Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Integration > Indefinite integration > Involving functions of the direct function, trigonometric and exponential functions > Involving rational functions of the direct function, trigonometric and exponential functions > Involving cos and exp > Involving ep zcos(d z)/a+b sinh2(c z)





http://functions.wolfram.com/01.19.21.3282.01









  


  










Input Form





Integrate[(E^(p z) Cos[d z])/(a + b Sinh[c z]^2), z] == (-(1/(2 Sqrt[a] Sqrt[a - b]))) (b ((E^((2 c - I d + p) z) Hypergeometric2F1[(2 c - I d + p)/(2 c), 1, (4 c - I d + p)/(2 c), (b E^(2 c z))/(-2 a - 2 Sqrt[a] Sqrt[a - b] + b)])/((2 a + 2 Sqrt[a] Sqrt[a - b] - b) (2 c - I d + p)) + (E^((2 c - I d + p) z) Hypergeometric2F1[(2 c - I d + p)/(2 c), 1, (4 c - I d + p)/(2 c), (b E^(2 c z))/(-2 a + 2 Sqrt[a] Sqrt[a - b] + b)])/((-2 a + 2 Sqrt[a] Sqrt[a - b] + b) (2 c - I d + p)) + (1/(2 c + I d + p)) (E^((2 c + I d + p) z) (Hypergeometric2F1[(2 c + I d + p)/(2 c), 1, (4 c + I d + p)/(2 c), (b E^(2 c z))/(-2 a - 2 Sqrt[a] Sqrt[a - b] + b)]/ (2 a + 2 Sqrt[a] Sqrt[a - b] - b) + Hypergeometric2F1[(2 c + I d + p)/(2 c), 1, (4 c + I d + p)/(2 c), (b E^(2 c z))/(-2 a + 2 Sqrt[a] Sqrt[a - b] + b)]/ (-2 a + 2 Sqrt[a] Sqrt[a - b] + b)))))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], RowBox[List["Cos", "[", RowBox[List["d", " ", "z"]], "]"]]]], RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]]]]], RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], ")"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], ")"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], "]"]], "/", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]], ")"]]]], "+", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], "]"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]], ")"]]]]]], ")"]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> p </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]], &quot;+&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;d&quot;]], &quot; &quot;, &quot;+&quot;, &quot;p&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List[&quot;4&quot;, &quot; &quot;, &quot;c&quot;]], &quot;+&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;d&quot;]], &quot; &quot;, &quot;+&quot;, &quot;p&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot; &quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;, &quot; &quot;, &quot;z&quot;]]]]], RowBox[List[RowBox[List[RowBox[List[&quot;-&quot;, &quot;2&quot;]], &quot; &quot;, &quot;a&quot;]], &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, SqrtBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;b&quot;]]], &quot; &quot;, SqrtBox[&quot;a&quot;]]], &quot;+&quot;, &quot;b&quot;]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]], &quot;+&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;d&quot;]], &quot; &quot;, &quot;+&quot;, &quot;p&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List[&quot;4&quot;, &quot; &quot;, &quot;c&quot;]], &quot;+&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;d&quot;]], &quot; &quot;, &quot;+&quot;, &quot;p&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot; &quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;, &quot; &quot;, &quot;z&quot;]]]]], RowBox[List[RowBox[List[RowBox[List[&quot;-&quot;, &quot;2&quot;]], &quot; &quot;, &quot;a&quot;]], &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, SqrtBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;b&quot;]]], &quot; &quot;, SqrtBox[&quot;a&quot;]]], &quot;+&quot;, &quot;b&quot;]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> - </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]], &quot;-&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;d&quot;]], &quot;+&quot;, &quot;p&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List[&quot;4&quot;, &quot; &quot;, &quot;c&quot;]], &quot;-&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;d&quot;]], &quot;+&quot;, &quot;p&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot; &quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;, &quot; &quot;, &quot;z&quot;]]]]], RowBox[List[RowBox[List[RowBox[List[&quot;-&quot;, &quot;2&quot;]], &quot; &quot;, &quot;a&quot;]], &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, SqrtBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;b&quot;]]], &quot; &quot;, SqrtBox[&quot;a&quot;]]], &quot;+&quot;, &quot;b&quot;]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]], &quot;-&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;d&quot;]], &quot;+&quot;, &quot;p&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List[&quot;4&quot;, &quot; &quot;, &quot;c&quot;]], &quot;-&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;d&quot;]], &quot;+&quot;, &quot;p&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot; &quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;, &quot; &quot;, &quot;z&quot;]]]]], RowBox[List[RowBox[List[RowBox[List[&quot;-&quot;, &quot;2&quot;]], &quot; &quot;, &quot;a&quot;]], &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, SqrtBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;b&quot;]]], &quot; &quot;, SqrtBox[&quot;a&quot;]]], &quot;+&quot;, &quot;b&quot;]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> p </ci> <ci> z </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> b </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Cos", "[", RowBox[List["d_", " ", "z_"]], "]"]]]], RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], ")"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], ")"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], "]"]], RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]], "+", FractionBox[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], ")"]]]], RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "p"]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]]]]]]]]]










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





2002-12-18