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; }

 Beta

 http://functions.wolfram.com/06.20.21.0010.01

 Input Form

 Integrate[a^(\[Alpha] - 1) Beta[Subscript[z, 1], Subscript[z, 2], a, b], a] == (a^(-1 + \[Alpha]) (Subscript[z, 2]^a - Subscript[z, 1]^a))/ (\[Alpha] - 1) + (a^\[Alpha]/(\[Alpha] - 1)) ((Log[Subscript[z, 1]] Gamma[\[Alpha], 0, (-a) Log[Subscript[z, 1]]])/ ((-a) Log[Subscript[z, 1]])^\[Alpha] - (Log[Subscript[z, 2]] Gamma[\[Alpha], 0, (-a) Log[Subscript[z, 2]]])/ ((-a) Log[Subscript[z, 2]])^\[Alpha]) + ((1 - b)/\[Alpha]) a^\[Alpha] Sum[((Pochhammer[2 - b, k] (Subscript[z, 2]^(k + a + 1) - Subscript[z, 1]^(k + a + 1)))/((1 + k)^2 k!)) Hypergeometric2F1[\[Alpha], 1, 1 + \[Alpha], -(a/(1 + k))], {k, 0, Infinity}] /; Abs[Subscript[z, 1]] < 1 && Abs[Subscript[z, 2]] < 1 && !(Element[-a, Integers] && -a >= 0)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["a", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], RowBox[List["\[DifferentialD]", "a"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]]], RowBox[List["(", RowBox[List[SubsuperscriptBox["z", "2", "a"], "-", SubsuperscriptBox["z", "1", "a"]]], ")"]]]], RowBox[List["\[Alpha]", "-", "1"]]], "+", RowBox[List[FractionBox[SuperscriptBox["a", "\[Alpha]"], RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", "0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]]]]]], "]"]]]], " ", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", "0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["1", "-", "b"]], "\[Alpha]"], SuperscriptBox["a", "\[Alpha]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]], RowBox[List["(", RowBox[List[SubsuperscriptBox["z", "2", RowBox[List["k", "+", "a", "+", "1"]]], "-", SubsuperscriptBox["z", "1", RowBox[List["k", "+", "a", "+", "1"]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", "1", ",", RowBox[List["1", "+", "\[Alpha]"]], ",", RowBox[List["-", FractionBox["a", RowBox[List["1", "+", "k"]]]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", SubscriptBox["z", "1"], "]"]], "<", "1"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["z", "2"], "]"]], "<", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], "]"]]]]]]]]

 MathML Form

 a α - 1 Β TagBox["\[CapitalBeta]", BetaRegularized] ( z 1 , z 2 ) ( a , b ) a ( z 2 a - z 1 a ) a α - 1 α - 1 + ( 1 - b ) a α α k = 0 ( 2 - b ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]], "k"], Pochhammer] ( z 2 a + k + 1 - z 1 a + k + 1 ) ( k + 1 ) 2 k ! 2 F 1 ( α , 1 ; α + 1 ; - a k + 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["\[Alpha]", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Alpha]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["a", RowBox[List["k", "+", "1"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + a α α - 1 ( ( - a log ( z 1 ) ) - α log ( z 1 ) Γ ( α , 0 , - a log ( z 1 ) ) - ( - a log ( z 2 ) ) - α log ( z 2 ) Γ ( α , 0 , - a log ( z 2 ) ) ) /; "\[LeftBracketingBar]" z 1 "\[RightBracketingBar]" < 1 "\[LeftBracketingBar]" z 2 "\[RightBracketingBar]" < 1 - a a α - 1 Β TagBox["\[CapitalBeta]", BetaRegularized] ( z 1 , z 2 ) ( a , b ) a ( z 2 a - z 1 a ) a α - 1 α - 1 + ( 1 - b ) a α α k = 0 ( 2 - b ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]], "k"], Pochhammer] ( z 2 a + k + 1 - z 1 a + k + 1 ) ( k + 1 ) 2 k ! 2 F 1 ( α , 1 ; α + 1 ; - a k + 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["\[Alpha]", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Alpha]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["a", RowBox[List["k", "+", "1"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + a α α - 1 ( ( - a log ( z 1 ) ) - α log ( z 1 ) Γ ( α , 0 , - a log ( z 1 ) ) - ( - a log ( z 2 ) ) - α log ( z 2 ) Γ ( α , 0 , - a log ( z 2 ) ) ) /; "\[LeftBracketingBar]" z 1 "\[RightBracketingBar]" < 1 "\[LeftBracketingBar]" z 2 "\[RightBracketingBar]" < 1 - a [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["a_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", "a_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["zz", "2", "a"], "-", SubsuperscriptBox["zz", "1", "a"]]], ")"]]]], RowBox[List["\[Alpha]", "-", "1"]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", "0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", "0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]]]]]], "]"]]]]]], ")"]]]], RowBox[List["\[Alpha]", "-", "1"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", SuperscriptBox["a", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["zz", "2", RowBox[List["k", "+", "a", "+", "1"]]], "-", SubsuperscriptBox["zz", "1", RowBox[List["k", "+", "a", "+", "1"]]]]], ")"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", "1", ",", RowBox[List["1", "+", "\[Alpha]"]], ",", RowBox[List["-", FractionBox["a", RowBox[List["1", "+", "k"]]]]]]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]]]]]], "\[Alpha]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", SubscriptBox["zz", "1"], "]"]], "<", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["zz", "2"], "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]]

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

 2001-10-29