Jonas Lara jonas1ara

I'm Jonas Lara

A mathematician

Fourier series, a poem ...

The Fourier series is a branch of Fourier analysis that aims to decompose a periodic function into a sum of exponentials (or trigonometric functions) with different frequencies and magnitudes, in this particular demonstration, we are defining $f(t)$ to be a periodic complex function with $t\in[0, 1]$

Represent $f(t)$ as a sum of exponential functions rotating at frequencies of $0, 1, -1, 2, -2, ...$ rotations per $t$:

$$ f(t) = \dots + c_{-2}e^{-2\cdot 2\pi it} + c_{-1}e^{-1\cdot 2\pi it} + c_{0}e^{0\cdot 2\pi it} + c_{1}e^{1\cdot 2\pi it} + c_{2}e^{2\cdot 2\pi it} + \dots $$

Taking the integral of the function across its domain:

$$ \int_0^1 f(t) dt $$

Expanding $f(t)$ in terms of its Fourier series:

$$ \int_0^1 f(t) dt = \int_0^1 (\dots + c_{-1}e^{-1\cdot 2\pi it} + c_{0}e^{0\cdot 2\pi it} + c_{1}e^{1\cdot 2\pi it} + \dots)dt $$

Distributing the definite integral:

$$ \int_0^1 f(t) dt = \dots + \int_0^1c_{-1}e^{-1\cdot 2\pi it}dt + \int_0^1c_{0}e^{0\cdot 2\pi it}dt + \int_0^1c_{1}e^{1\cdot 2\pi it}dt + \dots $$

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