E

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

General characteristics

The Euler number e is a constant. It is irrational and transcendental over ℚ positive real number.

Series representations

Generalized power series

Expansions for e

H. J. Brothers

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Expansions for 1/e

H. J. Brothers

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Expansions for (Sqrt[e])

H. J. Brothers

Product representations

Limit representations

Continued fraction representations

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 hypergeometric functions

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Through Meijer G

Classical cases

Through other functions

Representations through equivalent functions

Inequalities

History

– John Napier (1618) in the work on logarithms mentioned existence of special convinient constant for calculation of logarithms on its base but he did not evaluate it;
– William Oughtred (1622) found famouse slide rule, which is used for multiplication, division, evaluation of roots, logarithms and other functions;
– Isaac Newton (1669)published series 2+1/2!+1/3!+…==2.718281828459045`…;
– Jacob Bernoulli tried to find the limit of  (((1+(1/n)))^n), when n->∞;
– Leibniz (1690)was the first to recognize e as a constant, but he used notation b;
– L. Euler (1727, 1728)denoted limit (lim_(x->0)) (((1+x))^(1/x)) by the letter e; 
– L. Euler (1731) introduced the notation e in a letter to Goldbach;
 – L. Euler (1737) proved that e and (e^2) are irrational numbers and represented e through continued fractions;
– L. Euler (1748) represented e as sum of series 1+(1/1)+(1/(2!))+(1/(3!))+(1/(4!))+… and found its 23 digits;
– D. Bernoulli (1760) used e, as base of the natural logarithms;
– J. H. Lambert (1768) proved that (e^(p/q))is irrational number, if p/q is nonzero rational number;
– A. L. Cauchy (1823) determined e==(lim_(z->∞)) (((1+(1/z)))^z);
– J. Liouville (1844) proved that e does not satisfy any quadratic equation with integral coefficients;
–  Ch. Hermite (1873) proved that e is a transcendental number;
–  E. Catalan (1873) represented e through infinite product;

The only constant appearing more frequently in mathematics than e is π.

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