The bestknown properties and formulas for classical constants and the imaginary unit
Values
The imaginary unit satisfies the following relation:
Evaluation of specific values
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.
Analyticity
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.
Series representations
The five classical constants , , , , and have numerous series representations, for example, the following:
Product representations
The four classical constants , , , and can be represented by the following formulas:
Integral representations
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:
.
Limit representations
The six classical constants , , , , , and have numerous limit representations, for example:
Continued fraction representations
The four classical constants , , , and have numerous closed‐form continued fraction representations, for example:
Functional identities
The golden ratio satisfies the following special functional identities:
Complex characteristics
The eight classical constants (, , , , , , , and ) and the imaginary unit have the following complex characteristics:
Differentiation
Derivatives of the eight classical constants (, , , , , , , and ) and imaginary unit constant satisfy the following relations:
Integration
Simple indefinite integrals of the eight classical constants (, , , , , , , and ) and imaginary unit constant have the following values:
Integral transforms
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:
Inequalities
The eight classical constants (, , , , , , , and ) satisfy numerous inequalities, for example:
