26 lines
787 B
Markdown
26 lines
787 B
Markdown
#Math #Calculus
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# Proof
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Let's express a Fourier Series as:
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$$
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v = \frac {2\pi nx} P \\
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f(x) = \sum _{n = 0}^\infty A_n \cos v + B_n \sin v
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$$
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We can deduce:
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$$
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f(x) = \sum _{n = 0}^{\infty} \frac {A_n e^{iv} + A_n e^{-iv} - iB_n e^{iv} + iB_n e^{-iv}} 2 \\
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= \sum _{n = 0}^{\infty} 0.5(A_n + iB_n)e^{-iv} + 0.5(A_n - iB_n)e^{iv} \\
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= \sum _{n = 0}^{\infty} \frac {e^{-iv}} P \int _{-P/2}^{P/2} f(x) (\cos v + i\sin v) dx + \frac {e^{iv}} P \int _{-P/2}^{P/2} f(x) (\cos -v + i\sin -v) dx \\
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= \sum _{n = 0}^{\infty} \frac {e^{-iv}} P \int _{-P/2}^{P/2} f(x)e^{iv} dx + \frac {e^{iv}} P \int _{-P/2}^{P/2} f(x)e^{-iv} dx \\
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= \sum _{n = -\infty}^{\infty} \frac {e^{iv}} P \int _{-P/2}^{P/2} f(x)e^{-iv} dx
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$$
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## Definitions
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Definitions of $A_n$ and $B_n$:
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[[Fourier Series Proof]] |