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ArcCot






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCot[z] > Series representations > Generalized power series > Expansions at z==i > For the function itself





http://functions.wolfram.com/01.16.06.0046.01









  


  










Input Form





ArcCot[z] == Subscript[F, Infinity][z] /; Subscript[F, n][z] == -((I Log[2])/2) + (I/2) Log[(-I) (z - I)] - ((z - I)/4) Sum[((I/2)^k (z - I)^k)/(1 + k), {k, 0, n}] == ((2^(-n - 3) I^(n + 1) (z - I)^(2 + n))/(n + 2)) Hypergeometric2F1[1, 2 + n, 3 + n, (I (z - I))/2] + (I/2) Log[(-I) (z - I)] - (I/2) Log[(-I) (z + I)] && Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCot", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Log", "[", "2", "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["\[ImaginaryI]", "2"], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]], "k"]]], RowBox[List["1", "+", "k"]]]]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "-", "3"]]], " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "+", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]], RowBox[List["2", "+", "n"]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["3", "+", "n"]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]]]]]], "]"]]]], RowBox[List["n", "+", "2"]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]]]], "]"]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02