?A finite Fourier series is given by the sum\(f(x)=\sum_{n=1}^{N} a_{n} \sin n

Chapter 6, Problem 78

(choose chapter or problem)

A finite Fourier series is given by the sum

$f(x)=\sum_{n=1}^{N} a_{n} \sin n x$

$=a_{1} \sin x+a_{2} \sin 2 x+\cdots+a_{N} \sin N x$

Use the result of Exercise 76 to show that the $mth$ coefficient ${a}_{m}$ is given by the formula

$a_{m}=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin m x d x$

Equation Transcription:

$f(x)=\sum\limits_{n=1}^{N}{a}_{n\ }sin\ nx$

$={a}_{1\ }sin\ x\ +\ {a}_{x\ }sin\ 2x+$ ⠂⠂⠂+ ${a}_{N}\ sin\ Nx$

${a}_{m}=\frac{1}{\pi{}}\int\limits_{-\pi{}}^{\pi{}}f(x)\ sin\ mx\ dx$

Text Transcription:

f(x) = subscript n=1 N a_n sin nx

=a_1 sin x + a_x sin 2x + dot dot dot + a_N sin Nx

A_m = 1/pi integral -pi pi f(x) sin mx dx

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