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 BesselI

 http://functions.wolfram.com/03.02.21.0087.01

 Input Form

 Integrate[(t^(\[Alpha] - 1) BesselI[\[Nu], t])/E^t, {t, 0, Infinity}] == (1/2) ((2^(\[Alpha] - 1) Cos[(1/2) Pi (\[Alpha] + \[Nu])] Gamma[(\[Alpha] + \[Nu])/2] Hypergeometric2F1[(\[Alpha] - \[Nu])/2, (\[Alpha] + \[Nu])/2, 1/2, 1])/Gamma[(1/2) (2 - \[Alpha] + \[Nu])] + (Gamma[\[Alpha] + \[Nu]] Hypergeometric2F1[(\[Alpha] + \[Nu])/2, (1/2) (1 + \[Alpha] + \[Nu]), 1 + \[Nu], 1])/ (2^\[Nu] Gamma[1 + \[Nu]]) + 2^\[Alpha] Gamma[(1/2) (1 + \[Alpha] + \[Nu])] Hypergeometric2F1[ (1/2) (1 + \[Alpha] - \[Nu]), (1/2) (1 + \[Alpha] + \[Nu]), 3/2, 1] (Sin[(1/2) Pi (\[Alpha] + \[Nu])]/ Gamma[(1/2) (1 - \[Alpha] + \[Nu])])) /; Re[\[Alpha] + \[Nu]] > 0 && Re[\[Alpha]] < 1/2

 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["-", "t"]]], RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"], ",", FractionBox["1", "2"], ",", "1"]], "]"]]]], ")"]], "/", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", "1"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], "+", RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], ",", FractionBox["3", "2"], ",", "1"]], "]"]], " ", RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]], "/", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Nu]"]], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", FractionBox["1", "2"]]]]]]]]]

 MathML Form

 0 t α - 1 - t I ν ( t ) t 1 2 ( 2 - ν Γ ( α + ν ) Γ ( ν + 1 ) 2 F 1 ( α + ν 2 , α + ν + 1 2 ; ν + 1 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 2 α Γ ( 1 2 ( - α + ν + 1 ) ) Γ ( α + ν + 1 2 ) sin ( π ( α + ν ) 2 ) 2 F 1 ( α - ν + 1 2 , α + ν + 1 2 ; 3 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Alpha]", "-", "\[Nu]", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 2 α - 1 Γ ( 1 2 ( ν - α + 2 ) ) cos ( π ( α + ν ) 2 ) Γ ( α + ν 2 ) 2 F 1 ( α - ν 2 , α + ν 2 ; 1 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Alpha]", "-", "\[Nu]"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) /; Re ( α + ν ) > 0 Re ( α ) < 1 2 Condition t 0 t α -1 -1 t BesselI ν t 1 2 2 -1 ν Gamma α ν Gamma ν 1 -1 Hypergeometric2F1 α ν 2 -1 α ν 1 2 -1 ν 1 1 2 α Gamma 1 2 -1 α ν 1 -1 Gamma α ν 1 2 -1 α ν 2 -1 Hypergeometric2F1 α -1 ν 1 2 -1 α ν 1 2 -1 3 2 1 2 α -1 Gamma 1 2 ν -1 α 2 -1 α ν 2 -1 Gamma α ν 2 -1 Hypergeometric2F1 α -1 ν 2 -1 α ν 2 -1 1 2 1 α ν 0 α 1 2 [/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["-", "t_"]]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"], ",", FractionBox["1", "2"], ",", "1"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", "1"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], ",", FractionBox["3", "2"], ",", "1"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Nu]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", FractionBox["1", "2"]]]]]]]]]]]

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

 2001-10-29