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variants of this functions
HypergeometricPFQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQ[{a1},{b1,b2},z] > Series representations > Asymptotic series expansions > Expansions for any z in trigonometric form





http://functions.wolfram.com/07.22.06.0011.01









  


  










Input Form





HypergeometricPFQ[{Subscript[a, 1]}, {Subscript[b, 1], Subscript[b, 2]}, z] \[Proportional] ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ (Sqrt[Pi] Gamma[Subscript[a, 1]])) (-z)^((1/2) (1/2 + Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2])) (Cos[2 Sqrt[-z] + (1/2) Pi (1/2 + Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2])] (1 + (1/(512 z)) (-15 + 144 Subscript[a, 1]^4 + 16 Subscript[b, 1]^4 + 16 Subscript[b, 2] + 56 Subscript[b, 2]^2 - 64 Subscript[b, 2]^3 + 16 Subscript[b, 2]^4 - 64 Subscript[b, 1]^3 (1 + Subscript[b, 2]) - 64 Subscript[a, 1]^3 (7 + 3 Subscript[b, 1] + 3 Subscript[b, 2]) + 8 Subscript[b, 1]^2 (7 + 8 Subscript[b, 2] + 12 Subscript[b, 2]^2) + 16 Subscript[b, 1] (1 + 25 Subscript[b, 2] + 4 Subscript[b, 2]^2 - 4 Subscript[b, 2]^3) - 8 Subscript[a, 1]^2 (-43 + 4 Subscript[b, 1]^2 - 72 Subscript[b, 2] + 4 Subscript[b, 2]^2 - 8 Subscript[b, 1] (9 + 5 Subscript[b, 2])) + 16 Subscript[a, 1] (-1 + 4 Subscript[b, 1]^3 - 25 Subscript[b, 2] - 4 Subscript[b, 2]^2 + 4 Subscript[b, 2]^3 - 4 Subscript[b, 1]^2 (1 + Subscript[b, 2]) - Subscript[b, 1] (25 + 40 Subscript[b, 2] + 4 Subscript[b, 2]^2))) + \[Ellipsis]) + (1/(8 Sqrt[-z])) Sin[2 Sqrt[-z] + (1/2) Pi (1/2 + Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2])] (-3 + 12 Subscript[a, 1]^2 - 4 Subscript[b, 1]^2 + 8 Subscript[b, 2] - 4 Subscript[b, 2]^2 + 8 Subscript[b, 1] (1 + Subscript[b, 2]) - 8 Subscript[a, 1] (1 + Subscript[b, 1] + Subscript[b, 2]) + (1/(1536 z)) (-315 + 1728 Subscript[a, 1]^6 - 64 Subscript[b, 1]^6 + 216 Subscript[b, 2] + 1364 Subscript[b, 2]^2 - 960 Subscript[b, 2]^3 - 400 Subscript[b, 2]^4 + 384 Subscript[b, 2]^5 - 64 Subscript[b, 2]^6 + 384 Subscript[b, 1]^5 (1 + Subscript[b, 2]) - 1152 Subscript[a, 1]^5 (11 + 3 Subscript[b, 1] + 3 Subscript[b, 2]) - 16 Subscript[b, 1]^4 (25 + 72 Subscript[b, 2] + 60 Subscript[b, 2]^2) + 64 Subscript[b, 1]^3 (-15 - 71 Subscript[b, 2] + 12 Subscript[b, 2]^2 + 20 Subscript[b, 2]^3) + Subscript[b, 1]^2 (1364 + 13248 Subscript[b, 2] + 9888 Subscript[b, 2]^2 + 768 Subscript[b, 2]^3 - 960 Subscript[b, 2]^4) + 8 Subscript[b, 1] (27 + 2155 Subscript[b, 2] + 1656 Subscript[b, 2]^2 - 568 Subscript[b, 2]^3 - 144 Subscript[b, 2]^4 + 48 Subscript[b, 2]^5) + 48 Subscript[a, 1]^4 (693 + 12 Subscript[b, 1]^2 + 488 Subscript[b, 2] + 12 Subscript[b, 2]^2 + 8 Subscript[b, 1] (61 + 21 Subscript[b, 2])) + 64 Subscript[a, 1]^3 (-585 + 28 Subscript[b, 1]^3 - 853 Subscript[b, 2] - 108 Subscript[b, 2]^2 + 28 Subscript[b, 2]^3 - 12 Subscript[b, 1]^2 (9 + 5 Subscript[b, 2]) - Subscript[b, 1] (853 + 696 Subscript[b, 2] + 60 Subscript[b, 2]^2)) - 12 Subscript[a, 1]^2 (-1323 + 16 Subscript[b, 1]^4 - 4304 Subscript[b, 2] - 1432 Subscript[b, 2]^2 + 448 Subscript[b, 2]^3 + 16 Subscript[b, 2]^4 + 64 Subscript[b, 1]^3 (7 + 3 Subscript[b, 2]) - 8 Subscript[b, 1]^2 (179 + 184 Subscript[b, 2] + 52 Subscript[b, 2]^2) + 16 Subscript[b, 1] (-269 - 449 Subscript[b, 2] - 92 Subscript[b, 2]^2 + 12 Subscript[b, 2]^3)) - 8 Subscript[a, 1] (27 + 48 Subscript[b, 1]^5 + 2155 Subscript[b, 2] + 1656 Subscript[b, 2]^2 - 568 Subscript[b, 2]^3 - 144 Subscript[b, 2]^4 + 48 Subscript[b, 2]^5 - 144 Subscript[b, 1]^4 (1 + Subscript[b, 2]) + 8 Subscript[b, 1]^3 (-71 - 72 Subscript[b, 2] + 12 Subscript[b, 2]^2) + 24 Subscript[b, 1]^2 (69 + 141 Subscript[b, 2] + 60 Subscript[b, 2]^2 + 4 Subscript[b, 2]^3) + Subscript[b, 1] (2155 + 8112 Subscript[b, 2] + 3384 Subscript[b, 2]^2 - 576 Subscript[b, 2]^3 - 144 Subscript[b, 2]^4))) + \[Ellipsis])) /; (Abs[z] -> Infinity)










Standard Form





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MathML Form







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</mrow> <mo> - </mo> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> a </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 72 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 43 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> a </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 40 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 25 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 25 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 56 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 25 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 15 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msubsup> <mi> a </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 1536 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1728 </mn> <mo> &#8290; </mo> <msubsup> <mi> a </mi> <mn> 1 </mn> <mn> 6 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 1152 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 11 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> a </mi> <mn> 1 </mn> <mn> 5 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 48 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 21 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 61 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 488 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 693 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> a </mi> <mn> 1 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 28 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 696 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 853 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 28 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 108 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 853 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 585 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> a </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 52 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 184 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 179 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 92 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 449 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 269 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 448 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 1432 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 4304 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 1323 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> a </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 48 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 5 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 144 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 72 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 71 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 141 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 69 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 144 </mn> </mrow> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 576 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 3384 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 8112 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 2155 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 48 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 5 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 144 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 568 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 1656 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 2155 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 27 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 6 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 6 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 384 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 5 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 400 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 960 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 1364 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 216 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 384 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 5 </mn> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 4 </mn> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 72 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 25 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 20 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 71 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 15 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 960 </mn> </mrow> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 768 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 9888 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 13248 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 1364 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 48 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 5 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 144 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 568 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 1656 </mn> <mo> &#8290; </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 2155 </mn> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 27 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 315 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 512 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 144 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 9 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 72 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -43 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 40 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 25 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> 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<power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 56 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> 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Rule Form





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





2001-10-29





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