物理量の固有関数/メモ の履歴(No.5)

解答†

(1) のとき、

　&math( \int_0^a\varphi_n^*(x)\varphi_m(x)dx &=\frac{2}{\,a\,}\int_0^a\sin(n\pi x/a)\sin(m\pi x/a)dx\\ &=\frac{1}{\,a\,}\int_0^a-\cos\Big((n+m)\pi x/a\Big)+\cos\Big((n-m)\pi x/a\Big)dx\\ &=\frac{1}{\,a\,}\bigg[-\frac{a}{(n+m)\pi}\sin\Big((n+m)\pi x/a\Big)+\\ &\hspace{17.6mm}\frac{a}{(n-m)\pi}\sin\Big((n-m)\pi x/a\Big)\bigg]_0^a\\ &=0 );

(2) のとき、上式の右辺第一項はやはりゼロになるが、第二項は積分内が になって、

　&math( \int_0^a\varphi_n^*(x)\varphi_m(x)dx=\frac{1}{\,a\,}\int_0^a 1\,dx=1 );

(1) と合わせれば、

　&math( \int_0^a\varphi_n^*(x)\varphi_m(x)dx=\delta_{nm} );

(3)

　&math(c_n&=\int_0^1\sqrt 2\sin(n\pi x)f(x)dx\\ &=\int_0^{1/2}\sqrt 2\sin(n\pi x)\,x\,dx+\int_{1/2}^1\sqrt 2\sin(n\pi x)\,(x-1)\,dx\\ &=\int_0^{1}\sqrt 2\sin(n\pi x)\,x\,dx-\int_{1/2}^1\sqrt 2\sin(n\pi x)\,dx\\ &=\Big[-\frac{\sqrt 2}{n\pi}\cos(n\pi x)x\Big]_0^1+\int_0^{1}\frac{\sqrt 2}{n\pi}\cos(n\pi x)\,dx-\Big[-\frac{\sqrt 2}{n\pi}\cos(n\pi x)\Big]_{1/2}^1\\ &=-\frac{\sqrt 2}{n\pi}\cos(n\pi)+\Big[\frac{\sqrt 2}{n^2\pi^2}\sin n\pi\Big]_0^1+ \frac{\sqrt 2}{n\pi}\cos(n\pi)-\frac{\sqrt 2}{n\pi}\cos(n\pi/2)\\ &=-\frac{\sqrt 2}{n\pi}\cos(n\pi/2)\\ &=\begin{cases} 0&(n=2m+1)\\ \displaystyle(-1)^m\frac{\sqrt 2}{n\pi}&(n=2m)\\ \end{cases});

したがって、

　&math(f(x)&=-\sum_{n=1}^\infty\frac{2}{n\pi}\cos(n\pi/2)\sin(n\pi x)\\ &=\sum_{m=1}^\infty \frac{(-1)^m}{m\pi}\sin(2m\pi x)\\ &=-\frac{1}{\pi}\sin(2\pi x)+\frac{1}{2\pi}\sin(4\pi x)-\frac{1}{3\pi}\sin(6\pi x)+\dots);

解説†

Mathematica ソース

LANG:mathematica
FourierSinCoefficient[
  If[x < 1/2, x, x - 1], x, n, 
  FourierParameters -> {1, Pi}
]
c[n_Integer] =
  Integrate[
    Sqrt[2] Sin[n Pi xx] If[xx < 1/2, xx, -1 + xx],
    {xx, 0, 1}
  ]
approx = 
  Table[
    Sum[
      c[n] Sqrt[2] Sin[n Pi x],
      {n, 1, nmax}
    ],
    {nmax, {4, 16, 64, 256}}
];
Plot[
  {approx, If[x < 1/2, x, -1 + x]} // Flatten // Evaluate,
  {x, 0, 1}, ImageSize -> Large, PlotStyle -> {Thick}, 
  BaseStyle -> {FontSize -> 20}, 
  PlotLegends -> (Style[#, FontSize -> 20] & /@ { 
    "n \[LessEqual] 4", "n \[LessEqual] 16", "n \[LessEqual] 64", "n \[LessEqual] 256", 
    "Target"})
]

LANG:mathematica
Plot[
  Table[Sin[a k]/(Pi k), {a, {200, 50, 10}}] // Evaluate, 
  {k, -1, 1}, PlotRange -> {Full, {-15, 65}}, PlotStyle -> Thick, 
  PlotLegends -> {"a=10", "a=50", "a=200"}, PlotPoints -> 200,
  PlotLabel -> "\!\(\*SubscriptBox[\(\[Delta]\), \(a\)]\)(k)"
]