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http://functions.wolfram.com/01.24.21.0346.01
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Integrate[(z^n Sech[c z]^\[Nu])/E^(c z (\[Nu] + 2 q)), z] ==
n! (1 + E^(2 c z))^\[Nu] Sech[c z]^\[Nu]
(((-1)^q Gamma[q + \[Nu]] z^(1 + n))/(E^(c z \[Nu])
((1 + n)! q! Gamma[\[Nu]])) + (((-1)^q Pochhammer[\[Nu], 1 + q])/
(E^(c z (-2 + \[Nu])) (1 + q)!))
Sum[(z^(-j + n)/((-2 c)^(1 + j) (-j + n)!)) HypergeometricPFQ[
{Subscript[a, 1], \[Ellipsis], Subscript[a, 2 + j], 1 + q + \[Nu]},
{1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, 1 + j], 2 + q},
-E^(2 c z)], {j, 0, n}] -
Sum[((-1)^k Pochhammer[\[Nu], k] E^(c z (2 k - 2 q - \[Nu])) z^(-j + n))/
((2 c (-k + q))^(1 + j) k! (-j + n)!), {j, 0, n}, {k, 0, q - 1}]) /;
Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 2] ==
1 && Element[n, Integers] && n >= 0 && Element[q, Integers] && q >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "q"]]]], ")"]]]]], SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Nu]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", " ", RowBox[List[RowBox[List["n", "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], RowBox[List["Gamma", "[", RowBox[List["q", "+", "\[Nu]"]], "]"]], SuperscriptBox["z", RowBox[List["1", "+", "n"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z", " ", "\[Nu]"]]], " "]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], "!"]], " ", RowBox[List["q", "!"]], " ", RowBox[List["Gamma", "[", "\[Nu]", "]"]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["1", "+", "q"]]]], "]"]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]]]]]]], RowBox[List[" ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "q"]], ")"]], "!"]]]]], RowBox[List["Sum", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " "]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "c"]], ")"]], RowBox[List["1", "+", "j"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["2", "+", "j"]]], ",", RowBox[List["1", "+", "q", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["1", "+", "j"]]]]], ",", RowBox[List["2", "+", "q"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], " ", ",", RowBox[List["{", RowBox[List["j", ",", "0", ",", "n"]], "}"]]]], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "k"]], "]"]], SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["2", " ", "q"]], "-", "\[Nu]"]], ")"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " "]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "+", "q"]], ")"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "2"]]], "\[Equal]", "1"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["q", "\[Element]", "Integers"]], "\[And]", RowBox[List["q", "\[GreaterEqual]", "0"]]]]]]]]
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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> <mo> ∫ </mo> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mi> ν </mi> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> ν 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</mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> q </mi> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], RowBox[List["q", "+", "1"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> j </mi> <mo> + </mo> <mn> 3 </mn> </mrow> </msub> <msub> <mi> F </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> q </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "3"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ], ",", TagBox[RowBox[List["q", "+", "\[Nu]", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox["2", HypergeometricPFQ], ",", TagBox[RowBox[List["q", "+", "2"]], HypergeometricPFQ]]], 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<msup> <mi> ⅇ </mi> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> q </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> <ci> ν </ci> </apply> </apply> </apply> <apply> <power /> <apply> <sech /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> ν </ci> </apply> </apply> </apply> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <sech /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> ν </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> z </ci> <ci> ν </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> q </ci> <ci> ν </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <ci> q </ci> </apply> <apply> <ci> Gamma </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> ν </ci> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> z </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> c </ci> </apply> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <ci> … </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> q </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <cn type='integer'> 2 </cn> <ci> … </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> q </ci> <cn type='integer'> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> ν </ci> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <apply> <plus /> <ci> q </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> q </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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