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 StruveH

 http://functions.wolfram.com/03.09.20.0016.01

 Input Form

 Derivative[1, 0][StruveH][n + 1/2, z] == (-2 SinIntegral[z] + SinIntegral[2 z]) BesselJ[n + 1/2, z] + (-1)^n (CosIntegral[2 z] - 2 CosIntegral[z]) BesselJ[-n - 1/2, z] + (1/(n! Sqrt[Pi])) (z/2)^(n - 1/2) Log[z/2] HypergeometricPFQ[{-n, 1/2, 1}, {}, -(4/z^2)] + (1/(2 Pi)) (z/2)^(-(1/2) - n) Gamma[n + 1/2] (3 EulerGamma + Log[4] + PolyGamma[1/2 - n]) - (n!/2) (2/z)^n Sum[(1/((n - k) k!)) (-(z/2))^k BesselJ[-k - 1/2, z], {k, 0, n - 1}] - (n!/(2 Sqrt[Pi])) (z/2)^(-(1/2) - n) Sum[Pochhammer[1/2, k]/(k! (n - k)), {k, 0, n - 1}] - (1/Sqrt[Pi]) (z/2)^(n - 1/2) Sum[((2/z)^(2 k) Pochhammer[1/2, k] PolyGamma[n - k + 1])/(n - k)!, {k, 0, n - 1}] - (1/2) n! Sqrt[Pi] (z/2)^(1/2 - n) Sum[(1/(k! (n - k))) (z/2)^k Sum[(1/p!) (z/2)^p ((-1)^(p + 1) BesselJ[k + 1/2, z] (2 BesselJ[1/2 - p, z] - 2^(1/2 + p) BesselJ[1/2 - p, 2 z]) - (-1)^k BesselJ[-k - 1/2, z] (2 BesselJ[p - 1/2, z] - 2^(1/2 + p) BesselJ[p - 1/2, 2 z])), {p, 0, n - k - 1}], {k, 0, n - 1}] /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 H StruveH n + 1 2 ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) 1 n ! π log ( z 2 ) ( z 2 ) n - 1 2 3 F 0 ( - n , 1 2 , 1 ; ; - 4 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - 1 π ( z 2 ) n - 1 2 k = 0 n - 1 ψ TagBox["\[Psi]", PolyGamma] ( - k + n + 1 ) ( n - k ) ! ( 2 z ) 2 k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] - n ! π 2 ( z 2 ) 1 2 - n k = 0 n - 1 1 k ! ( n - k ) ( z 2 ) k p = 0 n - k - 1 ( z 2 ) p p ! ( ( - 1 ) p + 1 J k + 1 2 ( z ) ( 2 J 1 2 - p ( z ) - 2 p + 1 2 J 1 2 - p ( 2 z ) ) - ( - 1 ) k J - k - 1 2 ( z ) ( 2 J p - 1 2 ( z ) - 2 p + 1 2 J p - 1 2 ( 2 z ) ) ) + 1 2 π Γ ( n + 1 2 ) ( log ( 4 ) + ψ TagBox["\[Psi]", PolyGamma] ( 1 2 - n ) + 3 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ( z 2 ) - n - 1 2 - n ! 2 π ( z 2 ) - n - 1 2 k = 0 n - 1 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! ( n - k ) + ( - 1 ) n J - n - 1 2 ( z ) ( Ci ( 2 z ) - 2 Ci ( z ) ) + J n + 1 2 ( z ) ( Si ( 2 z ) - 2 Si ( z ) ) - n ! 2 ( 2 z ) n k = 0 n - 1 1 ( n - k ) k ! ( - z 2 ) k J - k - 1 2 ( z ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition 1 0 Subscript StruveH n 1 2 z 1 n 1 2 -1 z 2 -1 z 2 -1 n -1 1 2 HypergeometricPFQ -1 n 1 2 1 -1 4 z 2 -1 -1 1 1 2 -1 z 2 -1 n -1 1 2 k 0 n -1 PolyGamma -1 k n 1 n -1 k -1 2 z -1 2 k Pochhammer 1 2 k -1 n 1 2 2 -1 z 2 -1 1 2 -1 n k 0 n -1 1 k n -1 k -1 z 2 -1 k p 0 n -1 k -1 z 2 -1 p p -1 -1 p 1 BesselJ k 1 2 z 2 BesselJ 1 2 -1 p z -1 2 p 1 2 BesselJ 1 2 -1 p 2 z -1 -1 k BesselJ -1 k -1 1 2 z 2 BesselJ p -1 1 2 z -1 2 p 1 2 BesselJ p -1 1 2 2 z 1 2 -1 Gamma n 1 2 4 PolyGamma 1 2 -1 n 3 z 2 -1 -1 n -1 1 2 -1 n 2 1 2 -1 z 2 -1 -1 n -1 1 2 k 0 n -1 Pochhammer 1 2 k k n -1 k -1 -1 n BesselJ -1 n -1 1 2 z CosIntegral 2 z -1 2 CosIntegral z BesselJ n 1 2 z SinIntegral 2 z -1 2 SinIntegral z -1 n 2 -1 2 z -1 n k 0 n -1 1 n -1 k k -1 -1 z 2 -1 k BesselJ -1 k -1 1 2 z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["StruveH", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["n_", "+", FractionBox["1", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["SinIntegral", "[", "z", "]"]]]], "+", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["CosIntegral", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["BesselJ", "[", 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 Contributed by

 Brychkov Yu.A. (2005)

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

 2007-05-02