{ "definitions" : { "m" : [ { "Q28920052" : "non-negative number", "Q1413083" : "parameter" } ], "n" : [ { "Q28920052" : "non-negative number", "Q1413083" : "parameter" } ], "\\xi" : [ { "Q3176558" : "positive real number", "Q50701" : "variable" } ] }, "constraints" : [ ], "math_inputtex" : "\\[P^{m}_{n-\\frac{1}{2}}\\left(\\cosh\\xi\\right)=\\frac{\\Gamma\\left(n+m+\\tfrac{1}{2}%\n\\right)}{\\Gamma\\left(n-m+\\tfrac{1}{2}\\right)}\\*\\left(\\frac{2}{\\pi\\sinh\\xi}%\n\\right)^{1/2}\\boldsymbol{Q}^{n}_{m-\\frac{1}{2}}\\left(\\coth\\xi\\right),\\]", "math_inputtex_semantic" : "\\wf{Q215405}{\\LegendreP[\\w{Q28920052}{m}]{\\w{Q28920052}{n}-\\frac{1}{2}}}@{\\cosh@@{\\w{Q3176558}{\\xi}}} = \\frac{\\wf{Q190573}{\\EulerGamma}@{n+m+\\tfrac{1}{2}}} {\\EulerGamma@{n-m+\\tfrac{1}{2}}} \\* \\left( \\frac{2}{\\cpi \\sinh@@{\\xi}} \\right)^{\\ifrac{1}{2}} \\wf{Q215405}{\\LegendreBlackQ[n]{m-\\frac{1}{2}}}@{\\coth@@{\\xi}}", "correct_tex" : "P^{m}_{n-\\frac{1}{2}}\\left(\\cosh\\xi\\right)=\\frac{\\Gamma\\left(n+m+\\tfrac{1}{2}\\right)}{\\Gamma\\left(n-m+\\tfrac{1}{2}\\right)} \\left(\\frac{2}{\\pi\\sinh\\xi}\\right)^{1/2}\\boldsymbol{Q}^{n}_{m-\\frac{1}{2}}\\left(\\coth\\xi\\right)", "correct_mml" : "\n \n \n \n \n \n P\n \n n\n -\n \n 1\n 2\n \n \n m\n \n \n ⁡\n \n (\n \n cosh\n ⁡\n ξ\n \n )\n \n \n =\n \n \n \n \n Γ\n \n ⁡\n \n (\n \n n\n +\n m\n +\n \n \n 1\n 2\n \n \n \n )\n \n \n \n \n Γ\n \n ⁡\n \n (\n \n \n n\n -\n m\n \n +\n \n \n 1\n 2\n \n \n \n )\n \n \n \n ⁢\n \n \n (\n \n 2\n \n π\n ⁢\n \n sinh\n ⁡\n ξ\n \n \n \n )\n \n \n 1\n /\n 2\n \n \n ⁢\n \n \n \n \uD835\uDC78\n \n m\n -\n \n 1\n 2\n \n \n n\n \n \n ⁡\n \n (\n \n coth\n ⁡\n ξ\n \n )\n \n \n \n \n \n \n \n \n \n Legendre-P-first-kind\n Q28920052\n \n \n Q28920052\n \n \n 1\n 2\n \n \n \n \n \n Q3176558\n \n \n \n \n \n \n \n Euler-Gamma\n \n \n Q28920052\n Q28920052\n \n \n 1\n 2\n \n \n \n \n Euler-Gamma\n \n \n \n \n Q28920052\n Q28920052\n \n \n \n 1\n 2\n \n \n \n \n \n \n \n \n 2\n \n \n \n \n \n Q3176558\n \n \n \n \n \n 1\n 2\n \n \n \n \n associated-Legendre-black-Q\n Q28920052\n \n \n Q28920052\n \n \n 1\n 2\n \n \n \n \n hyperbolic-cotangent\n Q3176558\n \n \n \n \n \n \\LegendreP[m]{n-\\frac{1}{2}}@{\\cosh@{\\xi}}=\\frac{\\EulerGamma@{n+m+\\tfrac{1}{2}%\n}}{\\EulerGamma@{n-m+\\tfrac{1}{2}}}\\*\\left(\\frac{2}{\\pi\\sinh@{\\xi}}\\right)^{%\n\\ifrac{1}{2}}\\LegendreBlackQ[n]{m-\\frac{1}{2}}@{\\coth@{\\xi}}\n \n", "uri" : "http://dlmf.nist.gov/14.19#E7", "title" : "DLMF:_14.19_Toroidal_(or_Ring)_Functions", "type" : "definition", "substitutions" : { } }