fourier series short intro

7 mins read · March 9, 2024

Fourier series short intro: Proofs of some integral identities.

fourier_series_short_intro.pdf
- Intro 3
- Coefficients 7
- Related trigonometric identities 15
- Integral of sine and cosine 19
- Proofs a ≠ b 22
- Proofs a = b 26
- Integral identities 30
- Interval changes 32
- Other forms 38
- Closing 42

int sin

$$\tag{21} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \sin mx \ dx & = & \displaystyle \left[ -\frac{1}{m} \cos mx \right]_{-\pi}^\pi \newline \newline & = & \displaystyle -\frac{1}{m} \left[ \cos m \pi - \cos m(-\pi) \right] \newline \newline & = & \displaystyle -\frac{1}{m} ( \cos m \pi - \cos m \pi ) \newline \newline & = & 0. \end{array} $$

int cos

$$\tag{22} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \cos mx \ dx & = & \displaystyle \left[ -\frac{1}{m} \sin mx \right]_{-\pi}^\pi \newline \newline & = & \displaystyle -\frac{1}{m} \left[ \sin m \pi - \sin m(-\pi) \right] \newline \newline & = & \displaystyle -\frac{1}{m} ( \sin m \pi + \sin m \pi ) \newline \newline & = & \displaystyle -\frac{1}{m} ( 0 + 0 ) \ \ = \ \ 0. \newline \newline \end{array} $$

int sin · sin, $a \ne b$

$$\tag{23a} \begin{array}{rcl} \sin ax \sin bx & = & \frac12 \cos (ax - bx) - \frac12 \cos (ax + bx) \newline \newline & = & \frac12 \cos (a - b)x - \frac12 \cos (a + b)x \end{array} $$

$$\tag{23b} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \sin ax \sin bx \ dx & = & \displaystyle \tfrac12 \int_{-\pi}^\pi \cos (a - b)x \ dx \newline \newline & & \displaystyle - \tfrac12 \int_{-\pi}^\pi \cos (a + b)x \ dx \newline \newline & = & \displaystyle \tfrac12 \int_{-\pi}^\pi \cos nx \ dx - \tfrac12 \int_{-\pi}^\pi \cos mx \ dx \newline \newline & = & 0 - 0 \ \ = \ \ 0. \end{array} $$

int cos · cos; sin, $a \ne b$

$$\tag{24a} \begin{array}{rcl} \cos ax \cos bx & = & \frac12 \cos (ax + bx) + \frac12 \cos (ax - bx) \newline \newline & = & \frac12 \cos (a + b)x + \frac12 \cos (a - b)x \end{array} $$

$$\tag{24b} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \cos ax \cos bx \ dx & = & \displaystyle \tfrac12 \int_{-\pi}^\pi \cos (a + b)x \ dx \newline \newline & & \displaystyle + \tfrac12 \int_{-\pi}^\pi \cos (a - b)x \ dx \newline \newline & = & \displaystyle \tfrac12 \int_{-\pi}^\pi \cos mx \ dx + \tfrac12 \int_{-\pi}^\pi \cos nx \ dx \newline \newline & = & 0 + 0 \ \ = \ \ 0. \end{array} $$

int sin · cos; sin, $a \ne b$

$$\tag{25a} \begin{array}{rcl} \sin ax \cos bx & = & \frac12 \sin (ax + bx) + \frac12 \sin (ax - bx) \newline \newline & = & \frac12 \sin (a + b)x + \frac12 \sin (a - b)x \end{array} $$

$$\tag{25b} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \sin ax \cos bx \ dx & = & \displaystyle \tfrac12 \int_{-\pi}^\pi \sin (a + b)x \ dx \newline \newline & & \displaystyle + \tfrac12 \int_{-\pi}^\pi \sin (a - b)x \ dx \newline \newline & = & \displaystyle \tfrac12 \int_{-\pi}^\pi \sin mx \ dx + \tfrac12 \int_{-\pi}^\pi \sin nx \ dx \newline \newline & = & 0 + 0 \ \ = \ \ 0. \end{array} $$

int sin · sin, $a = b$

$$\tag{26a} \begin{array}{rcl} \sin ax \sin ax & = & \frac12 \cos (ax - ax) - \frac12 \cos (ax + ax) \newline \newline & = & \frac12 - \frac12 \cos 2ax \end{array} $$

$$\tag{26b} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \sin ax \sin ax \ dx & = & \displaystyle \tfrac12 \int_{-\pi}^\pi dx - \tfrac12 \int_{-\pi}^\pi \cos 2ax \ dx \newline \newline & = & \displaystyle \tfrac12 \int_{-\pi}^\pi dx - \tfrac12 \int_{-\pi}^\pi \cos mx \ dx \newline \newline & = & \pi - 0 \ \ = \ \ \pi. \end{array} $$

int cos · cos; sin, $a = b$

$$\tag{27a} \begin{array}{rcl} \cos ax \cos ax & = & \frac12 \cos (ax + ax) + \frac12 \cos (ax - ax) \newline \newline & = & \frac12 \cos 2ax + \frac12 \end{array} $$

$$\tag{27b} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \cos ax \cos bx \ dx & = & \displaystyle \tfrac12 \int_{-\pi}^\pi \cos 2ax \ dx + \tfrac12 \int_{-\pi}^\pi dx \newline \newline & = & \displaystyle \tfrac12 \int_{-\pi}^\pi \cos mx \ dx + \tfrac12 \int_{-\pi}^\pi dx \newline \newline & = & 0 + \pi \ \ = \ \ \pi. \end{array} $$

int sin · cos; sin, $a = b$

$$\tag{28a} \begin{array}{rcl} \sin ax \cos ax & = & \frac12 \sin (ax + ax) + \frac12 \sin (ax - ax) \newline \newline & = & \frac12 \sin 2ax + 0 \end{array} $$

$$\tag{28b} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \sin ax \cos bx \ dx & = & \displaystyle \tfrac12 \int_{-\pi}^\pi \sin 2ax \ dx \newline \newline & = & \displaystyle \tfrac12 \int_{-\pi}^\pi \sin mx \ dx \newline \newline & = & 0. \end{array} $$

int sin · sin, cos · cos, sin · cos

$$\tag{29} \int_{-\pi}^\pi \sin ax \sin bx \ dx = \left\{ \begin{array}{rcl} 0, & a \ne b \newline \pi, & a = b. \end{array} \right. $$

$$\tag{30} \int_{-\pi}^\pi \cos ax \cos bx \ dx = \left\{ \begin{array}{rcl} 0, & a \ne b \newline \pi, & a = b. \end{array} \right. $$

$$\tag{31} \int_{-\pi}^\pi \cos ax \sin bx \ dx = 0. $$

interval changes

$$\tag{32} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \sin mx \ dx & = & \displaystyle \int_{-\tfrac12 \lambda}^{\tfrac12 \lambda} \sin \left( m\frac{2\pi}{\lambda} x \right) \ dx \newline \newline \displaystyle -\left[ \frac{1}{m} \cos mx \right]^\pi_{-\pi} & = & \displaystyle -\left[ \frac{\lambda}{m2\pi} \cos \left( m \frac{2\pi}{\lambda} x \right) \right]_{-\tfrac12 \lambda}^{\tfrac12 \lambda} \newline \newline \displaystyle \frac{1}{m} [ \cos m \pi - \cos m(-\pi) ] & = & \displaystyle \frac{\lambda}{m2\pi} \left[ \cos \left( m \frac{2\pi}{\lambda} \right) (\tfrac12 \lambda) - \cos \left( m \frac{2\pi}{\lambda} \right) (-\tfrac12 \lambda) \right] \newline \newline & = & \displaystyle \frac{\lambda}{m2\pi} [ \cos m\pi - \cos m(-\pi) ] \newline \newline \displaystyle \frac{1}{m} (\cos m\pi - \cos m\pi) & = & \displaystyle \frac{\lambda}{m2\pi} (\cos m\pi - \cos m\pi) \newline \newline 0 & = & 0. \end{array} $$

$$\tag{33} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \sin mt \ dt & = & \displaystyle \int_{-\tfrac12 T}^{\tfrac12 T} \sin \left( m\frac{2\pi}{T} t \right) \ dt \newline \newline \displaystyle -\left[ \frac{1}{m} \cos mt \right]^\pi_{-\pi} & = & \displaystyle -\left[ \frac{T}{m2\pi} \cos \left( m \frac{2\pi}{T} t \right) \right]_{-\tfrac12 T}^{\tfrac12 T} \newline \newline \displaystyle \frac{1}{m} [ \cos m \pi - \cos m(-\pi) ] & = & \displaystyle \frac{T}{m2\pi} \left[ \cos \left( m \frac{2\pi}{T} \right) (\tfrac12 T) - \cos \left( m \frac{2\pi}{T} \right) (-\tfrac12 T) \right] \newline \newline & = & \displaystyle \frac{\lambda}{m2\pi} [ \cos m\pi - \cos m(-\pi) ] \newline \newline \displaystyle \frac{1}{m} (\cos m\pi - \cos m\pi) & = & \displaystyle \frac{\lambda}{m2\pi} (\cos m\pi - \cos m\pi) \newline \newline 0 & = & 0. \end{array} $$

period

$$\tag{34} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \sin mx \ dx & = & \displaystyle \int_{-\tfrac12 \lambda}^{\tfrac12 \lambda} \sin \left( m\frac{2\pi}{\lambda} x \right) \ dx \newline \newline & = & \displaystyle \int_0^{\lambda} \sin \left( m\frac{2\pi}{\lambda} x \right) \ dx \newline \newline & = & \displaystyle \int_0^{\lambda} \sin ( m k x ) \ dx \ \ = \ \ \int_0^{\lambda} \sin ( k_m x ) \ dx \end{array} $$

$$\tag{35} \begin{array}{rcl} \displaystyle \int_{-\pi}^\pi \sin mt \ dt & = & \displaystyle \int_{-\tfrac12 T}^{\tfrac12 T} \sin \left( m\frac{2\pi}{T} t \right) \ dt \newline \newline & = & \displaystyle \int_0^T \sin \left( m\frac{2\pi}{T} t \right) \ dt \newline \newline & = & \displaystyle \int_0^T \sin ( m \omega t ) \ dt \ \ = \ \ \int_0^T \sin ( \omega_m t) \ dt \end{array} $$

wavenumber and angular requency

$$\tag{36} k_m = m k_1, \ \ \ \ m = 1, 2, 3, .. $$ $$ \lambda_m = \frac{2\pi}{k_m} $$

$$\tag{37} \omega_m = m \omega_1, \ \ \ \ m = 1, 2, 3, .. $$ $$ T_m = \frac{2\pi}{\omega_m} $$

other forms

$$\tag{38} f(x) = \sum_{m=1}^\infty a_m \cos \left[ m \left( \frac{2\pi}{L} \right) x \right] + \sum_{m=1}^\infty b_m \sin \left[ m \left( \frac{2\pi}{L} \right) x \right] + c_0 $$

$$\tag{39} a_m = \frac{2}{L}\int_0^L f(x) \ \cos \left[ m \left( \frac{2\pi}{L} \right) x \right] \ dx $$

$$\tag{40} b_m = \frac{2}{L}\int_0^L f(x) \ \cos \left[ m \left( \frac{2\pi}{L} \right) x \right] \ dx $$

$$\tag{41} c_0 = \frac{1}{L} \int_0^L f(x) \ dx. $$

$$\tag{42} f(t) = \sum_{m=1}^\infty a_m \cos \left[ m \left( \frac{2\pi}{L} \right) t \right] + \sum_{m=1}^\infty b_m \sin \left[ m \left( \frac{2\pi}{L} \right) t \right] + c_0 $$

$$\tag{43} a_m = \frac{2}{T}\int_0^T f(t) \ \cos \left[ m \left( \frac{2\pi}{L} \right) t \right] \ dt $$

$$\tag{44} b_m = \frac{2}{T}\int_0^T f(t) \ \cos \left[ m \left( \frac{2\pi}{L} \right) t \right] \ dt $$

$$\tag{45} c_0 = \frac{1}{T} \int_0^T f(t) \ dt. $$

$$\tag{46} k_m = m \frac{2\pi}{L}, \ \ \ \ m = 1, 2, 3, .. $$

$$\tag{47} \omega_m = m \frac{2\pi}{T}, \ \ \ \ m = 1, 2, 3, .. $$

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