
11.3E: Fourier Series II (Exercises)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

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 $$\label{eq:11.2.8} F(x)=a_0+\sum_{n=1}^\infty\left(a_n\cos{n\pi x\over L}+b_n\sin{n\pi x\over L}\right)$$ 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.$$