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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:
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.
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 ]
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]
In addition, Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of complex analysis .
A famous relationship is Euler's reflection formula () = (), for the gamma function (), due to Leonhard Euler. There is also a reflection formula for the general n-th order polygamma function ψ (n) (z),
Euler's totient function in number theory; the argument of a complex number in mathematics; the value of a plane angle in physics and mathematics; the angle to the z axis in spherical coordinates (mathematics) epoch or phase difference between two waves or vectors; the angle to the x axis in the xy-plane in spherical or cylindrical coordinates ...
Euler's gamma function is strictly logarithmically convex when restricted to the positive real numbers. In fact, by the Bohr–Mollerup theorem , this property can be used to characterize Euler's gamma function among the possible extensions of the factorial function to real arguments.