Relates logarithm and harmonic series
Not to be confused with
Euler's number ,
e ≈ 2.71828 , the base of the natural logarithm.
Constant value used in mathematics
Euler's constant Type Unknown Fields Discovered 1734 By Leonhard Euler First mention De Progressionibus harmonicis observationes Named after
The area of the blue region converges to Euler's constant.
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 :
γ
=
lim
n
→
∞
(
−
log
n
+
∑
k
=
1
n
1
k
)
=
∫
1
∞
(
−
1
x
+
1
⌊
x
⌋
)
d
x
.
{\displaystyle {\begin{aligned}\gamma &=\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)\\[5px]&=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,\mathrm {d} x.\end{aligned}}}
Here, ⌊·⌋ represents the floor function .
The numerical value of Euler's constant, to 50 decimal places, is:[ 1]
0.5772156649 01532 86060 65120 90082 40243 10421 59335 93992 ...
Unsolved problem in mathematics :
Is Euler's constant irrational? If so, is it transcendental?
^ a b Cite error: The named reference A001620
was invoked but never defined (see the help page ).