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Mathematics LibreTexts

11.3E: Fourier Series II (Exercises)

  • Page ID
    18286
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    In Exercises [exer:11.3.1}- [exer:11.3.10} find the Fourier cosine series.

    [exer:11.3.1] \(f(x)=x^2\); \([0,L]\)

    [exer:11.3.2] \(f(x)=1-x\); \([0,1]\)

    [exer:11.3.3] \(f(x)=x^2-2Lx\); \([0,L]\)

    [exer:11.3.4] \(f(x)=\sin kx\)  (\(k\ne\) integer);\([0,\pi]\)

    [exer:11.3.5] \(f(x)= \left\{\begin{array}{cl} 1,&0\le x\le{L\over2}\\0,&{L\over2}<x<L; \end{array}\right.\) \([0,L]\)

    [exer:11.3.6] \(f(x)=x^2-L^2\); \([0,L]\)

    [exer:11.3.7] \(f(x)=(x-1)^2\); \([0,1]\)

    [exer:11.3.8] \(f(x)=e^x\); \([0,\pi]\)

    [exer:11.3.9] \(f(x)=x(L-x)\); \([0,L]\)

    [exer:11.3.10] \(f(x)=x(x-2L)\); \([0,L]\)

    [exer:11.3.11] \(f(x)=1\); \([0,L]\)

    [exer:11.3.12] \(f(x)=1-x\); \([0,1]\)

    [exer:11.3.13] \(f(x)=\cos kx\)  (\(k\ne\) integer); \([0,\pi]\)

    [exer:11.3.14] \(f(x)= \left\{\begin{array}{cl} 1,&0\le x\le{L\over2}\\0,&{L\over2}<x<L; \end{array}\right.\) \([0,L]\)

    [exer:11.3.15] \(f(x)= \left\{\begin{array}{cl} x,&0\le x\le{L\over2},\\L-x,&{L\over2}\le x\le L; \end{array}\right.\) \([0,L]\).

    [exer:11.3.16] \(f(x)=x\sin x\); \([0,\pi]\)

    [exer:11.3.17] \(f(x)=e^x\); \([0,\pi]\)

    [exer:11.3.18] \(f(x)=1\); \([0,L]\)

    [exer:11.3.19] \(f(x)=x^2\); \([0,L]\)

    [exer:11.3.20] \(f(x)=x\); \([0,1]\)

    [exer:11.3.21] \(f(x)= \left\{\begin{array}{cl} 1,&0\le x\le{L\over2}\\0,&{L\over2}<x<L; \end{array}\right.\) \([0,L]\)

    [exer:11.3.22] \(f(x)=\cos x\); \([0,\pi]\)

    [exer:11.3.23] \(f(x)=\sin x\); \([0,\pi]\)

    [exer:11.3.24] \(f(x)=x(L-x)\); \([0,L]\)

    [exer:11.3.25] \(f(x)=1\); \([0,L]\)

    [exer:11.3.26] \(f(x)=x^2\); \([0,L]\)

    [exer:11.3.27] \(f(x)= \left\{\begin{array}{cl} 1,&0\le x\le{L\over2}\\0,&{L\over2}<x<L; \end{array}\right.\)\([0,L]\)

    [exer:11.3.28] \(f(x)=\cos x\); \([0,\pi]\)

    [exer:11.3.29] \(f(x)=\sin x\); \([0,\pi]\)

    [exer:11.3.30] \(f(x)=x(L-x)\); \([0,L]\).

    [exer:11.3.31] \(f(x)=3x^2(x^2-2L^2)\)

    [exer:11.3.32] \(f(x)=x^3(3x-4L)\)

    [exer:11.3.33] \(f(x)=x^2(3x^2-8Lx+6L^2)\)

    [exer:11.3.34] \(f(x)=x^2(x-L)^2\)

    [exer:11.3.35]

    Prove Theorem [thmtype:11.3.5}

    b

    .

    In addition to the assumptions of Theorem [thmtype:11.3.5}

    b

    , suppose \(f''(0)=f''(L)=0\), \(f'''\) is continuous, and \(f^{(4)}\) is piecewise continuous on \([0,L]\). Show that

    \[b_n={2L^3\over n^4\pi^4}\int_0^L f^{(4)}(x)\sin{n\pi x\over L}\,dx, \quad n\ge1.\]

    [exer:11.3.36] \(f(x)=x(L-x)\)

    [exer:11.3.37] \(f(x)=x^2(L-x)\)

    [exer:11.3.38] \(f(x)=x(L^2-x^2)\)

    [exer:11.3.39] \(f(x)=x(x^3-2Lx^2+L^3)\)

    [exer:11.3.40] \(f(x)=x(3x^4-10L^2x^2+7L^4)\)

    [exer:11.3.41] \(f(x)=x(3x^4-5Lx^3+2L^4)\)

    [exer:11.3.42]

    Prove Theorem [thmtype:11.3.5}

    c

    .

    In addition to the assumptions of Theorem [thmtype:11.3.5}

    c

    , suppose \(f''(L)=0\), \(f''\) is continuous, and \(f'''\) is piecewise continuous on \([0,L]\). Show that

    \[c_n={16L^2\over(2n-1)^3\pi^3}\int_0^L f'''(x)\sin{(2n-1)\pi x\over2L} \,dx,\quad n\ge1.\]

    [exer:11.3.43] \(f(x)=x^2(L-x)\)

    [exer:11.3.44] \(f(x)=L^2-x^2\)

    [exer:11.3.45] \(f(x)=L^3-x^3\)

    [exer:11.3.46] \(f(x)=2x^3+3Lx^2-5L^3\)

    [exer:11.3.47] \(f(x)=4x^3+3Lx^2-7L^3\)

    [exer:11.3.48] \(f(x)=x^4-2Lx^3+L^4\)

    [exer:11.3.49] \(f(x)=x^4-4Lx^3+6L^2x^2-3L^4\)

    [exer:11.3.50]

    Prove Theorem [thmtype:11.3.5}

    d

    .

    In addition to the assumptions of Theorem [thmtype:11.3.5}

    d

    , suppose \(f''(0)=0\), \(f''\) is continuous, and \(f'''\) is piecewise continuous on \([0,L]\). Show that

    \[d_n=-{16L^2\over(2n-1)^3\pi^3}\int_0^L f'''(x)\cos{(2n-1)\pi x\over2L} \,dx,\quad n\ge1.\]

    [exer:11.3.51] \(f(x)=x(2L -x)\)

    [exer:11.3.52] \(f(x)=x^2(3L-2x)\)

    [exer:11.3.53] \(f(x)=(x-L)^3+L^3\)

    [exer:11.3.54] \(f(x)=x(x^2-3L^2)\)

    [exer:11.3.55] \(f(x)=x^3(3x-4L)\)

    [exer:11.3.56] \(f(x)=x(x^3-2Lx^2+2L^3)\)

    [exer:11.3.57] Show that the mixed Fourier cosine series of \(f\) on \([0,L]\) is the restriction to \([0,L]\) of the Fourier cosine series of

    \[f_3(x)= \left\{\begin{array}{cl} f(x),&0\le x\le L,\\-f(2L-x),&L< x\le 2L \end{array}\right.\]

    on \([0,2L]\). Use this to prove Theorem [thmtype:11.3.3}.

    [exer:11.3.58] Show that the mixed Fourier sine series of \(f\) on \([0,L]\) is the restriction to \([0,L]\) of the Fourier sine series of

    \[f_4(x)= \left\{\begin{array}{cl} f(x),&0\le x\le L,\\f(2L-x),&L< x\le 2L \end{array}\right.\]

    on \([0,2L]\). Use this to prove Theorem [thmtype:11.3.4}.

    [exer:11.3.59] Show that the Fourier sine series of \(f\) on \([0,L]\) is the restriction to \([0,L]\) of the Fourier sine series of

    \[f_3(x)= \left\{\begin{array}{cl} f(x),&0\le x\le L,\\-f(2L-x),&L< x\le 2L \end{array}\right.\]

    on \([0,2L]\).

    [exer:11.3.60] Show that the Fourier cosine series of \(f\) on \([0,L]\) is the restriction to \([0,L]\) of the Fourier cosine series of

    \[f_4(x)= \left\{\begin{array}{cl} f(x),&0\le x\le L,\\f(2L-x),&L< x\le 2L \end{array}\right.\]

    on \([0,2L]\). The next theorem gives sufficient conditions for convergence of a Fourier series. The proof is beyond the scope of this book. \begin{theorem}\color{blue} \label{thmtype:11.2.4} If $f$ is piecewise smooth on $[-L,L]$, then the Fourier series \begin{equation} \label{eq:11.2.8} F(x)=a_0+\sum_{n=1}^\infty\left(a_n\cos{n\pi x\over L}+b_n\sin{n\pi x\over L}\right) \end{equation} of $f$ on $[-L,L]$ converges for all $x$ in $[-L,L];$ moreover$,$ $$ F(x)= \left\{\begin{array}{cl} f(x)&\mbox{if $-L<l$>\[5pt] \{f(x-)+f(x+)\over2}&\mbox{if $-L<l$>\[5pt] \{f(-L+)+f(L-)\over2}&\mbox{if $x=L$ or $x=-L$.} \end{array}\right. $$ \end{theorem} $$ C(x)= \left\{\begin{array}{cl} f(0+)&\mbox{if }x=0 \\ f(x)& if $0<l$>\[5pt] \{f(x-)+f(x+)\over2}&\mbox{if $0<l$>\[5pt] f(L-)&\mbox{if }x=L. \end{array}\right. $$ $$ S(x)= \left\{\begin{array}{cl} 0&\mbox{if }x=0 \\ f(x)&\mbox{if $0<l$>\[5pt] \{f(x-)+f(x+)\over2}&\mbox{if $0<l$>\[5pt] 0&\mbox{if }x=L. \end{array}\right. $$ $$ C_M(x)= \left\{\begin{array}{cl} f(0+) & if $x=0$ \\ f(x)&\mbox{if $0<l$>\[5pt] \{f(x-)+f(x+)\over2}&\mbox{if $0<l$>\[5pt] 0&\mbox{if $x=L$.} \end{array}\right. $$ $$ S_M(x)= \left\{\begin{array}{cl} 0&\mbox{if $x=0$} \\ f(x)&\mbox{if $0<l$>\[5pt] \{f(x-)+f(x+)\over2}&\mbox{if $0<l$>\[5pt] f(L-)&\mbox{if $x=L$.} \end{array}\right. $$