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http://functions.wolfram.com/01.13.27.0657.01
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ArcCos[1/Sqrt[a z^c + 1]] == Pi/2 - ((Sqrt[a] z^(c/2))/Sqrt[a z^c])
ArcTan[1/(Sqrt[a] z^(c/2))] +
((I Pi)/2) (Sqrt[(-I) Sqrt[a] Sqrt[z^c] - 1]/
Sqrt[I Sqrt[a] Sqrt[z^c] + 1] + Sqrt[I Sqrt[a] Sqrt[z^c] - 1]/
Sqrt[1 - I Sqrt[a] Sqrt[z^c]])
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Cell[BoxData[RowBox[List[RowBox[List["ArcCos", "[", FractionBox["1", SqrtBox[RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "c"]]], "+", "1"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List[FractionBox[RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], SqrtBox[RowBox[List["a", " ", SuperscriptBox["z", "c"]]]]], RowBox[List["ArcTan", "[", FractionBox[StyleBox["1", "Program"], RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "+", "1"]]]], "+", FractionBox[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]]]]]]]], ")"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> c </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <msqrt> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> c </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> <mo> ⁢ </mo> <msqrt> <msup> <mi> z </mi> <mi> c </mi> </msup> </msqrt> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> <mo> ⁢ </mo> <msqrt> <msup> <mi> z </mi> <mi> c </mi> </msup> </msqrt> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> + </mo> <mfrac> <msqrt> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> <mo> ⁢ </mo> <msqrt> <msup> <mi> z </mi> <mi> c </mi> </msup> </msqrt> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> <mo> ⁢ </mo> <msqrt> <msup> <mi> z </mi> <mi> c </mi> </msup> </msqrt> </mrow> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccos /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctan /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCos", "[", FractionBox["1", SqrtBox[RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "c_"]]], "+", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["\[Pi]", "2"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox["1", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]]], "]"]]]], SqrtBox[RowBox[List["a", " ", SuperscriptBox["z", "c"]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "+", "1"]]]], "+", FractionBox[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]]]]]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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