Infinity

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

∞ is a symbol that represents a positive infinite quantity.

General characteristics

∞ is a special symbol. On the Riemann sphere it is the north pole approached from exactly East. In the projective complex plane it is a point at the line at infinity.

Limit representations

Transformations

Products, sums, and powers of the direct function

Products involving the direct function

Sums of the direct function

Related transformations

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Differentiation

Low-order differentiation

Integration

Indefinite integration

Summation

Finite summation

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Inverse Laplace transforms

Representations through more general functions

Through other functions

Representations through equivalent functions

History

– John Wallis (1655) introduced the sign ∞ to signify infinite number
– K. Weierstrass (1876) used symbol ∞ to represent an actual infinity, which is prototype ofsymbol ComplexInfinity (OverTilde[∞]) in Mathematica

The symbol ∞ is encountered often in mathematics and the natural sciences.

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