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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:
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 interpolating function is in fact closely related to the digamma function = (+) +, where ψ(x) is the digamma function, and γ is the Euler–Mascheroni constant. The integration process may be repeated to obtain H x , 2 = ∑ k = 1 ∞ ( − 1 ) k − 1 k ( x k ) H k . {\displaystyle H_{x,2}=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}{x ...
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 area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant. In mathematics , the Stieltjes constants are the numbers γ k {\displaystyle \gamma _{k}} that occur in the Laurent series expansion of the Riemann zeta function :
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Euler's summation formula, a theorem about integrals. Cauchy–Euler equation (or Euler equation), a second-order linear differential equation; Cauchy–Euler operator; Euler–Maclaurin formula – relation between integrals and sums; Euler–Mascheroni constant or Euler's constant γ ≈ 0.577216; Integration using Euler's formula; Euler ...
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