Im[CoshIntegral[x + I y]] == ArcTan[x, y] + y Sum[(x^(2 k + 1)/(2 k + 2)!) HypergeometricPFQ[{k + 1}, {3/2, k + 2}, -(y^2/4)], {k, 0, Infinity}]

Cell[BoxData[RowBox[List[RowBox[List["Im", "[", RowBox[List["CoshIntegral", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]], "+", RowBox[List["y", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["x", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "2"]], ")"]], "!"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["k", "+", "2"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], "4"]]]]], "]"]]]]]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Chi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> y </mi> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mi> x </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;1&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;3&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;2&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[SuperscriptBox[&quot;y&quot;, &quot;2&quot;], &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <imaginary /> <apply> <ci> CoshIntegral </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> <apply> <times /> <ci> y </ci> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <ci> x </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Im", "[", RowBox[List["CoshIntegral", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]], "+", RowBox[List["y", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["x", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["k", "+", "2"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], "4"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "2"]], ")"]], "!"]]]]]]]]]]]]]

2001-10-29