{ "definitions" : { "x" : [ { "Q12916" : "real number", "Q50701" : "variable" } ], "s" : [ { "Q11567" : "complex number", "Q50701" : "variable" } ] }, "constraints" : [ ], "math_inputtex" : "\\[F\\left(x,s\\right)=\\frac{\\Gamma\\left(1-s\\right)}{(2\\pi)^{1-s}}\\*\\left(e^{\\pi i(%\n1-s)/2}\\zeta\\left(1-s,x\\right)+e^{\\pi i(s-1)/2}\\zeta\\left(1-s,1-x\\right)\\right),\\]", "math_inputtex_semantic" : "\\wf{Q196822}{\\PeriodicZeta}@{\\w{Q12916}{x}}{\\w{Q11567}{s}} = \\frac{\\wf{Q190573}{\\EulerGamma}@{1-s}}{(2 \\cpi)^{1-s}} \\* \\left( \\expe^{\\cpi \\iunit \\ifrac{(1-s)}{2}} \\wf{Q196822}{\\HurwitzZeta}@{1-s}{x} + \\expe^{\\cpi \\iunit \\ifrac{(s-1)}{2}} \\HurwitzZeta@{1-s}{1-x} \\right)", "correct_tex" : "F\\left(x,s\\right)=\\frac{\\Gamma\\left(1-s\\right)}{(2\\pi)^{1-s}} \\left(e^{\\pi i(1-s)/2}\\zeta\\left(1-s,x\\right)+e^{\\pi i(s-1)/2}\\zeta\\left(1-s,1-x\\right)\\right)", "correct_mml" : "\n \n \n \n \n F\n \n ⁡\n \n (\n x\n ,\n s\n )\n \n \n =\n \n \n \n \n Γ\n \n ⁡\n \n (\n \n 1\n -\n s\n \n )\n \n \n \n \n (\n \n 2\n ⁢\n π\n \n )\n \n \n 1\n -\n s\n \n \n \n ⁢\n \n (\n \n \n \n e\n \n π\n ⁢\n i\n ⁢\n \n \n (\n \n 1\n -\n s\n \n )\n \n /\n 2\n \n \n \n ⁢\n \n \n ζ\n \n ⁡\n \n (\n \n 1\n -\n s\n \n ,\n x\n )\n \n \n \n +\n \n \n e\n \n π\n ⁢\n i\n ⁢\n \n \n (\n \n s\n -\n 1\n \n )\n \n /\n 2\n \n \n \n ⁢\n \n \n ζ\n \n ⁡\n \n (\n \n 1\n -\n s\n \n ,\n \n 1\n -\n x\n \n )\n \n \n \n \n )\n \n \n \n \n \n \n \n periodic-zeta\n Q12916\n Q11567\n \n \n \n \n \n \n Euler-Gamma\n \n \n 1\n Q11567\n \n \n \n \n \n \n 2\n \n \n \n \n 1\n Q11567\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 1\n Q11567\n \n 2\n \n \n \n \n Hurwitz-zeta\n \n \n 1\n Q11567\n \n Q12916\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Q11567\n 1\n \n 2\n \n \n \n \n Hurwitz-zeta\n \n \n 1\n Q11567\n \n \n \n 1\n Q12916\n \n \n \n \n \n \n \n \\PeriodicZeta@{x}{s}=\\frac{\\EulerGamma@{1-s}}{(2\\pi)^{1-s}}\\*\\left(e^{\\pi i%\n\\ifrac{(1-s)}{2}}\\HurwitzZeta@{1-s}{x}+e^{\\pi i\\ifrac{(s-1)}{2}}\\HurwitzZeta@{%\n1-s}{1-x}\\right)\n \n", "uri" : "http://dlmf.nist.gov/25.13#E2", "title" : "DLMF:_25.13_Periodic_Zeta_Function", "type" : "definition" }