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Definitions

\operatorname{pg }\,\,x =\frac 1\pi \psi\left(\frac x\pi\right)


\operatorname{cpg }\,\,x=-\frac 1\pi \psi\left(1-\frac x\pi\right)


\operatorname{gn}\,\,x=\frac 1\sqrt{2\pi}\Gamma\left(\frac x\pi\right)


\operatorname{cgn}\,\,x=\frac 1\sqrt{2\pi}\Gamma\left(\frac {\pi-x}\pi\right)


\operatorname{lgn}\,\,x=\ln \left(\frac 1\sqrt{2\pi}\Gamma\left(\frac x\pi\right)\right)


\operatorname{clgn}\,\,x=\ln \left(\frac 1\sqrt{2\pi}\Gamma\left(1-\frac x\pi\right)\right)


\operatorname{zn}\,\,x=\frac{2x}\pi \left(\frac {2x}\pi-1\right)\zeta\left(\frac{2x}{\pi}\right)

Properties

\operatorname{pg}\,\, x + \operatorname{cpg}\,\, x = - \cot x


\operatorname{pg}\,\, (-x) + \operatorname{cpg}\,\, x = \frac 1x


\operatorname{gn}\,\, x\, \operatorname{cgn}\,\,x\, = \frac12\csc x


(\operatorname{gn}\,\, x)'=\operatorname{gn}\,\, x\, \operatorname{pg}\,\, x


(\operatorname{lgn}\,\, x)'=\operatorname{pg}\,\, x


(\operatorname{clgn}\,\, x)'=\operatorname{cpg}\,\, x

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