html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 MeijerG

 http://functions.wolfram.com/07.34.21.0009.01

 Input Form

 Integrate[t^(\[Alpha] - 1) MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, t z], {t, 0, Infinity}] == Product[Gamma[Subscript[b, k] + \[Alpha]] Product[Gamma[1 - Subscript[a, k] - \[Alpha]], {k, 1, n}], {k, 1, m}]/ Product[Gamma[Subscript[a, k] + \[Alpha]] Product[Gamma[1 - Subscript[b, k] - \[Alpha]], {k, m + 1, q}], {k, n + 1, p}]/z^\[Alpha] /; SuperStar[c] == m + n - (p + q)/2 && m^2 + n^2 > 0 && z != 0 && ((SuperStar[c] > 0 && Abs[Arg[z]] < SuperStar[c] Pi && -Min[Re[Subscript[b, 1]], \[Ellipsis], Re[Subscript[b, m]]] < Re[\[Alpha]] < 1 - Max[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, n]]]) || (p != q && SuperStar[c] >= 0 && Abs[Arg[z]] == SuperStar[c] Pi && -Min[Re[Subscript[b, 1]], \[Ellipsis], Re[Subscript[b, m]]] < Re[\[Alpha]] < 1 - Max[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, n]]] && Re[\[Mu] + (q - p) \[Alpha]] < 3/2 && \[Mu] == Sum[Subscript[b, j], {j, 1, q}] - Sum[Subscript[a, j], {j, 1, p}] + (p - q)/2 + 1) || (p == q && SuperStar[c] == 0 && z > 0 && -Min[Re[Subscript[b, 1]], \[Ellipsis], Re[Subscript[b, m]]] < Re[\[Alpha]] < 1 - Max[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, n]]] && Re[Sum[Subscript[b, j] - Subscript[a, j], {j, 1, q}]] < 0))

 Standard Form

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FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "m"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "k"], "+", "\[Alpha]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["a", "k"], "-", "\[Alpha]"]], "]"]]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", RowBox[List["n", "+", "1"]]]], "p"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "+", "\[Alpha]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", RowBox[List["m", "+", "1"]]]], "q"], RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["b", "k"], "-", "\[Alpha]"]], "]"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SuperscriptBox["c", "*"], "\[Equal]", RowBox[List["m", "+", "n", "-", FractionBox[RowBox[List["p", "+", "q"]], "2"]]]]], "\[And]", RowBox[List[RowBox[List[SuperscriptBox["m", "2"], "+", SuperscriptBox["n", "2"]]], ">", "0"]], "\[And]", RowBox[List["z", "\[NotEqual]", "0"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["c", "*"], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", RowBox[List[SuperscriptBox["c", "*"], "\[Pi]"]]]], "\[And]", RowBox[List[RowBox[List["-", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["b", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["b", "m"], "]"]]]], "]"]]]], "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["1", "-", RowBox[List["Max", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["a", "n"], "]"]]]], "]"]]]]]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["p", "\[NotEqual]", "q"]], "\[And]", RowBox[List[SuperscriptBox["c", "*"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["c", "*"], "\[Pi]"]]]], "\[And]", RowBox[List[RowBox[List["-", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["b", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["b", "m"], "]"]]]], "]"]]]], "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["1", "-", RowBox[List["Max", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["a", "n"], "]"]]]], "]"]]]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "+", RowBox[List[RowBox[List["(", RowBox[List["q", "-", "p"]], ")"]], "\[Alpha]"]]]], "]"]], "<", FractionBox["3", "2"]]], "\[And]", RowBox[List["\[Mu]", "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", 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 MathML Form

 0 t α - 1 G p , q m , n ( t z a 1 , , a n , a n + 1 , , a p b 1 , , b m , b m + 1 , , b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["t", " ", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] t k = 1 m Γ ( α + b k ) k = 1 n Γ ( 1 - α - a k ) k = n + 1 p Γ ( α + a k ) k = m + 1 q Γ ( 1 - α - b k ) z - α /; c * m + n - p + q 2 m 2 + n 2 > 0 z 0 ( ( c * > 0 "\[LeftBracketingBar]" arg ( z ) "\[RightBracketingBar]" < c * π - min ( Re ( b 1 ) , , Re ( b m ) ) < Re ( α ) < 1 - max ( Re ( a 1 ) , , Re ( a n ) ) ) ( p q c * 0 "\[LeftBracketingBar]" arg ( z ) "\[RightBracketingBar]" c * π - min ( Re ( b 1 ) , , Re ( b m ) ) < Re ( α ) < 1 - max ( Re ( a 1 ) , , Re ( a n ) ) Re ( ( q - p ) α + μ ) < 3 2 μ j = 1 q b j - j = 1 p a j + p - q 2 + 1 ) ( p q c * 0 z > 0 - min ( Re ( b 1 ) , , Re ( b m ) ) < Re ( α ) < 1 - max ( Re ( a 1 ) , , Re ( a n ) ) Re ( j = 1 q ( b j - a j ) ) < 0 ) ) 0 t α - 1 G p , q m , n ( t z a 1 , , a n , a n + 1 , , a p b 1 , , b m , b m + 1 , , b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["t", " ", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] t k = 1 m Γ ( α + b k ) k = 1 n Γ ( 1 - α - a k ) k = n + 1 p Γ ( α + a k ) k = m + 1 q Γ ( 1 - α - b k ) z - α /; c * m + n - p + q 2 m 2 + n 2 > 0 z 0 ( ( c * > 0 "\[LeftBracketingBar]" arg ( z ) "\[RightBracketingBar]" < c * π - min ( Re ( b 1 ) , , Re ( b m ) ) < Re ( α ) < 1 - max ( Re ( a 1 ) , , Re ( a n ) ) ) ( p q c * 0 "\[LeftBracketingBar]" arg ( z ) "\[RightBracketingBar]" c * π - min ( Re ( b 1 ) , , Re ( b m ) ) < Re ( α ) < 1 - max ( Re ( a 1 ) , , Re ( a n ) ) Re ( ( q - p ) α + μ ) < 3 2 μ j = 1 q b j - j = 1 p a j + p - q 2 + 1 ) ( p q c * 0 z > 0 - min ( Re ( b 1 ) , , Re ( b m ) ) < Re ( α ) < 1 - max ( Re ( a 1 ) , , Re ( a n ) ) Re ( j = 1 q ( b j - a j ) ) < 0 ) ) [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "n_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a_", RowBox[List["n_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "m_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", RowBox[List["m_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]]]], "}"]], ",", RowBox[List["t_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", 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 Date Added to functions.wolfram.com (modification date)

 2001-10-29