Euler–Mascheroni constant

The constant γ is the limit of the difference between the nth harmonic number and ln n as n tends to infinity. It is approximately 0.57721 and appears in analysis, number theory, and asymptotic formulas.

Harmonic number: The nth harmonic number is the sum of the reciprocals of the first n positive integers. These finite sums form one side of the difference whose limit defines γ.
Stirling's approximation: Stirling's approximation estimates factorials using powers, exponentials, and a square-root factor. Its refined forms connect γ with asymptotic estimates involving factorials and harmonic sums.
Leonhard Euler: Leonhard Euler was an eighteenth-century mathematician whose work shaped analysis, number theory, and mathematical notation. Euler studied the limiting difference and introduced the notation γ.
Irrational number: An irrational number cannot be expressed as a ratio of two integers. Unlike many familiar constants, γ has no known proof of irrationality.
Harmonic series: The harmonic series is the divergent infinite sum of reciprocals of the positive integers. Its partial sums grow logarithmically, with γ measuring their persistent offset.
Digamma function: The digamma function is the logarithmic derivative of the gamma function. At the positive integers, its values relate harmonic numbers to γ.
Giulio Carlo Fagnano: Giulio Carlo Fagnano was an Italian mathematician known for work on lemniscates and elliptic integrals. His earlier work included a related limiting constant later associated with γ.
Transcendental number: A transcendental number is not a root of any nonzero polynomial with integer coefficients. Whether γ is transcendental remains unknown, a stronger unresolved question than irrationality.
Natural logarithm: The natural logarithm is the logarithm with base e, inverse to the exponential function. Subtracting ln n from the nth harmonic number produces the sequence converging to γ.
Gamma function: The gamma function extends the factorial to complex and noninteger arguments. The constant appears in expansions of the gamma function near zero and one.

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