Integrate[((1/Sqrt[x]) BesselI[0, (b x)/2] BesselK[0, (b x)/2])/E^(a x), {x, 0, Infinity}] == 2 Sqrt[2/Pi] (Sqrt[a - Sqrt[a^2 - b^2]]/b) Sech[\[Alpha]]^2 EllipticK[Sech[\[Alpha]]^2] EllipticK[Tanh[\[Alpha]]^2] /; Inequality[Re[a], GreaterEqual, Re[b], Greater, 0] && ArcCosh[Sqrt[b + Sqrt[2 a^2 - 2 a Sqrt[a^2 - b^2]]]/Sqrt[2 b]]

Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox["1", SqrtBox["x"]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", "x"]]], " ", RowBox[List["BesselI", "[", RowBox[List["0", ",", FractionBox[RowBox[List["b", " ", "x"]], "2"]]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List["0", ",", FractionBox[RowBox[List["b", " ", "x"]], "2"]]], "]"]], RowBox[List["\[DifferentialD]", "x"]]]]]], "\[Equal]", RowBox[List["2", SqrtBox[FractionBox["2", "\[Pi]"]], FractionBox[RowBox[List[SqrtBox[RowBox[List["a", "-", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]], " "]], "b"], SuperscriptBox[RowBox[List["Sech", "[", "\[Alpha]", "]"]], "2"], " ", RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["Sech", "[", "\[Alpha]", "]"]], "2"], "]"]], " ", RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["Tanh", "[", "\[Alpha]", "]"]], "2"], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], "\[GreaterEqual]", RowBox[List["Re", "[", "b", "]"]], ">", "0"]], "\[And]", RowBox[List["ArcCosh", "[", FractionBox[SqrtBox[RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["2", " ", "a", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]]], SqrtBox[RowBox[List["2", " ", "b"]]]], "]"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> x </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> K </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> x </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mfrac> <mrow> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </msqrt> <mtext> </mtext> </mrow> <mi> b </mi> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> sech </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> tanh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8805; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> </msqrt> </mrow> </msqrt> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> x </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> x </ci> </apply> </apply> <apply> <ci> BesselI </ci> <cn type='integer'> 0 </cn> <apply> <times /> <ci> b </ci> <ci> x </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> BesselK </ci> <cn type='integer'> 0 </cn> <apply> <times /> <ci> b </ci> <ci> x </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <sech /> <ci> &#945; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <power /> <apply> <sech /> <ci> &#945; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> EllipticK </ci> <apply> <power /> <apply> <tanh /> <ci> &#945; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Inequality </ci> <apply> <real /> <ci> a </ci> </apply> <geq /> <apply> <real /> <ci> b </ci> </apply> <gt /> <cn type='integer'> 0 </cn> </apply> <apply> <arccosh /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a_"]], " ", "x_"]]], " ", RowBox[List["BesselI", "[", RowBox[List["0", ",", FractionBox[RowBox[List["b_", " ", "x_"]], "2"]]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List["0", ",", FractionBox[RowBox[List["b_", " ", "x_"]], "2"]]], "]"]]]], SqrtBox["x_"]], RowBox[List["\[DifferentialD]", "x_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", SqrtBox[RowBox[List["a", "-", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]], " ", SuperscriptBox[RowBox[List["Sech", "[", "\[Alpha]", "]"]], "2"], " ", RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["Sech", "[", "\[Alpha]", "]"]], "2"], "]"]], " ", RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["Tanh", "[", "\[Alpha]", "]"]], "2"], "]"]]]], "b"], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], "\[GreaterEqual]", RowBox[List["Re", "[", "b", "]"]], ">", "0"]], "&&", RowBox[List["ArcCosh", "[", FractionBox[SqrtBox[RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["2", " ", "a", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]]], SqrtBox[RowBox[List["2", " ", "b"]]]], "]"]]]]]]]]]]

2002-12-18