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 Gamma

 http://functions.wolfram.com/06.05.21.0006.01

 Input Form

 Integrate[(Gamma[a + t] Gamma[1 + a/2 + t] Gamma[b + t] Gamma[c + t] Gamma[d + t] Gamma[e + t] Gamma[f + t] Gamma[b - a - t] Gamma[-t])/ (Gamma[a/2 + t] Gamma[1 + a - c + t] Gamma[1 + a - d + t] Gamma[1 + a - e + t] Gamma[1 + a - f + t]), {t, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] == Pi I ((Gamma[b] Gamma[c] Gamma[d] Gamma[e] Gamma[f] Gamma[b + c - a] Gamma[b + d - a] Gamma[b + e - a] Gamma[b + f - a])/ (Gamma[1 + a - c - e] Gamma[1 + a - d - e] Gamma[1 + a - c - d] Gamma[1 + a - c - f] Gamma[1 + a - d - f] Gamma[1 + a - e - f])) /; 2 a == b + c + d + e + f - 1 && -Min[Re[a], Re[a]/2 + 1, Re[b], Re[c], Re[d], Re[e], Re[f]] < \[Gamma] < Min[Re[b - a], 0]

 Standard Form

 Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List["a", "/", "2"]], "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["b", "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["c", "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["d", "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["e", "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["f", "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["b", "-", "a", "-", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["-", "t"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["a", "/", "2"]], "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "d", "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "e", "+", "t"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "f", "+", "t"]], "]"]]]], ")"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], RowBox[List["Gamma", "[", "c", "]"]], RowBox[List["Gamma", "[", "d", "]"]], RowBox[List["Gamma", "[", "e", "]"]], RowBox[List["Gamma", "[", "f", "]"]], RowBox[List["Gamma", "[", RowBox[List["b", "+", "c", "-", "a"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["b", "+", "d", "-", "a"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["b", "+", "e", "-", "a"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["b", "+", "f", "-", "a"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "-", "e"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "d", "-", "e"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "-", "d"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "-", "f"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "d", "-", "f"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "e", "-", "f"]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["2", "a"]], "\[Equal]", RowBox[List["b", "+", "c", "+", "d", "+", "e", "+", "f", "-", "1"]]]], "\[And]", RowBox[List[RowBox[List["-", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ",", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], "/", "2"]], "+", "1"]], ",", RowBox[List["Re", "[", "b", "]"]], ",", RowBox[List["Re", "[", "c", "]"]], ",", RowBox[List["Re", "[", "d", "]"]], ",", RowBox[List["Re", "[", "e", "]"]], ",", RowBox[List["Re", "[", "f", "]"]]]], "]"]]]], "<", "\[Gamma]", "<", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", RowBox[List["b", "-", "a"]], "]"]], ",", "0"]], "]"]]]]]]]]]]]]

 MathML Form

 γ - γ + Γ ( a + t ) Γ ( a 2 + 1 + t ) Γ ( b + t ) Γ ( c + t ) Γ ( d + t ) Γ ( e + t ) Γ ( f + t ) Γ ( b - a - t ) Γ ( - t ) Γ ( a 2 + t ) Γ ( a - c + 1 + t ) Γ ( a - d + 1 + t ) Γ ( a - e + 1 + t ) Γ ( a - f + 1 + t ) t π Γ ( b ) Γ ( c ) Γ ( d ) Γ ( e ) Γ ( f ) Γ ( b + c - a ) Γ ( b + d - a ) Γ ( b + e - a ) Γ ( b + f - a ) / ( Γ ( a - c - e + 1 ) Γ ( a - d - e + 1 ) Γ ( a - c - d + 1 ) Γ ( a - c - f + 1 ) Γ ( a - d - f + 1 ) Γ ( a - e - f + 1 ) ) /; 2 a b + c + d + e + f - 1 - min ( Re ( a ) , Re ( a ) 2 + 1 , Re ( b ) , Re ( c ) , Re ( d ) , Re ( e ) , Re ( f ) ) < γ < min ( Re ( b - a ) , 0 ) Condition t γ -1 γ Gamma a t Gamma a 2 -1 1 t Gamma b t Gamma c t Gamma d t Gamma e t Gamma f t Gamma b -1 a -1 t Gamma -1 t Gamma a 2 -1 t Gamma a -1 c 1 t Gamma a -1 d 1 t Gamma a -1 e 1 t Gamma a -1 f 1 t -1 Gamma b Gamma c Gamma d Gamma e Gamma f Gamma b c -1 a Gamma b d -1 a Gamma b e -1 a Gamma b f -1 a Gamma a -1 c -1 e 1 Gamma a -1 d -1 e 1 Gamma a -1 c -1 d 1 Gamma a -1 c -1 f 1 Gamma a -1 d -1 f 1 Gamma a -1 e -1 f 1 -1 2 a b c d e f -1 -1 a a 2 -1 1 b c d e f γ b -1 a 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]_", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]_", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a_", "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["a_", "2"], "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b_", "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["c_", "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["d_", "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["e_", "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["f_", "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b_", "-", "a_", "-", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "t_"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["a_", "2"], "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a_", "-", "c_", "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a_", "-", "d_", "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a_", "-", "e_", "+", "t_"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a_", "-", "f_", "+", "t_"]], "]"]]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["Gamma", "[", "d", "]"]], " ", RowBox[List["Gamma", "[", "e", "]"]], " ", RowBox[List["Gamma", "[", "f", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "c", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "d", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "e", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "f", "-", "a"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "-", "e"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "d", "-", "e"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "-", "d"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "-", "f"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "d", "-", "f"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "e", "-", "f"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["2", " ", "a"]], "\[Equal]", RowBox[List["b", "+", "c", "+", "d", "+", "e", "+", "f", "-", "1"]]]], "&&", RowBox[List[RowBox[List["-", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ",", RowBox[List[FractionBox[RowBox[List["Re", "[", "a", "]"]], "2"], "+", "1"]], ",", RowBox[List["Re", "[", "b", "]"]], ",", RowBox[List["Re", "[", "c", "]"]], ",", RowBox[List["Re", "[", "d", "]"]], ",", RowBox[List["Re", "[", "e", "]"]], ",", RowBox[List["Re", "[", "f", "]"]]]], "]"]]]], "<", "\[Gamma]", "<", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", RowBox[List["b", "-", "a"]], "]"]], ",", "0"]], "]"]]]]]]]]]]]]

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

 2001-10-29