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Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
Euler's identity is a direct result of Euler's formula, published in his monumental 1748 work of mathematical analysis, Introductio in analysin infinitorum, [16] but it is questionable whether the particular concept of linking five fundamental constants in a compact form can be attributed to Euler himself, as he may never have expressed it.
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
In mathematics, the Gompertz constant or Euler–Gompertz constant, [1] [2] denoted by , appears in integral evaluations and as a value of special functions. It is named after Benjamin Gompertz . It can be defined via the exponential integral as: [ 3 ]
Euler's critical load or Euler's buckling load is the compressive load at which a slender column will suddenly bend ... The four constants ,,, are determined by ...
The Euler–Mascheroni constant γ: In 2010 it has been shown that an infinite list of Euler-Lehmer constants (which includes γ/4) contains at most one algebraic number. [51] [52] In 2012 it was shown that at least one of γ and the Gompertz constant δ is transcendental. [53]