EulerGamma

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

General characteristics

The Euler-Mascheroni number ℽ  is a constant. It is a positive real number. Whether ℽ is irrational or transcendental over ℚ are not known.

Series representations

Generalized power series

Other series representations

Product representations

Integral representations

On the real axis

Of the direct function

Multiple integral representations

Limit representations

A. Radović

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Fractional integro-differentiation

Integration

Indefinite integration

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 Meijer G

Through other functions

Inequalities

Theorems

The value of the sum of all integers whose squares divide an integer

The expected value of the sum of all integers whose squares divide an integer n is given asymptotically by (1/2)log(n)+(3/2)ℽ.

History

– L. Euler (1735, 1740) introduced this constant, denoted it through symbol C, and initially calculated its value to 6 decimal places;
– L. Euler (1781) calculated it to 16 digits;
– Lorenzo Mascheroni (1790) first used symbol γ for this constant and calculated it to 19 correct digits;
– Soldner (1809) calculated γ to 40 correct digits;
– Gauss and Nicolai (1812) verified calculation γ to 40 correct digits;

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