html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 StruveH

 http://functions.wolfram.com/03.09.17.0017.01

 Input Form

 StruveH[\[Nu], z] == (2^(-1 + n) Pochhammer[1 - \[Nu], -1 + n] (z HypergeometricPFQ[{1, (1 - n)/2, 1 - n/2}, {1, 1 - n, 1 - \[Nu], 1 - n + \[Nu]}, -z^2] StruveH[\[Nu] - n - 1, z] + 2 (n - \[Nu]) HypergeometricPFQ[{1, (1 - n)/2, -(n/2)}, {1, -n, 1 - \[Nu], -n + \[Nu]}, -z^2] StruveH[\[Nu] - n, z]))/ (-z)^n + ((2^(1 - \[Nu]) z^(-1 + \[Nu]))/Sqrt[Pi]) Sum[((4^j Pochhammer[1 - \[Nu], j])/((-z^2)^j Gamma[1/2 - j + \[Nu]])) HypergeometricPFQ[{1, (1 - j)/2, -(j/2)}, {1, -j, 1 - \[Nu], -j + \[Nu]}, -z^2], {j, 0, -1 + n}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["1", "-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "n"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "-", "n", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n", "-", "1"]], ",", "z"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "-", "\[Nu]"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "n"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "n"]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " "]], SqrtBox["\[Pi]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "j"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "j", "+", "\[Nu]"]], "]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "j"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "j"]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]]]]]], " ", "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 H StruveH ν ( z ) 2 n - 1 ( 1 - ν ) n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], RowBox[List["n", "-", "1"]]], Pochhammer] ( - z ) - n ( z 3 F 4 ( 1 , 1 - n 2 , 1 - n 2 ; 1 , 1 - n , 1 - ν , ν - n + 1 ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "n"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["n", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "-", "n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] H StruveH - n + ν - 1 ( z ) + 2 ( n - ν ) 3 F 4 ( 1 , 1 - n 2 , - n 2 ; 1 , - n , 1 - ν , ν - n ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "n"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["n", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "-", "n"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] H StruveH ν - n ( z ) ) + 2 1 - ν z ν - 1 π j = 0 n - 1 4 j ( - z 2 ) - j ( 1 - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] Γ ( ν - j + 1 2 ) 3 F 4 ( 1 , 1 - j 2 , - j 2 ; 1 , - j , 1 - ν , ν - j ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "j"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["j", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "j"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "-", "j"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition StruveH ν z 2 n -1 Pochhammer 1 -1 ν n -1 -1 z -1 n z HypergeometricPFQ 1 1 -1 n 2 -1 1 -1 n 2 -1 1 1 -1 n 1 -1 ν ν -1 n 1 -1 z 2 StruveH -1 n ν -1 z 2 n -1 ν HypergeometricPFQ 1 1 -1 n 2 -1 -1 n 2 -1 1 -1 n 1 -1 ν ν -1 n -1 z 2 StruveH ν -1 n z 2 1 -1 ν z ν -1 1 2 -1 j 0 n -1 4 j -1 z 2 -1 j Pochhammer 1 -1 ν j Gamma ν -1 j 1 2 -1 HypergeometricPFQ 1 1 -1 j 2 -1 -1 j 2 -1 1 -1 j 1 -1 ν ν -1 j -1 z 2 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StruveH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["1", "-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "n"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "-", "n", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n", "-", "1"]], ",", "z"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "-", "\[Nu]"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "n"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "n"]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "z"]], "]"]]]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "j"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "j"]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "j", "+", "\[Nu]"]], "]"]]]]]]], SqrtBox["\[Pi]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Contributed by

 Brychkov Yu.A. (2005)

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

 2007-05-02

© 1998-2013 Wolfram Research, Inc.