# 함수값의 계산

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$\cos {\frac{2\pi}{3}} = -\frac{1}{2}$

$\cos\frac{2\pi}{5}=\frac{\sqrt5 -1}{4}$$z^4+z^3+z^2+z^1+1=0$ 복소평면상에서 $$z$$ 의 $$x$$ 좌표는 $$\frac{-1+\sqrt{5}}{4} , \frac{-1-\sqrt{5}}{4}$$ 로 주어짐.

$\cos \frac{2\pi}{17}= \frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \sqrt{68+12\sqrt{17}-4{\sqrt{170+38\sqrt{17}}}} }{16}$ $16\cos{2\pi\over17} = -1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ 2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}$

$j(\sqrt{-1})=1728$$j(\frac {-1+\sqrt{-3}}{2})=0$ $j(\frac {-1+\sqrt{-43}} {2})=-884736744$ $j(\frac {-1+\sqrt{-67}} {2})=147197952744$$j(\frac {-1+\sqrt{-163}} {2})=-262537412640768744$

$\zeta(2n) =(-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}, n \ge 1$여기서 $$B_{2n}$$은 베르누이 수. $\zeta(-n)=-\frac{B_{n+1}}{n+1}, n \ge 1$$\zeta(0)=-\frac{1}{2}$

$\beta(0)= \frac{1}{2}, \beta(1)\;=\;\tan^{-1}(1)\;=\;\frac{\pi}{4}, \beta(3)\;=\;\frac{\pi^3}{32}, \beta(5)\;=\;\frac{5\pi^5}{1536}, \beta(7)\;=\;\frac{61\pi^7}{184320}$

$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$

$\psi(1) = -\gamma\,\!$ $\psi\left(\frac{1}{2}\right) = -2\ln{2} - \gamma$ $\psi\left(\frac{1}{3}\right) = -\frac{\pi}{2\sqrt{3}} -\frac{3}{2}\ln{3} - \gamma$$\psi\left(\frac{1}{4}\right) = -\frac{\pi}{2} - 3\ln{2} - \gamma$$\psi\left(\frac{1}{6}\right) = -\frac{\pi}{2}\sqrt{3} -2\ln{2} -\frac{3}{2}\ln(3) - \gamma$ $\psi\left(\frac{1}{8}\right) = -\frac{\pi}{2} - 4\ln{2} - \frac{1}{\sqrt{2}} \left\{\pi + \ln(2 + \sqrt{2}) - \ln(2 - \sqrt{2})\right\} - \gamma$

$\mbox{Li}_{2}(0)=0$ $\mbox{Li}_{2}(1)=\frac{\pi^2}{6}$ $\mbox{Li}_{2}(-1)=-\frac{\pi^2}{12}$ $\mbox{Li}_{2}(\frac{1}{2})=\frac{\pi^2}{12}-\frac{1}{2}\log^2(2)$ $\mbox{Li}_{2}(\frac{3-\sqrt{5}}{2})=\frac{\pi^2}{15}-\log^2(\frac{1+\sqrt{5}}{2})$ $\mbox{Li}_{2}(\frac{-1+\sqrt{5}}{2})=\frac{\pi^2}{10}-\log^2(\frac{1+\sqrt{5}}{2})$ $\mbox{Li}_{2}(\frac{1-\sqrt{5}}{2})=-\frac{\pi^2}{15}+\frac{1}{2}\log^2(\frac{1+\sqrt{5}}{2})$ $\mbox{Li}_{2}(\frac{-1-\sqrt{5}}{2})=-\frac{\pi^2}{10}+\frac{1}{2}\log^2(\frac{1+\sqrt{5}}{2})$