Relation to gamma function is related to the
digamma function (not to be confused with
wave function), and hence the
derivative of the
gamma function , when both functions are evaluated at 1. Thus: -\gamma = \Gamma'(1) = \psi(1). This is equal to the limits: \begin{align}-\gamma &= \lim_{z\to 0}\left(\Gamma(z) - \frac1{z}\right) \\&= \lim_{z\to 0}\left(\psi(z) + \frac1{z}\right).\end{align} Further limit results are: \begin{align} \lim_{z\to 0}\frac1{z}\left(\frac1{\Gamma(1+z)} - \frac1{\Gamma(1-z)}\right) &= 2\gamma \\ \lim_{z\to 0}\frac1{z}\left(\frac1{\psi(1-z)} - \frac1{\psi(1+z)}\right) &= \frac{\pi^2}{3\gamma^2}. \end{align} A limit related to the
beta function (expressed in terms of
gamma functions) is \begin{align} \gamma &= \lim_{n\to\infty}\left(\frac{ \Gamma\left(\frac1{n}\right) \Gamma(n+1)\, n^{1+\frac1{n}}}{\Gamma\left(2+n+\frac1{n}\right)} - \frac{n^2}{n+1}\right) \\ &= \lim\limits_{m\to\infty}\sum_{k=1}^m{m \choose k}\frac{(-1)^k}{k}\log\big(\Gamma(k+1)\big). \end{align}
Relation to the zeta function can also be expressed as an
infinite sum whose terms involve the
Riemann zeta function evaluated at positive integers: \begin{align}\gamma &= \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{m} \\ &= \log\frac4{\pi} + \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{2^{m-1}m}.\end{align} The constant \gamma can also be expressed in terms of the sum of the reciprocals of
non-trivial zeros \rho of the zeta function: One of the earliest of these is a formula for the
harmonic number attributed to
Srinivasa Ramanujan where \gamma is related to \textstyle \log 2T_{k} in a series that considers the powers of \textstyle \frac{1}{T_{k}} (an earlier, less-generalizable proof by
Ernesto Cesàro gives the first two terms of the series, with an error term): :\begin{align} \gamma &= H_u - \frac{1}{2} \log 2T_u - \sum_{k=1}^{v}\frac{R(k)}{T_{u}^{k}}-\Theta_{v}\,\frac{R(v+1)}{T_{u}^{v+1}} \end{align} From
Stirling's approximation follows a similar series: :\gamma = \log 2\pi - \sum_{k=2}^{\infty} \frac{\zeta(k)}{T_{k}} The series of inverse triangular numbers also features in the study of the
Basel problem posed by
Pietro Mengoli. Mengoli proved that \textstyle \sum_{k = 1}^\infty \frac{1}{2T_k} = 1, a result
Jacob Bernoulli later used to estimate the
value of \zeta(2), placing it between 1 and \textstyle \sum_{k = 1}^\infty \frac{2}{2T_k} = \sum_{k = 1}^\infty \frac{1}{T_{k}} = 2. This identity appears in a formula used by
Bernhard Riemann to compute
roots of the zeta function, where \gamma is expressed in terms of the sum of roots \rho plus the difference between Boya's expansion and the series of exact
unit fractions \textstyle \sum_{k = 1}^{\infty} \frac{1}{T_{k}}: :\gamma - \log 2 = \log 2\pi + \sum_{\rho} \frac{2}{\rho} - \sum_{k = 1}^{\infty} \frac{1}{T_k}
Integrals equals the value of a number of definite
integrals: \begin{align} \gamma &= - \int_0^\infty e^{-x} \log x \,dx \\ &= -\int_0^1\log\left(\log\frac 1 x \right) dx \\ &= \int_0^\infty \left(\frac1{e^x-1}-\frac1{x\cdot e^x} \right)dx \\ &= \int_0^1\frac{1-e^{-x}}{x} \, dx -\int_1^\infty \frac{e^{-x}}{x}\, dx\\ &= \int_0^1\left(\frac1{\log x} + \frac1{1-x}\right)dx\\ &= \int_0^\infty \left(\frac1{1+x^k}-e^{-x}\right)\frac{dx}{x},\quad k>0\\ &= 2\int_0^\infty \frac{e^{-x^2}-e^{-x}}{x} \, dx ,\\ &= \log\frac{\pi}{4}-\int_0^\infty \frac{\log x}{\cosh^2x} \, dx ,\\ &= \int_0^1 H_x \, dx, \\ &= \frac{1}{2}+\int_{0}^{\infty}\log\left(1+\frac{\log\left(1+\frac{1}{t}\right)^{2}}{4\pi^{2}}\right)dt \\ &= 1-\int_0^1 \{1/x\} dx \\ &= \frac{1}{2}+\int_{0}^{\infty}\frac{2x\,dx}{(x^2+1)(e^{2\pi x}-1)} \\ &= \frac{1}{\pi}\int_{0}^{\pi}\frac{\sin x}{x}e^{x\cot x}\log\left(\frac{\sin x}{x}e^{x\cot x}\right)dx \end{align} where is the
fractional harmonic number, and \{1/x\} is the
fractional part of 1/x. The third formula in the integral list can be proved in the following way: \begin{align} &\int_0^{\infty} \left(\frac{1}{e^x - 1} - \frac{1}{x e^x} \right) dx = \int_0^{\infty} \frac{e^{-x} + x - 1}{x[e^x -1]} dx = \int_0^{\infty} \frac{1}{x[e^x - 1]} \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}x^{m+1}}{(m+1)!} dx \\[2pt] &= \int_0^{\infty} \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}x^m}{(m+1)![e^x -1]} dx = \sum_{m = 1}^{\infty} \int_0^{\infty} \frac{(-1)^{m+1}x^m}{(m+1)![e^x -1]} dx = \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{(m+1)!} \int_0^{\infty} \frac{x^m}{e^x - 1} dx \\[2pt] &= \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{(m+1)!} m!\zeta(m+1) = \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{m+1}\zeta(m+1) = \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{m+1} \sum_{n = 1}^{\infty}\frac{1}{n^{m+1}} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{(-1)^{m+1}}{m+1}\frac{1}{n^{m+1}} \\[2pt] &= \sum_{n = 1}^{\infty} \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{m+1}\frac{1}{n^{m+1}} = \sum_{n = 1}^{\infty} \left[\frac{1}{n} - \log\left(1+\frac{1}{n}\right)\right] = \gamma \end{align} The integral on the second line of the equation is the definition of the
Riemann zeta function, which is . Definite integrals in which appears include: \begin{align} \int_0^\infty e^{-x^2} \log x \,dx &= -\frac{(\gamma+2\log 2)\sqrt{\pi}}{4} \\ \int_0^\infty e^{-x} \log^2 x \,dx &= \gamma^2 + \frac{\pi^2}{6} \\ \int_0^\infty \frac{e^{-x}\log x}{e^x +1} \,dx &= \frac12 \log^2 2 - \gamma \end{align} We also have
Catalan's 1875 integral \gamma = \int_0^1 \left(\frac1{1+x}\sum_{n=1}^\infty x^{2^n-1}\right)\,dx. One can express using a special case of
Hadjicostas's formula as a
double integral with equivalent series: \begin{align} \gamma &= \int_0^1 \int_0^1 \frac{x-1}{(1-xy)\log xy}\,dx\,dy \\ &= \sum_{n=1}^\infty \left(\frac 1 n -\log\frac{n+1} n \right). \end{align} An interesting comparison by Sondow is the double integral and alternating series \begin{align} \log\frac 4 \pi &= \int_0^1 \int_0^1 \frac{x-1}{(1+xy)\log xy} \,dx\,dy \\ &= \sum_{n=1}^\infty \left((-1)^{n-1}\left(\frac 1 n -\log\frac{n+1} n \right)\right). \end{align} It shows that may be thought of as an "alternating Euler constant". The two constants are also related by the pair of series \begin{align} \gamma &= \sum_{n=1}^\infty \frac{N_1(n) + N_0(n)}{2n(2n+1)} \\ \log\frac4{\pi} &= \sum_{n=1}^\infty \frac{N_1(n) - N_0(n)}{2n(2n+1)} , \end{align} where and are the number of 1s and 0s, respectively, in the
base 2 expansion of .
Series expansions In general, \gamma = \lim_{n \to \infty}\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3} + \ldots + \frac{1}{n} - \log(n+\alpha) \right) \equiv \lim_{n \to \infty} \gamma_n(\alpha) for any . However, the
rate of convergence of this expansion depends significantly on . In particular, exhibits much more rapid convergence than the conventional expansion . This is because \frac{1}{2(n+1)} while \frac{1}{24(n+1)^2} Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below. Euler showed that the following
infinite series approaches : \gamma = \sum_{k=1}^\infty \left(\frac 1 k - \log\left(1+\frac 1 k \right)\right). The series for is equivalent to a series
Nielsen found in 1897: \gamma = 1 - \sum_{k=2}^\infty (-1)^k\frac{\left\lfloor\log_2 k\right\rfloor}{k+1}. In 1910,
Vacca found the closely related series \begin{align} \gamma & = \sum_{k=1}^\infty (-1)^k\frac{\left\lfloor\log_2 k\right\rfloor} k \\[5pt] & = \tfrac12-\tfrac13 + 2\left(\tfrac14 - \tfrac15 + \tfrac16 - \tfrac17\right) + 3\left(\tfrac18 - \tfrac19 + \tfrac1{10} - \tfrac1{11} + \cdots - \tfrac1{15}\right) + \cdots, \end{align} where is the
logarithm to base 2 and is the
floor function. This can be generalized to: \gamma= \sum_{k=1}^\infty \frac{\left\lfloor\log_B k\right\rfloor}{k} \varepsilon(k)where:\varepsilon(k)= \begin{cases} B-1, &\text{if } B\mid n \\ -1, &\text{if }B\nmid n \end{cases} In 1926 Vacca found a second series: \begin{align} \gamma + \zeta(2) & = \sum_{k=2}^\infty \left( \frac1{\left\lfloor\sqrt{k}\right\rfloor^2} - \frac1{k}\right) \\[5pt] & = \sum_{k=2}^\infty \frac{k - \left\lfloor\sqrt{k}\right\rfloor^2}{k \left\lfloor \sqrt{k} \right\rfloor^2} \\[5pt] &= \frac12 + \frac23 + \frac1{2^2}\sum_{k=1}^{2\cdot 2} \frac{k}{k+2^2} + \frac1{3^2}\sum_{k=1}^{3\cdot 2} \frac{k}{k+3^2} + \cdots \end{align} From the
Malmsten–
Kummer expansion for the logarithm of the gamma function \frac{\pi^2}{6e^\gamma}=\lim_{n\to\infty} \log p_n \prod_{i=1}^n \frac{p_i}{p_i+1}. Other
infinite products relating to include: \begin{align} \frac{e^{1+\frac{\gamma}{2}}}{\sqrt{2\pi}} &= \prod_{n=1}^\infty e^{-1+\frac1{2n}}\left(1+\frac1{n}\right)^n \\ \frac{e^{3+2\gamma}}{2\pi} &= \prod_{n=1}^\infty e^{-2+\frac2{n}}\left(1+\frac2{n}\right)^n. \end{align} These products result from the
Barnes -function. In addition, e^{\gamma} = \sqrt{\frac2{1}} \cdot \sqrt[3]{\frac{2^2}{1\cdot 3}} \cdot \sqrt[4]{\frac{2^3\cdot 4}{1\cdot 3^3}} \cdot \sqrt[5]{\frac{2^4\cdot 4^4}{1\cdot 3^6\cdot 5}} \cdots where the th factor is the th root of \prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}. This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using
hypergeometric functions. It also holds that \frac{e^\frac{\pi}{2}+e^{-\frac{\pi}{2}}}{\pi e^\gamma}=\prod_{n=1}^\infty\left(e^{-\frac{1}{n}}\left(1+\frac{1}{n}+\frac{1}{2n^2}\right)\right). ==Published digits==