|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/05.05.25.0003.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
f[x] == Sum[Subscript[c, n] Subscript[\[Psi], n][x], {n, 0, Infinity}] /;
Subscript[c, n] == Integrate[Subscript[\[Psi], n][t] f[t], {t, -1, 1}] &&
Subscript[\[Psi], n][x] == Sqrt[2/Pi] (1 - x^2)^(1/4) ChebyshevU[n, x] &&
-1 < x < 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["f", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["c", "n"], RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "x", "]"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "n"], "\[Equal]", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "t", "]"]], RowBox[List["f", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "x", "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["x", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List["ChebyshevU", "[", RowBox[List["n", ",", "x"]], "]"]]]]]], "\[And]", RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "1"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> n </mi> </msub> <mo> ⁢ </mo> <mrow> <msub> <mi> ψ </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <mrow> <mrow> <msub> <mi> ψ </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <msub> <mi> ψ </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mrow> <msub> <mi> U </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> < </mo> <mi> x </mi> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> f </ci> <ci> x </ci> </apply> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> n </ci> </apply> <apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> n </ci> </apply> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> n </ci> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> n </ci> </apply> <ci> t </ci> </apply> <apply> <ci> f </ci> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> n </ci> </apply> <ci> x </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <ci> ChebyshevU </ci> <ci> n </ci> <ci> x </ci> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["f", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["c", "n"], " ", RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "x", "]"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "n"], "\[Equal]", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "t", "]"]], " ", RowBox[List["f", "[", "t", "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "x", "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["x", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List["n", ",", "x"]], "]"]]]]]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "1"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|