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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:
where γ is the Euler–Mascheroni constant and ζ '(2) is the derivative of the Riemann zeta function evaluated at s = 2 1961 [ OEIS 67 ] Lochs constant [ 79 ]
where is the Euler–Mascheroni constant and denotes asymptotic equivalence. It is unknown whether these constants are transcendental in general, but Γ( 1 / 3 ) and Γ( 1 / 4 ) were shown to be transcendental by G. V. Chudnovsky.
The definition for the gamma function due to Weierstrass is also valid for all complex numbers except non-positive integers: = = (+) /, where is the Euler–Mascheroni constant. [1] This is the Hadamard product of 1 / Γ ( z ) {\displaystyle 1/\Gamma (z)} in a rewritten form.
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: = = + + + + +.. The first terms of the series sum to approximately +, where is the natural logarithm and is the Euler–Mascheroni constant.
Gamma (/ ˈ ɡ æ m ə /; [1] uppercase Γ, lowercase γ; Greek: γάμμα, romanized: gámma) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek , the letter gamma represented a voiced velar stop IPA: [ɡ] .
Euler's product formula for the gamma function, combined with the functional equation and an identity for the Euler–Mascheroni constant, yields the following expression for the digamma function, valid in the complex plane outside the negative integers (Abramowitz and Stegun 6.3.16): [1]
The circumference of a circle with diameter 1 is π.. A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]