Spezielle Weierstraßsche Sigma-Funktionen

Zusammenfassung

Für die zu den sechs speziellen Weierstrassschen Zeta-Funktionen gehörigen Sigma-Funk-tionen lauten die Definitionsgleichungen

$$ \left. \begin{array}{l} \ln {\sigma _1}(z,\;k) = \ln z + \int\limits_0^z {({\delta _1}(\bar z,k) - \frac{1}{2})d\bar z,\quad \frac{\partial }{{\partial z}}\ln {\sigma _1} = {\delta _1},\quad \frac{{{\partial ^2}}}{{\partial {z^2}}}\ln {\sigma _1} = - {\wp _1},} \\ \ln {\sigma _2}(z,k) = \quad \int\limits_0^z {({\delta _2}(\bar z,k) + {n_1}K)d\bar z,\quad \frac{\partial }{{\partial z}}\ln {\sigma _2} = {\delta _2} + {n_1}K,\quad \frac{{{\partial ^2}}}{{\partial {z^2}}}\ln {\sigma _2} = - {\wp _2},} \\ \ln {\sigma _3}(z,k) = \quad \int\limits_0^z {{\delta _3}(\bar z,k)d\bar z,\quad \frac{\partial }{{\partial z}}\ln {\sigma _3} = {\delta _3},\quad \frac{{{\partial ^2}}}{{\partial {z^2}}}\ln {\sigma _3} = - {\wp _3},} \\ \ln {\sigma _4}(z,k) = \quad \int\limits_0^z {({\delta _4}(\bar z,k) + {n_1}K)d\bar z,\quad \frac{\partial }{{\partial z}}\ln {\sigma _4} = {\delta _4} + {n_1}K,\quad \frac{{{\partial ^2}}}{{\partial {z^2}}}\ln {\sigma _4} = - {\wp _4},} \\ \ln {\sigma _5}(z,k) = \quad \ln z + \int\limits_0^z {({\delta _5}(\bar z,k) + \frac{1}{{\bar z}})d\bar z,\quad \frac{\partial }{{\partial z}}\ln {\sigma _5} = {\delta _5},\quad \frac{{{\partial ^2}}}{{\partial {z^2}}}\ln {\sigma _5} = - {\wp _5},} \\ \ln {\sigma _6}(z,k) = \quad \int\limits_0^z {({\delta _6}(\bar z,k) + {{\bar n}_1}K)d\bar z,\quad \frac{\partial }{{\partial z}}\ln {\sigma _6} = {\delta _6} + {{\bar n}_1}K,\quad \frac{{{\partial ^2}}}{{\partial {z^2}}}\ln {\sigma _6} = - {\wp _6}.} \\ \end{array} \right\}(1063) $$

((1063))