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 StruveH

 http://functions.wolfram.com/03.09.21.0011.01

 Input Form

 Integrate[(StruveH[\[Mu], z] StruveH[\[Nu], z])/z, z] == (1/(2 (\[Mu] - \[Nu]) (\[Mu] + \[Nu]))) ((2^(-\[Mu] - \[Nu]) z^(2 + \[Mu] + \[Nu]) Gamma[(1/2) (2 + \[Mu] + \[Nu])] (Gamma[1/2 + \[Mu]] HypergeometricPFQRegularized[ {1, (1/2) (2 + \[Mu] + \[Nu])}, {3/2 + \[Mu], 3/2, (1/2) (4 + \[Mu] + \[Nu])}, -(z^2/4)] - Gamma[1/2 + \[Nu]] HypergeometricPFQRegularized[{1, (1/2) (2 + \[Mu] + \[Nu])}, {3/2 + \[Nu], 3/2, (1/2) (4 + \[Mu] + \[Nu])}, -(z^2/4)]))/ (Sqrt[Pi] Gamma[1/2 + \[Mu]] Gamma[1/2 + \[Nu]]) + 2 z StruveH[-1 + \[Mu], z] StruveH[\[Nu], z] - 2 StruveH[\[Mu], z] (z StruveH[-1 + \[Nu], z] + (\[Mu] - \[Nu]) StruveH[\[Nu], z]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["StruveH", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "z"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]], ",", FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["4", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["4", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]]]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]"]], ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["StruveH", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 H StruveH μ ( z ) H StruveH ν ( z ) z z 1 2 ( μ - ν ) ( μ + ν ) ( 2 H StruveH μ - 1 ( z ) H StruveH ν ( z ) z + 1 π Γ ( μ + 1 2 ) Γ ( ν + 1 2 ) ( 2 - μ - ν z μ + ν + 2 Γ ( 1 2 ( μ + ν + 2 ) ) ( Γ ( μ + 1 2 ) 2 F ~ 3 ( 1 , 1 2 ( μ + ν + 2 ) ; μ + 3 2 , 3 2 , 1 2 ( μ + ν + 4 ) ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "2"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Mu]", "+", FractionBox["3", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "4"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] - Γ ( ν + 1 2 ) 2 F ~ 3 ( 1 , 1 2 ( μ + ν + 2 ) ; ν + 3 2 , 3 2 , 1 2 ( μ + ν + 4 ) ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "2"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "4"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) ) - 2 H StruveH μ ( z ) ( z H StruveH ν - 1 ( z ) + ( μ - ν ) H StruveH ν ( z ) ) ) z StruveH μ z StruveH ν z z -1 1 2 μ -1 ν μ ν -1 2 StruveH μ -1 z StruveH ν z z 1 1 2 Gamma μ 1 2 Gamma ν 1 2 -1 2 -1 μ -1 ν z μ ν 2 Gamma 1 2 μ ν 2 Gamma μ 1 2 HypergeometricPFQRegularized 1 1 2 μ ν 2 μ 3 2 3 2 1 2 μ ν 4 -1 z 2 4 -1 -1 Gamma ν 1 2 HypergeometricPFQRegularized 1 1 2 μ ν 2 ν 3 2 3 2 1 2 μ ν 4 -1 z 2 4 -1 -1 2 StruveH μ z z StruveH ν -1 z μ -1 ν StruveH ν z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["StruveH", "[", RowBox[List["\[Mu]_", ",", "z_"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "z_"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]], ",", FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["4", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["4", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]"]], ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["StruveH", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]], ")"]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]]]]]]

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

 2001-10-29