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; }

 WhittakerW

 http://functions.wolfram.com/07.45.20.0015.01

 Input Form

 D[WhittakerW[\[Nu], \[Mu], z], {z, \[Alpha]}] == (((-1)^(1 + 2 \[Mu]) z^(\[Mu] + 1/2 - \[Alpha]))/ Gamma[1/2 - \[Mu] - \[Nu]]) ((1/(2 \[Mu])!) Sum[(((-(1/2))^(k - j) Pochhammer[1/2 + \[Mu] - \[Nu], j])/ (j! (k - j)! Pochhammer[1 + 2 \[Mu], j])) FDLogConstant[z, k + \[Mu] + 1/2, \[Alpha]] z^k, {k, 0, Infinity}, {j, 0, k}] - Sum[(((-(1/2))^j (-1 - k + j + 2 \[Mu])!)/(j! (k - j)! Pochhammer[1/2 - \[Mu] + \[Nu], -k + j + 2 \[Mu]])) FDPowerConstant[z, k - \[Mu] + 1/2, \[Alpha]] z^k, {k, 0, 2 \[Mu] - 1}, {j, 0, k}]/z^(2 \[Mu]) + Sum[(((-1)^(j + k) 2^(-1 - j - k) j!)/((1 + j + k)! (2 \[Mu] - j - 1)! Pochhammer[1/2 - \[Mu] + \[Nu], 1 + j])) ((Gamma[k + \[Mu] + 3/2] z^k)/Gamma[k + \[Mu] + 3/2 - \[Alpha]]), {k, 0, Infinity}, {j, 0, 2 \[Mu] - 1}] - Sum[(((-(1/2))^(k - j) Pochhammer[1/2 + \[Mu] - \[Nu], j] Gamma[k + \[Mu] + 3/2])/(j! (k - j)! (j + 2 \[Mu])! Gamma[k + \[Mu] + 3/2 - \[Alpha]])) (PolyGamma[1 + j] + PolyGamma[1 + j + 2 \[Mu]] - PolyGamma[1/2 + j + \[Mu] - \[Nu]]) z^k, {k, 0, Infinity}, {j, 0, k}]) /; Element[2 \[Mu], Integers] && 2 \[Mu] >= 0

 Standard Form

 Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], " ", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]]], SuperscriptBox["z", RowBox[List["\[Mu]", "+", FractionBox["1", "2"], "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["2", "\[Mu]"]], ")"]], "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", "j"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "j"]], "]"]]]]], RowBox[List["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["1", "2"]]], ",", "\[Alpha]"]], "]"]], SuperscriptBox["z", "k"]]]]]]]]], "-", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "\[Mu]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "j"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "k", "+", "j", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], "!"]]]], RowBox[List[RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "k"]], "+", "j", "+", RowBox[List["2", " ", "\[Mu]"]]]]]], "]"]]]]], RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["1", "2"]]], ",", "\[Alpha]"]], "]"]], SuperscriptBox["z", "k"]]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["2", "\[Mu]"]], "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "j", "-", "k"]]], " ", RowBox[List["j", "!"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "j", "+", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "-", "j", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "j"]]]], "]"]]]]], FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]], SuperscriptBox["z", "k"]]], RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"], "-", "\[Alpha]"]], "]"]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["j", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"], "-", "\[Alpha]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j", "+", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "j", "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]], ")"]], SuperscriptBox["z", "k"]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["2", "\[Mu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["2", "\[Mu]"]], "\[GreaterEqual]", "0"]]]]]]]]]]

 MathML Form

 α W TagBox["W", WhittakerW] ν , μ ( z ) z α ( - 1 ) 2 μ + 1 z μ + 1 2 - α Γ ( 1 2 - μ - ν ) ( - z - 2 μ k = 0 2 μ - 1 j = 0 k ( - 1 2 ) j ( j - k + 2 μ - 1 ) ! j ! ( k - j ) ! ( ν - μ + 1 2 ) j - k + 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "-", "k", "+", RowBox[List["2", " ", "\[Mu]"]]]]], Pochhammer] ℱ𝒞 exp ( α ) ( z , k - μ + 1 2 ) z k + 1 ( 2 μ ) ! k = 0 j = 0 k ( - 1 2 ) k - j ( μ - ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] j ! ( k - j ) ! ( 2 μ + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]], "j"], Pochhammer] ℱ𝒞 log ( α ) ( z , k + μ + 1 2 ) z k - k = 0 j = 0 k ( - 1 2 ) k - j ( μ - ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] Γ ( k + μ + 3 2 ) j ! ( k - j ) ! ( j + 2 μ ) ! Γ ( k - α + μ + 3 2 ) ( ψ TagBox["\[Psi]", PolyGamma] ( j + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( j + 2 μ + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + μ - ν + 1 2 ) ) z k + k = 0 j = 0 2 μ - 1 ( - 1 ) j + k 2 - j - k - 1 j ! Γ ( k + μ + 3 2 ) z k ( j + k + 1 ) ! ( 2 μ - j - 1 ) ! ( ν - μ + 1 2 ) j + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "+", "1"]]], Pochhammer] Γ ( k - α + μ + 3 2 ) ) /; 2 μ Condition z α WhittakerW ν μ z -1 2 μ 1 z μ 1 2 -1 α Gamma 1 2 -1 μ -1 ν -1 -1 z -2 μ j 0 k k 0 2 μ -1 -1 1 2 j j -1 k 2 μ -1 j k -1 j Pochhammer ν -1 μ 1 2 j -1 k 2 μ -1 Subscript ℱ𝒞 exp α z k -1 μ 1 2 z k 1 2 μ -1 j 0 k k 0 -1 1 2 k -1 j Pochhammer μ -1 ν 1 2 j j k -1 j Pochhammer 2 μ 1 j -1 Subscript ℱ𝒞 log α z k μ 1 2 z k -1 j 0 k k 0 -1 1 2 k -1 j Pochhammer μ -1 ν 1 2 j Gamma k μ 3 2 j k -1 j j 2 μ Gamma k -1 α μ 3 2 -1 PolyGamma j 1 PolyGamma j 2 μ 1 -1 PolyGamma j μ -1 ν 1 2 z k j 0 2 μ -1 k 0 -1 j k 2 -1 j -1 k -1 j Gamma k μ 3 2 z k j k 1 2 μ -1 j -1 Pochhammer ν -1 μ 1 2 j 1 Gamma k -1 α μ 3 2 -1 2 μ [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]]], " ", SuperscriptBox["z", RowBox[List["\[Mu]", "+", FractionBox["1", "2"], "-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["1", "2"]]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "j"]], "]"]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[Mu]"]], ")"]], "!"]]], "-", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "j"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "k", "+", "j", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], "!"]]]], ")"]], " ", RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["1", "2"]]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "k"]], "+", "j", "+", RowBox[List["2", " ", "\[Mu]"]]]]]], "]"]]]]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "j", "-", "k"]]], " ", RowBox[List["j", "!"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]], " ", SuperscriptBox["z", "k"]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "j", "+", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "-", "j", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "j"]]]], "]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"], "-", "\[Alpha]"]], "]"]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j", "+", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "j", "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Mu]", "+", FractionBox["3", "2"], "-", "\[Alpha]"]], "]"]]]]]]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02