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 SinIntegral

 http://functions.wolfram.com/06.37.21.0066.01

 Input Form

 Integrate[(t^(\[Alpha] - 1) SinIntegral[t])/E^(z t), {t, 0, Infinity}] == z^(-1 - \[Alpha]) Gamma[1 + \[Alpha]] HypergeometricPFQ[ {1/2, (1 + \[Alpha])/2, 1 + \[Alpha]/2}, {3/2, 3/2}, -(1/z^2)] /; Re[z] > 0 && Re[\[Alpha]] > -1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], " ", "t"]]], " ", RowBox[List["SinIntegral", "[", " ", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Alpha]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]]

 MathML Form

 0 t α - 1 - z t Si ( t ) t z - α - 1 Γ ( α + 1 ) 3 F 2 ( 1 2 , α + 1 2 , α 2 + 1 ; 3 2 , 3 2 ; - 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; Re ( z ) > 0 Re ( α ) > - 1 Condition t 0 t α -1 -1 z t SinIntegral t z -1 α -1 Gamma α 1 HypergeometricPFQ 1 2 α 1 2 -1 α 2 -1 1 3 2 3 2 -1 1 z 2 -1 z 0 α -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z_"]], " ", "t_"]]], " ", RowBox[List["SinIntegral", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Alpha]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]]]]

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

 2001-10-29