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 HermiteH

 http://functions.wolfram.com/07.01.06.0027.01

 Input Form

 HermiteH[\[Nu], z] \[Proportional] Piecewise[ {{2^\[Nu] z^\[Nu] (Sum[((-1)^k Pochhammer[-(\[Nu]/2), k] Pochhammer[(1 - \[Nu])/2, k])/(z^(2 k) k!), {k, 0, n}] + O[z^(-2 n - 2)]), Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]}, {2^\[Nu] z^\[Nu] (Sum[((-1)^k Pochhammer[-(\[Nu]/2), k] Pochhammer[(1 - \[Nu])/2, k])/(z^(2 k) k!), {k, 0, n}] + O[z^(-2 n - 2)]) - ((E^(z^2 + I Pi \[Nu]) Sqrt[Pi] z^(-1 - \[Nu]))/ Gamma[-\[Nu]]) (Sum[(Pochhammer[(1 + \[Nu])/2, k] Pochhammer[(2 + \[Nu])/2, k])/(z^(2 k) k!), {k, 0, n}] + O[z^(-2 n - 2)]), Arg[z] > Pi/2}}, 2^\[Nu] z^\[Nu] (Sum[((-1)^k Pochhammer[-(\[Nu]/2), k] Pochhammer[(1 - \[Nu])/2, k])/(z^(2 k) k!), {k, 0, n}] + O[z^(-2 n - 2)]) - ((E^(z^2 - I Pi \[Nu]) Sqrt[Pi] z^(-1 - \[Nu]))/ Gamma[-\[Nu]]) (Sum[(Pochhammer[(1 + \[Nu])/2, k] Pochhammer[(2 + \[Nu])/2, k])/(z^(2 k) k!), {k, 0, n}] + O[z^(-2 n - 2)])] /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", "\n", " ", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "n"]], "-", "2"]]], "]"]]]], ")"]]]], ",", " ", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]], "}"]], ",", " ", "\n", " ", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "n"]], "-", "2"]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["z", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "n"]], "-", "2"]]], "]"]]]], ")"]]]]]], ",", " ", "\n", " ", RowBox[List[RowBox[List["Arg", "[", "z", "]"]], ">", FractionBox["\[Pi]", "2"]]]]], "}"]]]], "}"]], ",", " ", "\n", " ", RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "n"]], "-", "2"]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["z", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "n"]], "-", "2"]]], "]"]]]], ")"]]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 H ν ( z ) 2 ν z ν ( k = 0 n ( - 1 ) k ( - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ")"]], "k"], Pochhammer] ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] z - 2 k k ! + O ( z - 2 n - 2 ) ) - π 2 < arg ( z ) π 2 2 ν z ν ( k = 0 n ( - 1 ) k ( - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ")"]], "k"], Pochhammer] ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] z - 2 k k ! + O ( z - 2 n - 2 ) ) - z 2 + π ν π z - ν - 1 Γ ( - ν ) ( k = 0 n ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], ")"]], "k"], Pochhammer] ( ν + 2 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], ")"]], "k"], Pochhammer] z - 2 k k ! + O ( z - 2 n - 2 ) ) arg ( z ) > π 2 2 ν z ν ( k = 0 n ( - 1 ) k ( - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ")"]], "k"], Pochhammer] ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] z - 2 k k ! + O ( z - 2 n - 2 ) ) - z 2 - π ν π z - ν - 1 Γ ( - ν ) ( k = 0 n ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], ")"]], "k"], Pochhammer] ( ν + 2 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], ")"]], "k"], Pochhammer] z - 2 k k ! + O ( z - 2 n - 2 ) ) True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional HermiteH ν z 2 ν z ν k 0 n -1 k Pochhammer -1 ν 2 -1 k Pochhammer 1 -1 ν 2 -1 k z -2 k k -1 O z -2 n -2 Inequality -1 2 -1 z 2 -1 2 ν z ν k 0 n -1 k Pochhammer -1 ν 2 -1 k Pochhammer 1 -1 ν 2 -1 k z -2 k k -1 O z -2 n -2 -1 z 2 ν 1 2 z -1 ν -1 Gamma -1 ν -1 k 0 n Pochhammer ν 1 2 -1 k Pochhammer ν 2 2 -1 k z -2 k k -1 O z -2 n -2 z 2 -1 2 ν z ν k 0 n -1 k Pochhammer -1 ν 2 -1 k Pochhammer 1 -1 ν 2 -1 k z -2 k k -1 O z -2 n -2 -1 z 2 -1 ν 1 2 z -1 ν -1 Gamma -1 ν -1 k 0 n Pochhammer ν 1 2 -1 k Pochhammer ν 2 2 -1 k z -2 k k -1 O z -2 n -2 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List["k", "!"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", "2"]]]]], ")"]]]], RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]], List[RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List["k", "!"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", "2"]]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["z", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List["k", "!"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", "2"]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], RowBox[List[RowBox[List["Arg", "[", "z", "]"]], ">", FractionBox["\[Pi]", "2"]]]], List[RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List["k", "!"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", "2"]]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["z", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List["k", "!"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", "2"]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2007-05-02