ) and the imaginary unit can be evaluated in a distributional or classical sense and can include the Dirac delta function:

The eight classical constants (, , , , , , , and ) satisfy numerous inequalities, for example:





© 1998-2014 Wolfram Research, Inc.