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/06.20.20.0018.01

 Input Form

 D[Beta[Subscript[z, 1], Subscript[z, 2], a, b], {a, \[Alpha]}] == (Log[Subscript[z, 2]] HypergeometricPFQRegularized[{1, 1}, {2, 1 - \[Alpha]}, a Log[Subscript[z, 2]]] - Log[Subscript[z, 1]] HypergeometricPFQRegularized[{1, 1}, {2, 1 - \[Alpha]}, a Log[Subscript[z, 1]]])/a^\[Alpha] + ((1 - b) Sum[(Pochhammer[2 - b, k]/((k + 1)^2 k!)) HypergeometricPFQRegularized[{1, 1}, {1 - \[Alpha]}, -(a/(1 + k))] (Subscript[z, 2]^(k + a + 1) - Subscript[z, 1]^(k + a + 1)), {k, 0, Infinity}])/a^\[Alpha] /; 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[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]]], RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["a", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", RowBox[List["1", "-", "\[Alpha]"]]]], "}"]], ",", RowBox[List["a", " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", RowBox[List["1", "-", "\[Alpha]"]]]], "}"]], ",", RowBox[List["a", " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], SuperscriptBox["a", RowBox[List["-", "\[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[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Alpha]"]], "}"]], ",", RowBox[List["-", FractionBox["a", RowBox[List["1", "+", "k"]]]]]]], "]"]], RowBox[List["(", RowBox[List[SubsuperscriptBox["z", "2", RowBox[List["k", "+", "a", "+", "1"]]], "-", SubsuperscriptBox["z", "1", RowBox[List["k", "+", "a", "+", "1"]]]]], ")"]]]]]]]]]]]], "/;", 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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["a", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", RowBox[List["1", "-", "\[Alpha]"]]]], "}"]], ",", RowBox[List["a", " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", RowBox[List["1", "-", "\[Alpha]"]]]], "}"]], ",", RowBox[List["a", " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", SuperscriptBox["a", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Alpha]"]], "}"]], ",", RowBox[List["-", FractionBox["a", RowBox[List["1", "+", "k"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["zz", "2", RowBox[List["k", "+", "a", "+", "1"]]], "-", SubsuperscriptBox["zz", "1", RowBox[List["k", "+", "a", "+", "1"]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]]]]]]]], "/;", 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