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variants of this functions
ArcTan






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.14.06.0052.01









  


  










Input Form





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










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTan", "[", "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[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["(", RowBox[List["-", FractionBox["\[ImaginaryI]", "2"]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]], "k"]]], RowBox[List["k", "+", "1"]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ArcTan", "[", "z", "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], RowBox[List["n", "+", "1"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]], RowBox[List["n", "+", "2"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["2", "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List["3", "+", "n"]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", "z"]], ")"]]]]]], "]"]]]], RowBox[List["2", "+", "n"]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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