Die Transformation der elliptischen Funktionen

Zusammenfassung

Wenn

EquationSource$ % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaara % WaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaeqySdeMaam4DamaaBaaa % leaacaaIYaaabeaakiabgUcaRiabek7aIjaadEhadaWgaaWcbaGaaG % ymaaqabaGccaGGSaGaaGzbVlqadEhagaqeamaaBaaaleaacaaIXaaa % beaakiabg2da9iabeo7aNjaadEhadaWgaaWcbaGaaGOmaaqabaGccq % GHRaWkcqaH0oazcaWG3bWaaSbaaSqaaiaaigdaaeqaaaaa!4E43! \[{\bar w_2} = \alpha {w_2} + \beta {w_1},\quad {\bar w_1} = \gamma {w_2} + \delta {w_1}\]$$

((1))

gesetzt wird, wo α, β, γ, δ ganze Zahlen der Determinante αδ - βγ = 1 bezeichnen, so sagt man, (ω̄₂, ω̄₁) gehe aus (ω₂, ω₁) durch lineare Transformation hervor. Wir beschäftigen uns hier zunächst mit der Frage, wie sich die Weierstraßschen Funktionen bei linearer Transformation der Perioden verhalten. Es war

EquationSource$ % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aae % WaaeaacaWG1bGaai4laiaadEhadaWgaaWcbaGaaGymaaqabaGccaGG % SaGaam4DamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiabg2 % da9iaadwhacaWGjbGabmysayaafaWaaiWaaeaadaqadaqaaiaaigda % cqGHsisldaWcaaqaaiaadwhaaeaacaWG3baaaaGaayjkaiaawMcaai % aadwgadaahaaWcbeqaamaalaaabaGaamyDaaqaaiaadEhaaaGaey4k % aSYaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGaam % yDaaqaaiaadEhaaaaacaGLOaGaayzkaaWaaWbaaWqabeaacaaIYaaa % aaaaaOGaay5Eaiaaw2haamaabmaabaGaam4Daiabg2da9iaad2gada % WgaaWcbaGaaGymaaqabaGccaWG3bWaaSbaaSqaaiaaigdaaeqaaOGa % ey4kaSIaamyBamaaBaaaleaacaaIYaaabeaakiaadEhadaWgaaWcba % GaaGOmaaqabaaakiaawIcacaGLPaaaaaa!609E! \[\sigma \left( {u/{w_1},{w_2}} \right) = uII'\left\{ {\left( {1 - \frac{u} {w}} \right){e^{\frac{u} {w} + \frac{1} {2}{{\left( {\frac{u} {w}} \right)}^2}}}} \right\}\left( {w = {m_1}{w_1} + {m_2}{w_2}} \right)\] $$