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; }

 WhittakerM

 http://functions.wolfram.com/07.44.03.0019.01

 Input Form

 WhittakerM[n/2, \[Mu], z] == 2^(-2 - n + 2 \[Mu]) z^((1 + n)/2) (4 BesselI[-(n/2) + \[Mu], z/2] Gamma[1 - n/2 + \[Mu]] Sum[((-1)^(-j + k) 2^(-4 j + 2 k) z^(2 j - k) (-j + k)! Pochhammer[-n, k] Pochhammer[1 - n/2 + \[Mu], k] Pochhammer[-n + 2 \[Mu], k])/(j! k! (-2 j + k)! Pochhammer[1 - k + n/2 - \[Mu], j] Pochhammer[-(n/2) + \[Mu], j] Pochhammer[1 + 2 \[Mu], k]), {k, 0, n}, {j, 0, Floor[k/2]}] - z BesselI[-1 - n/2 + \[Mu], z/2] Gamma[-(n/2) + \[Mu]] Sum[((-1)^(-j + k) 2^(-4 j + 2 k) z^(2 j - k) (-1 - j + k)! Pochhammer[-n, k] Pochhammer[1 - n/2 + \[Mu], k] Pochhammer[-n + 2 \[Mu], k])/(j! k! (-1 - 2 j + k)! Pochhammer[1 - k + n/2 - \[Mu], j] Pochhammer[1 - n/2 + \[Mu], j] Pochhammer[1 + 2 \[Mu], k]), {k, 0, n}, {j, 0, Floor[(1/2) (-1 + k)]}]) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WhittakerM", "[", RowBox[List[FractionBox["n", "2"], ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "n", "+", RowBox[List["2", " ", "\[Mu]"]]]]], " ", SuperscriptBox["z", FractionBox[RowBox[List["1", "+", "n"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], "+", "\[Mu]"]], ",", FractionBox["z", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["n", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox["k", "2"], "]"]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "j"]], "+", RowBox[List["2", " ", "k"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["n", "2"], "+", "\[Mu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", FractionBox["n", "2"], "-", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], "+", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]]]], ")"]]]]]]]]]], "-", RowBox[List["z", " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["n", "2"], "+", "\[Mu]"]], ",", FractionBox["z", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], "+", "\[Mu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "k"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "j"]], "+", RowBox[List["2", " ", "k"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["n", "2"], "+", "\[Mu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "j"]], "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", FractionBox["n", "2"], "-", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["n", "2"], "+", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]]]], ")"]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 M TagBox["M", WhittakerM] n 2 , μ ( z ) 2 - n + 2 μ - 2 z n + 1 2 ( 4 I μ - n 2 ( z 2 ) Γ ( - n 2 + μ + 1 ) k = 0 n j = 0 k 2 ( - 1 ) k - j 2 2 k - 4 j z 2 j - k ( k - j ) ! ( - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], Pochhammer] ( - n 2 + μ + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], "+", "\[Mu]", "+", "1"]], ")"]], "k"], Pochhammer] ( 2 μ - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "-", "n"]], ")"]], "k"], Pochhammer] j ! k ! ( k - 2 j ) ! ( - k + n 2 - μ + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "+", FractionBox["n", "2"], "-", "\[Mu]", "+", "1"]], ")"]], "j"], Pochhammer] ( μ - n 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", FractionBox["n", "2"]]], ")"]], "j"], Pochhammer] ( 2 μ + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]], "k"], Pochhammer] - z I - n 2 + μ - 1 ( z 2 ) Γ ( μ - n 2 ) k = 0 n j = 0 k - 1 2 ( - 1 ) k - j 2 2 k - 4 j z 2 j - k ( - j + k - 1 ) ! ( - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], Pochhammer] ( - n 2 + μ + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], "+", "\[Mu]", "+", "1"]], ")"]], "k"], Pochhammer] ( 2 μ - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "-", "n"]], ")"]], "k"], Pochhammer] j ! k ! ( - 2 j + k - 1 ) ! ( - k + n 2 - μ + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "+", FractionBox["n", "2"], "-", "\[Mu]", "+", "1"]], ")"]], "j"], Pochhammer] ( - n 2 + μ + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], "+", "\[Mu]", "+", "1"]], ")"]], "j"], Pochhammer] ( 2 μ + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]], "k"], Pochhammer] ) /; n Condition WhittakerM n 2 -1 μ z 2 -1 n 2 μ -2 z n 1 2 -1 4 BesselI μ -1 n 2 -1 z 2 -1 Gamma -1 n 2 -1 μ 1 j 0 k 2 -1 k 0 n -1 k -1 j 2 2 k -1 4 j z 2 j -1 k k -1 j Pochhammer -1 n k Pochhammer -1 n 2 -1 μ 1 k Pochhammer 2 μ -1 n k j k k -1 2 j Pochhammer -1 k n 2 -1 -1 μ 1 j Pochhammer μ -1 n 2 -1 j Pochhammer 2 μ 1 k -1 -1 z BesselI -1 n 2 -1 μ -1 z 2 -1 Gamma μ -1 n 2 -1 j 0 k -1 2 -1 k 0 n -1 k -1 j 2 2 k -1 4 j z 2 j -1 k -1 j k -1 Pochhammer -1 n k Pochhammer -1 n 2 -1 μ 1 k Pochhammer 2 μ -1 n k j k -2 j k -1 Pochhammer -1 k n 2 -1 -1 μ 1 j Pochhammer -1 n 2 -1 μ 1 j Pochhammer 2 μ 1 k -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerM", "[", RowBox[List[FractionBox["n_", "2"], ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "n", "+", RowBox[List["2", " ", "\[Mu]"]]]]], " ", SuperscriptBox["z", FractionBox[RowBox[List["1", "+", "n"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], "+", "\[Mu]"]], ",", FractionBox["z", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["n", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox["k", "2"], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "j"]], "+", RowBox[List["2", " ", "k"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["n", "2"], "+", "\[Mu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", FractionBox["n", "2"], "-", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], "+", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]]]]]]]]]]], "-", RowBox[List["z", " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["n", "2"], "+", "\[Mu]"]], ",", FractionBox["z", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], "+", "\[Mu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "k"]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "j"]], "+", RowBox[List["2", " ", "k"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["n", "2"], "+", "\[Mu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "j"]], "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", FractionBox["n", "2"], "-", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["n", "2"], "+", "\[Mu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]]]]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02