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Fourier Series in Terms of e.md

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