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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 harmonic number with = ⌊ ⌋ (red line) with its asymptotic limit + (blue line) where is the Euler–Mascheroni constant.. In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] = + + + + = =.
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.
An example for such a particle [9] is the spin 1 / 2 companion to spin 3 / 2 in the D (½,1) ⊕ D (1,½) representation space of the Lorentz group. This particle has been shown to be characterized by g = − + 2 / 3 and consequently to behave as a truly quadratic fermion.
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In general terms of powers of or the following is true, and can be deduced using De Moivre's formula, Euler's formula and the binomial theorem. if n is ... cos n θ {\displaystyle \cos ^{n}\theta }