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; }

 StruveL

 http://functions.wolfram.com/03.10.20.0016.01

 Input Form

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

 Standard Form

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 MathML Form

 L StruveL n + 1 2 ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) ( - 1 ) n 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 ! 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[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 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( - k + n + 1 ) ( n - k ) ! ( - 4 z 2 ) k - ( - 1 ) n π n ! 2 ( z 2 ) 1 2 - n k = 0 n - 1 1 k ! ( n - k ) ( - z 2 ) k p = 0 n - k - 1 1 p ! ( - z 2 ) p ( ( 2 I p - 1 2 ( z ) - 2 p + 1 2 I p - 1 2 ( 2 z ) ) I - k - 1 2 ( z ) - ( 2 I 1 2 - p ( z ) - 2 p + 1 2 I 1 2 - p ( 2 z ) ) I k + 1 2 ( z ) ) + I - n - 1 2 ( z ) ( 2 Chi ( z ) - Chi ( 2 z ) ) + ( - 1 ) n + 1 2 π ( z 2 ) - n - 1 2 Γ ( n + 1 2 ) ( log ( 4 ) + ψ TagBox["\[Psi]", PolyGamma] ( 1 2 - n ) + 3 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) + I n + 1 2 ( z ) ( Shi ( 2 z ) - 2 Shi ( z ) ) + ( - 1 ) n n ! 2 ( 2 z ) n k = 0 n - 1 1 k ! ( n - k ) ( - z 2 ) k I - k - 1 2 ( z ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition 1 0 Subscript StruveL n 1 2 z -1 n 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 1 n 1 2 -1 z 2 -1 z 2 -1 n -1 1 2 HypergeometricPFQ -1 n 1 2 1 4 z 2 -1 1 1 2 -1 z 2 -1 n -1 1 2 k 0 n -1 Pochhammer 1 2 k PolyGamma -1 k n 1 n -1 k -1 -1 4 z 2 -1 k -1 -1 n 1 2 n 2 -1 z 2 -1 1 2 -1 n k 0 n -1 1 k n -1 k -1 -1 z 2 -1 k p 0 n -1 k -1 1 p -1 -1 z 2 -1 p 2 BesselI p -1 1 2 z -1 2 p 1 2 BesselI p -1 1 2 2 z BesselI -1 k -1 1 2 z -1 2 BesselI 1 2 -1 p z -1 2 p 1 2 BesselI 1 2 -1 p 2 z BesselI k 1 2 z BesselI -1 n -1 1 2 z 2 CoshIntegral z -1 CoshIntegral 2 z -1 n 1 2 -1 z 2 -1 -1 n -1 1 2 Gamma n 1 2 4 PolyGamma 1 2 -1 n 3 BesselI n 1 2 z SinhIntegral 2 z -1 2 SinhIntegral z -1 n n 2 -1 2 z -1 n k 0 n -1 1 k n -1 k -1 -1 z 2 -1 k BesselI -1 k -1 1 2 z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["StruveL", 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[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "1"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", "n"]], "-", FractionBox["1", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List["3", " ", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]]]], ")"]]]], RowBox[List["2", " ", "\[Pi]"]]], "+", 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 Contributed by

 Brychkov Yu.A. (2005)

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

 2007-05-02