html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 EllipticE

 http://functions.wolfram.com/08.04.06.0073.01

 Input Form

 EllipticE[z, m] == (-1)^Round[Re[z]/Pi] (Sqrt[(-m) Sin[z]^2]/(4 m Sin[z])) (2 m Sin[z]^2 Sqrt[1 - Csc[z]^2/m] - 2 Log[1 + Sqrt[1 - Csc[z]^2/m]] - Log[(-m) Sin[z]^2] + Sum[(Pochhammer[1/2, k]/(m^k k!)) ((Pochhammer[3/2, k]/(2 m Sin[z]^2)) HypergeometricPFQRegularized[ {1, 1, 3/2 + k}, {3 + k, 2}, Csc[z]^2/m] - (Pochhammer[1/2, k]/(1 + k)!) (Log[-4 m Sin[z]^2] + 1/(1 + k) - 1/k - Sum[2/(i + k + 1), {i, 0, k - 2}]) + 2 m Sin[z]^2 Sum[(Pochhammer[-(1/2), -i + k] m^i Sin[z]^(2 i))/ ((1 + i) (-i + k)!), {i, 0, k}]), {k, 1, Infinity}]) + 2 Round[Re[z]/Pi] EllipticE[m]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], RowBox[List["4", " ", "m", " ", RowBox[List["Sin", "[", "z", "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], "m"]]]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], "m"]]]]]], "]"]]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], SuperscriptBox["m", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]], " ", RowBox[List["(", " ", RowBox[List[RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], RowBox[List["2", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["3", "2"], "+", "k"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["3", "+", "k"]], ",", "2"]], "}"]], ",", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], "m"]]], "]"]]]], " ", "-", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]], " "]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], "+", FractionBox["1", RowBox[List["1", "+", "k"]]], "-", FractionBox["1", "k"], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "2"]]], FractionBox["2", RowBox[List["i", "+", "k", "+", "1"]]]]]]], ")"]]]], "+", " ", RowBox[List["2", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "i"]], "+", "k"]]]], "]"]], " ", SuperscriptBox["m", "i"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["2", " ", "i"]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "i"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "i"]], "+", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]]]]]]]]

 MathML Form

 E ( z m ) 2 Re ( z ) π E ( m ) + ( - 1 ) Re ( z ) π - m sin 2 ( z ) 4 m sin ( z ) ( 2 m sin 2 ( z ) 1 - csc 2 ( z ) m - log ( - m sin 2 ( z ) ) - 2 log ( 1 - csc 2 ( z ) m + 1 ) + k = 1 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] m - k k ! ( 2 m sin 2 ( z ) i = 0 k ( - 1 2 ) k - i TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "i"]]], Pochhammer] m i sin 2 i ( z ) ( i + 1 ) ( k - i ) ! - ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( k + 1 ) ! ( log ( - 4 m sin 2 ( z ) ) - i = 0 k - 2 2 i + k + 1 - 1 k + 1 k + 1 ) + ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] 2 m sin 2 ( z ) 3 F ~ 2 ( 1 , 1 , k + 3 2 ; k + 3 , 2 ; csc 2 ( z ) m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", FractionBox["3", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", "3"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SuperscriptBox["csc", "2"], "(", "z", ")"]], "m"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) ) E ( z m ) 2 Re ( z ) π E ( m ) + ( - 1 ) Re ( z ) π - m sin 2 ( z ) 4 m sin ( z ) ( 2 m sin 2 ( z ) 1 - csc 2 ( z ) m - log ( - m sin 2 ( z ) ) - 2 log ( 1 - csc 2 ( z ) m + 1 ) + k = 1 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] m - k k ! ( 2 m sin 2 ( z ) i = 0 k ( - 1 2 ) k - i TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "i"]]], Pochhammer] m i sin 2 i ( z ) ( i + 1 ) ( k - i ) ! - ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( k + 1 ) ! ( log ( - 4 m sin 2 ( z ) ) - i = 0 k - 2 2 i + k + 1 - 1 k + 1 k + 1 ) + ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] 2 m sin 2 ( z ) 3 F ~ 2 ( 1 , 1 , k + 3 2 ; k + 3 , 2 ; csc 2 ( z ) m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", FractionBox["3", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", "3"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SuperscriptBox["csc", "2"], "(", "z", ")"]], "m"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) ) [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], "m"]]]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], "m"]]]]]], "]"]]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["m", RowBox[List["-", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["3", "2"], "+", "k"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["3", "+", "k"]], ",", "2"]], "}"]], ",", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], "m"]]], "]"]]]], RowBox[List["2", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], "-", FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], "+", FractionBox["1", RowBox[List["1", "+", "k"]]], "-", FractionBox["1", "k"], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "2"]]], FractionBox["2", RowBox[List["i", "+", "k", "+", "1"]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]], "+", RowBox[List["2", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "i"]], "+", "k"]]]], "]"]], " ", SuperscriptBox["m", "i"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["2", " ", "i"]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "i"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "i"]], "+", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]], RowBox[List["k", "!"]]]]]]], ")"]]]], RowBox[List["4", " ", "m", " ", RowBox[List["Sin", "[", "z", "]"]]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02