The Euler-Mascheroni constant (often incorrectly called Euler's constant) is a
mathematical constant that constantly pops up in analysis and number theory.
It's defined as the limiting difference between the harmonic series and the natural
logarithm, and is usually denoted by the lowercase Greek letter gamma (γ).
Rounded to the nearest 10-8, the number is γ = 0.5772156 6 .
γ should not be confused with the base of the natural logarithm, e, which is
sometimes called Euler's number or Euler's constant.
That number, rounded to the nearest 10-8, is e = 2.71828 183 .
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It is approximately Euler's constant.The "true" value can be obtained from the infinite sum:e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...where n! denotes 1*2*3*...*n.
Eulers number Approx x^2.31
Euler published the formula, which relates complex exponentials to trigonometric functions in 1748. See related link.
I think the following Wikipedia link on Fourier Series (see related links below), has the information that you're looking for.
between what two square root of integers is Eulers number