The best-known 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: