Re: Integrate vs Nintegrate for impulsive functions
- To: mathgroup at smc.vnet.net
- Subject: [mg61760] Re: Integrate vs Nintegrate for impulsive functions
- From: Fred Bartoli <fred._canxxxel_this_bartoli at RemoveThatAlso_free.fr_AndThisToo>
- Date: Fri, 28 Oct 2005 03:25:46 -0400 (EDT)
- References: <djn3na$inr$1@smc.vnet.net>
- Reply-to: Fred Bartoli <fred._canxxxel_this_bartoli at RemoveThatAlso_free.fr_AndThisToo>
- Sender: owner-wri-mathgroup at wolfram.com
Try to evaluate the integral with symbolic coefficients. It gives the same result as Nintegrate. -- Thanks, Fred. Cell[CellGroupData[{Cell[BoxData[{ RowBox[{"Clear", "[", RowBox[{"h", ",", "h1"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"h", "[", "x_", "]"}], "=", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "0.24982234345508192"}], "-", RowBox[{"0.0429732983215806", "*", "\[ImaginaryI]"}]}], ")"}], "*", RowBox[{"Sin", "[", RowBox[{ RowBox[{"(", RowBox[{"3.1734427242687215", "+", RowBox[{ "0.3295480781081674", "*", "\[ImaginaryI]"}]}], ")"}], "*", "x"}], "]"}], "*", RowBox[{"(", RowBox[{ RowBox[{"Cosh", "[", RowBox[{"1000.", "*", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "0.4"}], "+", "x"}], ")"}], "^", "2"}]}], "]"}], "-", RowBox[{"Sinh", "[", RowBox[{"1000.", "*", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "0.4"}], "+", "x"}], ")"}], "^", "2"}]}], "]"}]}], ")"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"h1", "[", "x_", "]"}], "=", RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"\[ImaginaryI]", " ", "b"}]}], ")"}], RowBox[{"Sin", "[", RowBox[{ RowBox[{"(", RowBox[{"c", "+", RowBox[{"\[ImaginaryI]", " ", "d"}]}], ")"}], "x"}], "]"}], RowBox[{"(", RowBox[{ RowBox[{"Cosh", "[", RowBox[{"e", SuperscriptBox[ RowBox[{"(", RowBox[{"f", "+", "x"}], ")"}], "2"]}], "]"}], "-", RowBox[{"Sinh", "[", RowBox[{"e", SuperscriptBox[ RowBox[{"(", RowBox[{"f", "+", "x"}], ")"}], "2"]}], "]"}]}], ")"}]}]}], "\[IndentingNewLine]", RowBox[{"sol", "=", RowBox[{"Integrate", "[", RowBox[{ RowBox[{"h1", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}]}], "]"}]}], "\n", RowBox[{"sol", "/.", RowBox[{"{", RowBox[{ RowBox[{"a", "\[Rule]", RowBox[{"-", "0.24982234345508192"}]}], ",", RowBox[{"b", "\[Rule]", RowBox[{"-", "0.0429732983215806"}]}], ",", RowBox[{"c", "\[Rule]", "3.1734427242687215"}], ",", RowBox[{"d", "\[Rule]", "0.3295480781081674"}], ",", RowBox[{"e", "\[Rule]", "1000"}], ",", RowBox[{"f", "\[Rule]", FractionBox[ RowBox[{"-", "4"}], "10"]}]}], "}"}]}], "\[IndentingNewLine]", }], "Input", CellLabel->"In[7]:="], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "0.24982234345508192`"}], "-", RowBox[{"0.0429732983215806`", " ", "\[ImaginaryI]"}]}], ")"}], " ", RowBox[{"Sin", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"3.1734427242687215`", "\[InvisibleSpace]"}], "+", RowBox[{"0.3295480781081674`", " ", "\[ImaginaryI]"}]}], ")"}], " ", "x"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"Cosh", "[", RowBox[{"1000.`", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "0.4`"}], "+", "x"}], ")"}], "2"]}], "]"}], "-", RowBox[{"Sinh", "[", RowBox[{"1000.`", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "0.4`"}], "+", "x"}], ")"}], "2"]}], "]"}]}], ")"}]}]], "Output", CellLabel->"Out[8]="], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"\[ImaginaryI]", " ", "b"}]}], ")"}], " ", RowBox[{"Sin", "[", RowBox[{ RowBox[{"(", RowBox[{"c", "+", RowBox[{"\[ImaginaryI]", " ", "d"}]}], ")"}], " ", "x"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"Cosh", "[", RowBox[{"e", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"f", "+", "x"}], ")"}], "2"]}], "]"}], "-", RowBox[{"Sinh", "[", RowBox[{"e", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"f", "+", "x"}], ")"}], "2"]}], "]"}]}], ")"}]}]], "Output", CellLabel->"Out[9]="], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"\[ImaginaryI]", " ", "b"}]}], ")"}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ RowBox[{ SuperscriptBox["c", "2"], "-", RowBox[{"d", " ", RowBox[{"(", RowBox[{"d", "-", RowBox[{"4", " ", "e", " ", "f"}]}], ")"}]}], "+", RowBox[{"2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[{"(", RowBox[{"d", "+", RowBox[{"2", " ", "e", " ", "f"}]}], ")"}]}]}], RowBox[{"4", " ", "e"}]]}]], " ", SqrtBox["\[Pi]"], " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ "2", " ", "\[ImaginaryI]", " ", "c", " ", "f"}]], " ", RowBox[{"Erfi", "[", FractionBox[ RowBox[{"c", "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"(", RowBox[{"d", "-", RowBox[{"2", " ", "e", " ", "f"}]}], ")"}]}]}], RowBox[{"2", " ", SqrtBox["e"]}]], "]"}]}], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "d", " ", "f"}]], " ", RowBox[{"Erfi", "[", FractionBox[ RowBox[{"c", "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"(", RowBox[{"d", "+", RowBox[{"2", " ", "e", " ", "f"}]}], ")"}]}]}], RowBox[{"2", " ", SqrtBox["e"]}]], "]"}]}], "-", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ "2", " ", "\[ImaginaryI]", " ", "c", " ", "f"}]], " ", RowBox[{"Erfi", "[", FractionBox[ RowBox[{"c", "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"(", RowBox[{"d", "-", RowBox[{"2", " ", "e", " ", RowBox[{"(", RowBox[{"1", "+", "f"}], ")"}]}]}], ")"}]}]}], RowBox[{"2", " ", SqrtBox["e"]}]], "]"}]}], "-", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "d", " ", "f"}]], " ", RowBox[{"Erfi", "[", FractionBox[ RowBox[{"c", "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"(", RowBox[{"d", "+", RowBox[{"2", " ", "e", " ", RowBox[{"(", RowBox[{"1", "+", "f"}], ")"}]}]}], ")"}]}]}], RowBox[{"2", " ", SqrtBox["e"]}]], "]"}]}]}], ")"}]}], RowBox[{"4", " ", SqrtBox["e"]}]]], "Output", CellLabel->"Out[10]="], Cell[BoxData[ RowBox[{ RowBox[{"-", "0.013361219528374293`"}], "-", RowBox[{ "0.002855508349311959`", " ", "\[ImaginaryI]"}]}]], "Output", CellLabel->"Out[11]="] }, Open ]] "Pratik Desai" <pdesai1 at umbc.edu> a écrit dans le message de news:djn3na$inr$1 at smc.vnet.net... > Hi Folks > > > I have an expression resulting from a fourier series (for a 1D wave > equation for a string) (fourier coeffficient) of the form > > h[x_]=(-0.24982234345508192 - 0.0429732983215806*I)* > Sin[(3.1734427242687215 + 0.3295480781081674*I)*x]* > (Cosh[1000.*(-0.4 + x)^2] - Sinh[1000.*(-0.4 + x)^2]) > > I try to integrate this on the domain x(0,1) to get the fourier > coefficient. I get some results that I need help explaining > > > Integrate[h[x],{x,0,1}] > > >>0+0 *I > > NIntegrate[h[x],{x,0,1}] > > >>-0.0133612 - 0.00285551 \[ImaginaryI] > > Is the result from NIntegrate valid > > The initial condition is essentially a smoothed delta function at x=0.4 > > gxx[x_]=E^(-1000.*(-0.4 + x)^2) > > Please advise > > > Regards > > > Pratik . > > -- > Pratik Desai > Graduate Student > UMBC > Department of Mechanical Engineering > Phone: 410 455 8134 > >
- Follow-Ups:
- Re: Re: Integrate vs Nintegrate for impulsive functions
- From: Pratik Desai <pdesai1@umbc.edu>
- Re: Re: Integrate vs Nintegrate for impulsive functions