Toolbox

List of formulae often used in the resolution of divergent series, divergent infinite products and/or divergent integrals.

$\gamma = \zeta(1) \approx 0.5772156649015328606065120900824024310\dots$ denotes the Euler-Mascheroni constant.

Harmonic Numbers

$H_0 = 0 \quad \text{and} \quad H_{n} = H_{n-1} + \dfrac{1}{n} \qquad (n \geq 1)$

$H_n = \psi^{(0)}(n+1) + \gamma \qquad (n\in\mathbb{N})$

The reflection formulae for harmonic numbers at the integers are

$H_{-n} = \psi^{(0)}(1-n) + \gamma = H_{n-1} = \psi^{(0)}(n) + \gamma = H_{n} - \dfrac{1}{n} \quad (n \geq 1)$

$H_{-\frac{1}{n}} = H_{\frac{1}{n}} - n + \pi \cot\left(\dfrac{\pi}{n}\right) \qquad (n \geq 1)$

In particular, we have

$H_\frac{1}{2} = 2 + H_{-\frac{1}{2}} = 2 - 2\ln2$

$H_\frac{1}{3} = 3 + H_{-\frac{2}{3}} = 3 - \dfrac{3\ln3}{2} - \dfrac{\pi}{2\sqrt{3}}$

$H_\frac{2}{3} = \dfrac{3}{2} + H_{-\frac{1}{3}} = \dfrac{3}{2} - \dfrac{3\ln3}{2} + \dfrac{\pi}{2\sqrt{3}}$

$H_\frac{1}{4} = 4 + H_{-\frac{3}{4}} = 4 - 3\ln2 - \dfrac{\pi}{2}$

$H_\frac{3}{4} = \dfrac{4}{3} + H_{-\frac{1}{4}} = \dfrac{4}{3} - 3\ln2 + \dfrac{\pi}{2}$

$H_\frac{1}{5} = 5 + H_{-\frac{4}{5}} = 5 - \dfrac{5\ln5}{4} - \dfrac{\sqrt{5}}{4}\,\ln\dfrac{3+\sqrt{5}}{2} - \dfrac{\pi}{2}\,\sqrt{1+\dfrac{2}{\sqrt{5}}}$

$H_\frac{2}{5} = \dfrac{5}{2} + H_{-\frac{3}{5}} = \dfrac{5}{2} - \dfrac{5\ln5}{4} + \dfrac{\sqrt{5}}{4}\,\ln\dfrac{3+\sqrt{5}}{2} - \dfrac{\pi}{2}\,\sqrt{1-\dfrac{2}{\sqrt{5}}}$

$H_\frac{3}{5} = \dfrac{5}{3} + H_{-\frac{2}{5}} = \dfrac{5}{3} - \dfrac{5\ln5}{4} + \dfrac{\sqrt{5}}{4}\,\ln\dfrac{3+\sqrt{5}}{2} + \dfrac{\pi}{2}\,\sqrt{1-\dfrac{2}{\sqrt{5}}}$

$H_\frac{4}{5} = \dfrac{5}{4} + H_{-\frac{1}{5}} = \dfrac{5}{4} - \dfrac{5\ln5}{4} - \dfrac{\sqrt{5}}{4}\,\ln\dfrac{3+\sqrt{5}}{2} + \dfrac{\pi}{2}\,\sqrt{1+\dfrac{2}{\sqrt{5}}}$

$H_\frac{1}{6} = 6 + H_{-\frac{5}{6}} = 6 - 2\ln2 - \dfrac{3\ln3}{2} - \dfrac{\pi\sqrt{3}}{2}$

$H_\frac{5}{6} = \dfrac{6}{5} + H_{-\frac{1}{6}} = \dfrac{6}{5} - 2\ln2 - \dfrac{3\ln3}{2} + \dfrac{\pi\sqrt{3}}{2}$

$H_\frac{1}{8} = 8 + H_{-\frac{7}{8}} = 8 - 4\ln2 - \dfrac{\ln(3+\sqrt{2})}{\sqrt{2}} - \dfrac{\pi}{2}\,(\sqrt{2}+1)$

$H_\frac{3}{8} = \dfrac{8}{3} + H_{-\frac{5}{8}} = \dfrac{8}{3} - 4\ln2 + \dfrac{\ln(3+\sqrt{2})}{\sqrt{2}} - \dfrac{\pi}{2}\,(\sqrt{2}-1)$

$H_\frac{5}{8} = \dfrac{8}{5} + H_{-\frac{3}{8}} = \dfrac{8}{5} - 4\ln2 + \dfrac{\ln(3+\sqrt{2})}{\sqrt{2}} + \dfrac{\pi}{2}\,(\sqrt{2}-1)$

$H_\frac{7}{8} = \dfrac{8}{7} + H_{-\frac{1}{8}} = \dfrac{8}{7} - 4\ln2 - \dfrac{\ln(3+\sqrt{2})}{\sqrt{2}} + \dfrac{\pi}{2}\,(\sqrt{2}+1)$

Generalized Harmonic Numbers

$\displaystyle H_{\frac{q}{p},r} = \zeta(r) - p^r \, \sum_{n\geq1} \dfrac{1}{(pn+q)^r} \qquad (r\in\mathbb{N})$

$\displaystyle H_{\frac{1}{2},r} = 2^r - (2^r-2)\zeta(r) \qquad (r\in\mathbb{N}\setminus\{1\})$

In particular, we have

$H_{\frac{1}{2},0} = 0$

$H_{\frac{1}{2},1} = H_{\frac{1}{2}} = 2 - 2\ln{2}$

$H_{\frac{1}{2},2} = 4 - 2\zeta(2) = 4 - \dfrac{\pi^2}{3}$

$H_{\frac{1}{2},3} = 8 - 6\zeta(3)$

$H_{\frac{1}{2},4} = 16 - 14\zeta(4) = 16 - \dfrac{7\pi^4}{45}$

$H_{\frac{1}{2},5} = 32 - 30\zeta(5)$

$H_{\frac{1}{2},6} = 64 - 62\zeta(6) = 64 - \dfrac{62\pi^6}{945}$

$H_{\frac{1}{2},7} = 128 - 126\zeta(7)$

$H_{\frac{1}{2},8} = 256 - 254\zeta(8) = 256 - \dfrac{127\pi^8}{4725}$

$H_{\frac{1}{2},9} = 512 - 510\zeta(9)$

$H_{\frac{1}{2},10} = 1024 - 1022\zeta(10) = 1024 - \dfrac{146\pi^{10}}{13365}$

Log Function

$\ln0^n = - n\gamma \qquad (n\in\mathbb{Z})$

In particular, $\ln0 = - \gamma$ and $\ln\dfrac{1}{0} = \gamma$ .

Trigonometric Functions

$\dfrac{\pi}{\sin(n\pi)} = \pi\,\text{csc}(n\pi) = \dfrac{\pi}{\cos((n\!-\!\frac{1}{2})\pi)} = \pi\sec((n-\frac{1}{2})\pi) = (-1)^{n+1}\,\gamma$

$\dfrac{1}{\sin(n\pi)} = \text{csc}(n\pi) = \dfrac{1}{\cos((n\!-\!\frac{1}{2})\pi)} = \sec((n-\frac{1}{2})\pi) = (-1)^{n+1}\,\dfrac{\gamma}{\pi}$

Gamma Function

$\text{Res}(\Gamma,-n) = \dfrac{1}{0^n} = \dfrac{(-1)^n}{n!} \qquad (n\in\mathbb{N})$

$\Gamma(-n) = (-1)^n \, \dfrac{\gamma}{\Gamma(n\!+\!1)} = (-1)^n \, \dfrac{\gamma}{n!} \qquad (n\in\mathbb{N})$

$\Gamma(n\!+\!1) \, \Gamma(-n) = -\dfrac{\pi}{\sin(n\pi)} = (-1)^n \, \gamma \qquad (n\in\mathbb{N})$

$\Gamma^\prime(-n) = \Gamma(-n) \, \psi^{(0)}(-n) = (-1)^n \, \dfrac{\gamma}{\Gamma(n\!+\!1)} \, \left(H_n \!-\! \gamma\right) = (-1)^n \, \dfrac{\gamma}{n!} \, \left(H_n \!-\! \gamma\right) \quad (n\in \nolinebreak \mathbb{N})$

In particular, $\Gamma(0) = \gamma$ and $\Gamma^\prime(0) = -\gamma^2$

$\dfrac{\Gamma(s\!+\!1)}{0^s} = (-1)^s \qquad (s\!>\!0)$

$\dfrac{\Gamma(s)}{0^s} = \dfrac{(-1)^s}{s} \qquad (s\!>\!0)$

Digamma Function

$\psi^{(0)}(n+1) = \psi^{(0)}(n) + \dfrac{1}{n} \qquad (n\in\mathbb{N})$

$\psi^{(0)}(n) = H_{n-1} - \gamma \qquad (n\in\mathbb{N})$

In particular, $\psi^{(0)}(1) = \psi^{(0)}(0) = -\gamma$ .

The reflection formula for the digamma function at the integers is

$\psi^{(0)}(-n) = H_n + \psi^{(0)}(0) = H_n - \gamma \qquad (n\in\mathbb{N})$

$\psi^{(0)}(1-n) = H_{n-1} + \psi^{(0)}(0) = H_{n-1} - \gamma \qquad (n\in\mathbb{N})$

Riemann and Hurwitz Zeta Functions (s > 0)

$\displaystyle \zeta(s,0) = \sum_{n\geq0} \dfrac{1}{n^s} = \dfrac{1}{0^s} + \sum_{n\geq1} \dfrac{1}{n^s} = \dfrac{(-1)^s}{\Gamma(s\!+\!1)} + \zeta(s) = \dfrac{(-1)^s}{s\Gamma(s)} + \nolinebreak \zeta(s)$

$\displaystyle \zeta(s,1) = \zeta(s)$

$\displaystyle \zeta\!\left(s,\frac{1}{2}\right) = (2^s\!-\!1) \, \zeta(s)$

$\displaystyle \zeta\!\left(s,\frac{1}{3}\right) + \zeta\!\left(s,\frac{2}{3}\right) = (3^s\!-\!1) \, \zeta(s)$

$\displaystyle \zeta\!\left(s,\frac{1}{4}\right) + \zeta\!\left(s,\frac{3}{4}\right) = 2^s \, (2^s\!-\!1) \, \zeta(s)$

$\displaystyle \zeta\!\left(s,\frac{1}{5}\right) + \zeta\!\left(s,\frac{4}{5}\right) + \zeta\!\left(s,\frac{2}{5}\right) + \zeta\!\left(s,\frac{3}{5}\right) = (5^s\!-\!1) \, \zeta(s)$

$\displaystyle \zeta\!\left(s,\frac{1}{6}\right) + \zeta\!\left(s,\frac{5}{6}\right) = (3^s\!-\!1) \, (2^s\!-\!1) \, \zeta(s)$

$\displaystyle \zeta\!\left(s,\frac{1}{7}\right) + \zeta\!\left(s,\frac{6}{7}\right) + \zeta\!\left(s,\frac{2}{7}\right) + \zeta\!\left(s,\frac{5}{7}\right) + \zeta\!\left(s,\frac{3}{7}\right) + \zeta\!\left(s,\frac{4}{7}\right) \\ \\ = (7^s\!-\!1) \, \zeta(s)$

$\displaystyle \zeta\!\left(s,\frac{1}{8}\right) + \zeta\!\left(s,\frac{7}{8}\right) + \zeta\!\left(s,\frac{3}{8}\right) + \zeta\!\left(s,\frac{5}{8}\right) \\ \\ = 2^s \, 2^s \, (2^s\!-\!1) \, \zeta(s) = 4^s \, (2^s\!-\!1) \, \zeta(s)$

$\displaystyle \zeta\!\left(s,\frac{1}{9}\right) + \zeta\!\left(s,\frac{8}{9}\right) + \zeta\!\left(s,\frac{2}{9}\right) + \zeta\!\left(s,\frac{7}{9}\right) + \zeta\!\left(s,\frac{4}{9}\right) + \zeta\!\left(s,\frac{5}{9}\right) \\ \\ = 3^s \, (3^s\!-\!1) \, \zeta(s)$

$\displaystyle \zeta\!\left(s,\frac{1}{10}\right) + \zeta\!\left(s,\frac{9}{10}\right) + \zeta\!\left(s,\frac{3}{10}\right) + \zeta\!\left(s,\frac{7}{10}\right) = (5^s\!-\!1) \, (2^s\!-\!1) \, \zeta(s)$

Dirichlet Eta Function

Pages: 1 2