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Sec






Mathematica Notation

Traditional Notation









Elementary Functions > Sec[z] > Integration > Indefinite integration > Involving functions of the direct function, exponential and a power functions > Involving powers of the direct function, exponential and a power functions > Involving exp and power > Involving zn eb z





http://functions.wolfram.com/01.11.21.0124.01









  


  










Input Form





Integrate[(z^n Sec[c z]^\[Nu])/E^(I c z (\[Nu] + 2 q)), z] == n! (1 + E^(2 I c z))^\[Nu] Sec[c z]^\[Nu] (((-1)^q Gamma[q + \[Nu]] z^(1 + n))/(E^(I c z \[Nu]) ((1 + n)! q! Gamma[\[Nu]])) + (((-1)^q Pochhammer[\[Nu], 1 + q])/ (E^(I c z (-2 + \[Nu])) (1 + q)!)) Sum[(z^(-j + n)/((-2 I 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 I c z)], {j, 0, n}] - Sum[((-1)^k Pochhammer[\[Nu], k] E^(I c z (2 k - 2 q - \[Nu])) z^(-j + n))/((2 I 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










Standard Form





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MathML Form







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</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> <imaginaryi /> </apply> <ci> c </ci> <ci> z </ci> <apply> <plus /> <ci> &#957; </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> <imaginaryi /> <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> &#8230; </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> q </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <cn type='integer'> 2 </cn> <ci> &#8230; </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> <imaginaryi /> <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> &#957; </ci> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <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> &#957; </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> <imaginaryi /> <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> &#8469; </ci> </apply> <apply> <in /> <ci> q </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2002-10-15