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http://functions.wolfram.com/07.18.13.0003.01
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z Derivative[2][w][z] + b Derivative[1][w][z] - w[z] == 0 /;
w[z] == Subscript[c, 1] Hypergeometric0F1Regularized[b, z] +
Subscript[c, 2] z^((1 - b)/2) BesselK[1 - b, 2 Sqrt[z]]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["z", " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List["b", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "-", RowBox[List["w", "[", "z", "]"]]]], "\[Equal]", "0"]], " ", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b", ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], SuperscriptBox["z", FractionBox[RowBox[List["1", "-", "b"]], "2"]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]], " "]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mtext> </mtext> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mrow> <mrow> <msub> <mo>   </mo> <mn> 0 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mo>   </mo> <annotation encoding='Mathematica'> TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]] </annotation> </semantics> <mo> ; </mo> <semantics> <mi> b </mi> <annotation encoding='Mathematica'> TagBox[TagBox[TagBox["b", Hypergeometric0F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]] </annotation> </semantics> <mo> ; </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", Hypergeometric0F1Regularized, Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <msup> <mi> z </mi> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> w </ms> <ms> ′′ </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> b </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> w </ms> <ms> ′ </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> w </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ⩵ </ms> <ms> 0 </ms> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> w </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <ms> 1 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms>  </ms> </apply> <apply> <ci> FormBox </ci> <ms> 0 </ms> <ci> TraditionalForm </ci> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> OverscriptBox </ci> <ms> F </ms> <ms> ~ </ms> </apply> <apply> <ci> FormBox </ci> <ms> 1 </ms> <ci> TraditionalForm </ci> </apply> </apply> </list> </apply> <ms> ⁡ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <ms>  </ms> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric0F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <ms> b </ms> <ci> Hypergeometric0F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric0F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <ms> z </ms> <ci> Hypergeometric0F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <ms> 2 </ms> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> b </ms> </list> </apply> <ms> 2 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> K </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> b </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> SqrtBox </ci> <ms> z </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["z_", " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List["b_", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "-", RowBox[List["w", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b", ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", FractionBox[RowBox[List["1", "-", "b"]], "2"]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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HypergeometricPFQRegularized[{a},{b},z] | HypergeometricPFQRegularized[{a1,a2},{b1},z] | HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] | |
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