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 | | http://functions.wolfram.com/01.20.26.0112.01 | 
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 | | (Cosh[z] Hypergeometric1F1Regularized[a, b, 2 z])/E^z == 
 (Pi/(2 Gamma[a])) MeijerG[{{1 - a}, {1/2}}, {{0}, {1 - b, 1/2}}, 2 z] + 
  (1/(2 Gamma[b - a])) MeijerG[{{1 - b + a}, {}}, {{0}, {1 - b}}, 2 z] | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["Hypergeometric1F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List["2", "z"]]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", RowBox[List["2", RowBox[List["Gamma", "[", "a", "]"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", "a"]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "b"]], ",", FractionBox["1", "2"]]], "}"]]]], "}"]], ",", RowBox[List["2", "z"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", RowBox[List["Gamma", "[", RowBox[List["b", "-", "a"]], "]"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", "b", "+", "a"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "b"]], "}"]]]], "}"]], ",", RowBox[List["2", "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>  <msup>  <mi> ⅇ </mi>  <mrow>  <mo> - </mo>  <mi> z </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> cosh </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mn> 1 </mn>  </msub>  <msub>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <mn> 1 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> ; </mo>  <mi> b </mi>  <mo> ; </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["a", Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["2", " ", "z"]], Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] </annotation>  </semantics>  </mrow>  <mo>  </mo>  <mrow>  <mrow>  <mfrac>  <mi> π </mi>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> a </mi>  <mo> ) </mo>  </mrow>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 3 </mn>  </mrow>  <mrow>  <mn> 1 </mn>  <mo> , </mo>  <mn> 1 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> a </mi>  </mrow>  <mo> , </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> , </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "3"]], RowBox[List["1", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["2", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  </mrow>  <mo> + </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 1 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mn> 1 </mn>  <mo> , </mo>  <mn> 1 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> b </mi>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "2"]], RowBox[List["1", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["2", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[RowBox[List["a", "-", "b", "+", "1"]], MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <times />  <apply>  <power />  <exponentiale />  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> z </ci>  </apply>  </apply>  <apply>  <cosh />  <ci> z </ci>  </apply>  <apply>  <ci> Hypergeometric1F1Regularized </ci>  <ci> a </ci>  <ci> b </ci>  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> z </ci>  </apply>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <apply>  <times />  <pi />  <apply>  <power />  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <ci> Gamma </ci>  <ci> a </ci>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <ci> MeijerG </ci>  <list>  <list>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> a </ci>  </apply>  </apply>  </list>  <list>  <cn type='rational'> 1 <sep /> 2 </cn>  </list>  </list>  <list>  <list>  <cn type='integer'> 0 </cn>  </list>  <list>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </list>  </list>  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> z </ci>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <times />  <cn type='integer'> 1 </cn>  <apply>  <power />  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <ci> b </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> a </ci>  </apply>  </apply>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <ci> MeijerG </ci>  <list>  <list>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  </list>  <list />  </list>  <list>  <list>  <cn type='integer'> 0 </cn>  </list>  <list>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  </list>  </list>  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z_"]]], " ", RowBox[List["Cosh", "[", "z_", "]"]], " ", RowBox[List["Hypergeometric1F1Regularized", "[", RowBox[List["a_", ",", "b_", ",", RowBox[List["2", " ", "z_"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", "a"]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "b"]], ",", FractionBox["1", "2"]]], "}"]]]], "}"]], ",", RowBox[List["2", " ", "z"]]]], "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", "a", "]"]]]]], "+", FractionBox[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", "b", "+", "a"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "b"]], "}"]]]], "}"]], ",", RowBox[List["2", " ", "z"]]]], "]"]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "a"]], "]"]]]]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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