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/06.20.20.0006.01

 Input Form

 D[Beta[Subscript[z, 1], Subscript[z, 2], a, b], {a, 2}] == Beta[Subscript[z, 2], a, b] Log[Subscript[z, 2]]^2 - Beta[Subscript[z, 1], a, b] Log[Subscript[z, 1]]^2 + 2 Subscript[z, 2]^a Gamma[a]^2 (Gamma[a] HypergeometricPFQRegularized[{a, a, a, 1 - b}, {1 + a, 1 + a, 1 + a}, Subscript[z, 2]] - HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, Subscript[z, 2]] Log[Subscript[z, 2]]) - 2 Subscript[z, 1]^a Gamma[a]^2 (Gamma[a] HypergeometricPFQRegularized[{a, a, a, 1 - b}, {1 + a, 1 + a, 1 + a}, Subscript[z, 1]] - HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, Subscript[z, 1]] Log[Subscript[z, 1]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "2"]], "}"]]], RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]], "2"]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z", "1"], ",", "a", ",", "b"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]], "2"]]], "+", RowBox[List["2", " ", SubsuperscriptBox["z", "2", "a"], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["z", "2"]]], "]"]]]], "-", RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]]]]]], ")"]]]], "-", RowBox[List["2", " ", SubsuperscriptBox["z", "1", "a"], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["z", "1"]]], "]"]]]], "-", RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["z", "1"]]], "]"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]]]]]], ")"]]]]]]]]]]

 MathML Form

 2 Β Beta ( z 1 , z 2 , a , b ) a 2 2 Γ ( a ) 2 z 2 a ( Γ ( a ) 4 F ~ 3 ( a , a , a , 1 - b ; a + 1 , a + 1 , a + 1 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] - log ( z 2 ) 3 F ~ 2 ( a , a , 1 - b ; a + 1 , a + 1 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) - 2 Γ ( a ) 2 z 1 a ( Γ ( a ) 4 F ~ 3 ( a , a , a , 1 - b ; a + 1 , a + 1 , a + 1 ; z 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "1"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] - log ( z 1 ) 3 F ~ 2 ( a , a , 1 - b ; a + 1 , a + 1 ; z 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "1"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) - log 2 ( z 1 ) Β Beta z 1 ( a , b ) + log 2 ( z 2 ) Β Beta z 2 ( a , b ) a 2 Beta Subscript z 1 Subscript z 2 a b 2 Gamma a 2 Subscript z 2 a Gamma a HypergeometricPFQRegularized a a a 1 -1 b a 1 a 1 a 1 Subscript z 2 -1 Subscript z 2 HypergeometricPFQRegularized a a 1 -1 b a 1 a 1 Subscript z 2 -1 2 Gamma a 2 Subscript z 1 a Gamma a HypergeometricPFQRegularized a a a 1 -1 b a 1 a 1 a 1 Subscript z 1 -1 Subscript z 1 HypergeometricPFQRegularized a a 1 -1 b a 1 a 1 Subscript z 1 -1 Subscript z 1 2 Beta Subscript z 1 a b Subscript z 2 2 Beta Subscript z 2 a b [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "2"]], "}"]]]]], RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["zz", "2"], ",", "a", ",", "b"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]], "2"]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["zz", "1"], ",", "a", ",", "b"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]], "2"]]], "+", RowBox[List["2", " ", SubsuperscriptBox["zz", "2", "a"], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["zz", "2"]]], "]"]]]], "-", RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["zz", "2"]]], "]"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]]]]]], ")"]]]], "-", RowBox[List["2", " ", SubsuperscriptBox["zz", "1", "a"], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["zz", "1"]]], "]"]]]], "-", RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["zz", "1"]]], "]"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]]]]]], ")"]]]]]]]]]]

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

 2001-10-29