What Is Euler's Constant?

Euler's constant is a mathematical expression for the limit of the sum of 1 + 1/2 + 1/3 + 1/4 ... + 1/n, minus the natural log of n as n approaches infinity. Euler's constant is represented by the lower case gamma (γ) and appears in calculus as a derivative of a logarithmic function. It is the difference between a harmonic series and the natural logarithm (log base e). There is no closed-form expression for the harmonic number, but gamma can provide an estimate of it.

Euler's constant can often be found in analysis methods and number theory. It is also referred to as the Euler–Mascheroni constant.

Understanding Euler's Constant

Information on Euler's constant was presented by the Swiss mathematician Leonard Euler in the 18th century in his work "De Progressionibus Harmonicus Observations." Mathematicians are uncertain as to whether it is a rational, transcendental (like pi) or algebraic number. It is not the same as Euler's number, e, nor is it as well-known as pi or e.