Sin[z/2] == Sqrt[(1 - Cos[z])/2] /; 0 < Re[z] < 2 Pi || (Re[z] == 0 && Im[z] >= 0) || (Re[z] == 2 Pi && Im[z] <= 0)

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sin", "[", FractionBox["z", "2"], "]"]], "\[Equal]", " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["Cos", "[", "z", "]"]]]], "2"]]]], " ", "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Re", "[", "z", "]"]], "<", RowBox[List["2", "\[Pi]"]]]], "\[Or]", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[GreaterEqual]", "0"]]]], "||", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[Equal]", RowBox[List["2", " ", "\[Pi]"]]]], "&&", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[LessEqual]", "0"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> </msqrt> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8805; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sin /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <or /> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <real /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <geq /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <real /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <leq /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sin", "[", FractionBox["z_", "2"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Cos", "[", "z", "]"]]]], ")"]]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Re", "[", "z", "]"]], "<", RowBox[List["2", " ", "\[Pi]"]]]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[GreaterEqual]", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[Equal]", RowBox[List["2", " ", "\[Pi]"]]]], "&&", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[LessEqual]", "0"]]]], ")"]]]]]]]]]]

2001-10-29