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ArcCosh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCosh[z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself > In the whole plane





http://functions.wolfram.com/01.26.06.0065.01









  


  










Input Form





ArcCosh[z] == Subscript[F, Infinity][z] /; Subscript[F, n][z] == (Sqrt[z - 1]/Sqrt[1 - z]) (Pi/2 - Sum[(Pochhammer[1/2, k] z^(2 k + 1))/((1 + 2 k) k!), {k, 0, n}]) == ArcCosh[z] + (1/(2 Sqrt[Pi])) z^(3 + 2 n) Gamma[3/2 + n]^2 (Sqrt[z - 1]/Sqrt[1 - z]) HypergeometricPFQRegularized[ {1, 3/2 + n, 3/2 + n}, {2 + n, 5/2 + n}, z^2] && Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCosh", "[", "z", "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["z", "-", "1"]]], " "]], SqrtBox[RowBox[List["1", "-", "z"]]]], RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "/", "2"]], ",", "k"]], "]"]], SuperscriptBox["z", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], ")"]]]], "\[Equal]", RowBox[List[StyleBox[RowBox[List["ArcCosh", "[", "z", "]"]], Rule[FontWeight, "Plain"]], StyleBox["+", Rule[FontWeight, "Plain"]], RowBox[List[StyleBox[FractionBox["1", RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], Rule[FontWeight, "Plain"]], StyleBox[SuperscriptBox["z", RowBox[List["3", "+", RowBox[List["2", "n"]]]]], Rule[FontWeight, "Plain"]], StyleBox[" ", Rule[FontWeight, "Plain"]], StyleBox[SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]], "2"], Rule[FontWeight, "Plain"]], StyleBox[" ", Rule[FontWeight, "Plain"]], FractionBox[RowBox[List[SqrtBox[RowBox[List["z", "-", "1"]]], " "]], SqrtBox[RowBox[List["1", "-", "z"]]]], RowBox[List[StyleBox["HypergeometricPFQRegularized", Rule[FontWeight, "Plain"]], StyleBox["[", Rule[FontWeight, "Plain"]], RowBox[List[StyleBox[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]], Rule[FontWeight, "Plain"]], StyleBox[",", Rule[FontWeight, "Plain"]], StyleBox[RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]], Rule[FontWeight, "Plain"]], StyleBox[",", Rule[FontWeight, "Plain"]], SuperscriptBox[StyleBox["z", Rule[FontWeight, "Plain"]], "2"]]], StyleBox["]", Rule[FontWeight, "Plain"]]]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCosh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", FractionBox[RowBox[List[SqrtBox[RowBox[List["z", "-", "1"]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], ")"]]]], SqrtBox[RowBox[List["1", "-", "z"]]]], "\[Equal]", RowBox[List[RowBox[List["ArcCosh", "[", "z", "]"]], "+", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]], "2"], " ", SqrtBox[RowBox[List["z", "-", "1"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox["\[Pi]"]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02