List of mathematical formulas: “Fourier series. Examples of solutions

List of basic formulas related to Fourier series, as well as examples of solutions:

Basic formulas of Fourier series

1. Formula for expanding a function into a Fourier series:
   For a periodic function ( f(x) ) with period ( T ):
   [
   f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left (\frac{2\pi nx}{T}\right) + b_n \sin\left(\frac{2\pi nx}{T}\right) \right)
   ]
2. Fourier coefficients:

• Zero coefficient:
  [
  a_0 = \frac{1}{T} \int_{0}^{T} f(x) \, dx
  ]
• Coefficients ( a_n ):
  [
  a_n = \frac{2}{T} \int_{0}^{T} f(x) \cos\left(\frac{2\pi nx}{T}\right) \, dx
  ]
• Coefficients ( b_n ):
  [
  b_n = \frac{2}{T} \int_{0}^{T} f(x) \sin\left(\frac{2\pi nx}{T}\right) \, dx
  ]

Examples of solutions

1. Example 1: Rectangular signal function:
   Consider a function ( f(x) ) equal to 1 on the interval ( [0, T/2] ) and -1 on the interval ( [T/2, T] ).

• We calculate ( a_0 ), ( a_n ) and ( b_n ):
  [
  a_0 = 0
  ]
  [
  a_n = 0 \quad (n \geq 1)
  ]
  [
  b_n = \frac{2}{n\pi} (1 – (-1 )^n)
  ]
  Thus, the Fourier series will contain only odd harmonics.

1. Example 2: Function ( f(x) = x ) on the interval ( [-\pi, \pi] ):

• We calculate the coefficients:
  [
  a_0 = 0
  ]
  [
  a_n = 0 \quad (n \geq 1)
  ]
  [
  b_n = \frac{2(-1)^{n+1}}{n}
  ]
  The Fourier series will look like:
  [
  f(x) = \sum_{n=1}^{\infty} \frac{2(-1)^{n+1}}{n} \sin(nx)
  ]