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; }

 FresnelC

 http://functions.wolfram.com/06.33.21.0101.01

 Input Form

 Integrate[z^(\[Alpha] - 1) E^(b z^2) Cosh[c z^2] FresnelC[a z], z] == (1/4) (z^\[Alpha] FresnelC[a z] (Gamma[\[Alpha]/2, 0, (-(b + c)) z^2]/ ((-(b + c)) z^2)^(\[Alpha]/2) + Gamma[\[Alpha]/2, 0, (-(b - c)) z^2]/ ((-(b - c)) z^2)^(\[Alpha]/2)) + ((2 I)/((a^4 z^4)^(\[Alpha]/2) a)) (Sum[((2^((1/2) (-3 + 2 k + \[Alpha])) Pi^((1/2) (-1 - 2 k - \[Alpha])) z^(-1 - 2 k + \[Alpha]) (b + c)^k)/(a^(4 k) ((2 k + \[Alpha]) k!))) (Sqrt[(-I) a^2 z^2] (I a^2 z^2)^(k + \[Alpha]/2) Gamma[k + (\[Alpha] + 1)/2, (-(1/2)) I a^2 Pi z^2] - ((-I) a^2 z^2)^(k + \[Alpha]/2) Sqrt[I a^2 z^2] Gamma[k + (\[Alpha] + 1)/2, (1/2) I a^2 Pi z^2]), {k, 0, Infinity}] + Sum[((2^((1/2) (-3 + 2 k + \[Alpha])) Pi^((1/2) (-1 - 2 k - \[Alpha])) z^(-1 - 2 k + \[Alpha]) (b - c)^k)/ (a^(4 k) ((2 k + \[Alpha]) k!))) (Sqrt[(-I) a^2 z^2] (I a^2 z^2)^(k + \[Alpha]/2) Gamma[k + (\[Alpha] + 1)/2, (-(1/2)) I a^2 Pi z^2] - ((-I) a^2 z^2)^(k + \[Alpha]/2) Sqrt[I a^2 z^2] Gamma[k + (\[Alpha] + 1)/2, (1/2) I a^2 Pi z^2]), {k, 0, Infinity}]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "2"]]]], RowBox[List["Cosh", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "2"], ",", "0", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "2"], ",", "0", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "/", "2"]]]]], "a"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ")"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "k"]], "-", "\[Alpha]"]], ")"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "4"]], " ", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["k", "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], 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RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ")"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "k"]], "-", "\[Alpha]"]], ")"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "4"]], " ", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["k", "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["k", "+", FractionBox["\[Alpha]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["k", "+", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["k", "+", FractionBox["\[Alpha]", "2"]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["k", "+", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"]]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 b z 2 cosh ( c z 2 ) C TagBox["C", FresnelC] ( a z ) z 1 4 ( z α C TagBox["C", FresnelC] ( a z ) ( ( - ( b - c ) z 2 ) - α 2 Γ ( α 2 , 0 , - ( b - c ) z 2 ) + ( - ( b + c ) z 2 ) - α 2 Γ ( α 2 , 0 , - ( b + c ) z 2 ) ) + 2 ( a 4 z 4 ) - α 2 a ( k = 0 2 1 2 ( 2 k + α - 3 ) π 1 2 ( - 2 k - α - 1 ) z - 2 k + α - 1 a - 4 k ( b - c ) k ( 2 k + α ) k ! ( - a 2 z 2 ( a 2 z 2 ) k + α 2 Γ ( α + 1 2 + k , - 1 2 a 2 π z 2 ) - ( - a 2 z 2 ) k + α 2 a 2 z 2 Γ ( α + 1 2 + k , 1 2 a 2 π z 2 ) ) + k = 0 2 1 2 ( 2 k + α - 3 ) π 1 2 ( - 2 k - α - 1 ) z - 2 k + α - 1 a - 4 k ( b + c ) k ( 2 k + α ) k ! ( - a 2 z 2 ( a 2 z 2 ) k + α 2 Γ ( α + 1 2 + k , - 1 2 a 2 π z 2 ) - ( - a 2 z 2 ) k + α 2 a 2 z 2 Γ ( α + 1 2 + k , 1 2 a 2 π z 2 ) ) ) ) z z α -1 b z 2 c z 2 FresnelC a z 1 4 z α FresnelC a z -1 b -1 c z 2 -1 α 2 -1 Gamma α 2 -1 0 -1 b -1 c z 2 -1 b c z 2 -1 α 2 -1 Gamma α 2 -1 0 -1 b c z 2 2 a 4 z 4 -1 α 2 -1 a -1 k 0 2 1 2 2 k α -3 1 2 -2 k -1 α -1 z -2 k α -1 a -4 k b -1 c k 2 k α k -1 -1 a 2 z 2 1 2 a 2 z 2 k α 2 -1 Gamma α 1 2 -1 k -1 1 2 a 2 z 2 -1 -1 a 2 z 2 k α 2 -1 a 2 z 2 1 2 Gamma α 1 2 -1 k 1 2 a 2 z 2 k 0 2 1 2 2 k α -3 1 2 -2 k -1 α -1 z -2 k α -1 a -4 k b c k 2 k α k -1 -1 a 2 z 2 1 2 a 2 z 2 k α 2 -1 Gamma α 1 2 -1 k -1 1 2 a 2 z 2 -1 -1 a 2 z 2 k α 2 -1 a 2 z 2 1 2 Gamma α 1 2 -1 k 1 2 a 2 z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "2"], 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 Date Added to functions.wolfram.com (modification date)

 2001-10-29