Mathematics, Physics

Fourier Fun

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Fourier Series

Allows us to expand any periodic funciton on the range (-L,L) in terms of sinusoidal functions that are periodic on that interval

f(x)=\sum _{ n=0 }^{ \infty }{ A_{ n }\cos (\frac { n\pi /*x }{ L } ) } +\sum _{ n=0 }^{ \infty }{ B_{ n }\sin (\frac { n\pi x }{ L } ) }

  • Recall Euler’s formula: e^z = e^{s+it} = e^s e^{it} = e^{s}(\cos(t)+i\sin(t))
    • Euler’s identity gives e^{i\pi} + 1 = 0 (i.e., e^{i/pi}=-1)
    • This is a consequence of Euler’s formula:
      • e^{ix} = \cos x + i \sin x  ⇒ let x = \pi e^{i\pi} = \cos(\pi) + i\sin(\pi) = -1 + i(0) = -1
  • Since the sines and cosines can be combined into a complex exponential, we can use this equivalency to simplify f(x) into a single term:

f(x) = \sum_{n=-\infty}^{\infty}{a_n e^{in\pi x/L}}

where a_n=\frac{1}{2L} \int _{ -L}^{L}{f(x)e^{-in\pi x/L}dx}


Review of sine and cosine:

Cosine Function

  • Even function ⇒ \cos(-x) = cos(x) ⇒ symmetric
  • \int _{-\pi}^{\pi}{\cos(\theta)d\theta} = 0
cosine function
Figure 1 (generated using MATLAB)

Sine Function

  • Odd function ⇒ \sin(-x) = -\sin(x) 1 ⇒ antisymmetric
  • \int _{-\pi}^{\pi}{\sin(\theta)d\theta} = 0
sine function
Figure 2 (generated using MATLAB)

The Fourier Transform

Note: a majority of these equations I got from the first reference  I listed

Let g(t) be a function of time and G(\omega) be a function of frequency

  • Aside: \omega \equiv 2\pi \upsilon where \omega = angular frequency and \upsilon = oscillation frequency
  • Then, the FT of g(t) (if it exists) is …

G(\omega) = \mathcal{F} \{g(t) \} = \sqrt { \frac { |b| }{ (2\pi )^{ 1-a } } } \int _{ -\infty }^{ \infty }{g(t)e^{ib\omega t}dt}

g(t) = \mathcal{F}^{-1} \{g(t) \} = \mathcal{F} \{G(\omega ) \} = \sqrt { \frac { |b| }{ (2\pi )^{ 1+a } } } \int _{ -\infty }^{ \infty }{G(\omega)e^{-ib\omega t}d\omega}

Some common parameter choices

  • Physics and Mathematica default: a = 0, b =1

G(\omega) = \sqrt { \frac { |1| }{ (2\pi )^{ 1-0 } } } \int _{ -\infty }^{ \infty }{g(t)e^{i(1)\omega t}dt} =  \sqrt { \frac { 1 }{ (2\pi ) } } \int _{ -\infty }^{ \infty }{g(t)e^{i\omega t}dt} 

G(\omega) = \sqrt { \frac { 1 }{ (2\pi ) } } \int _{ -\infty }^{ \infty }{g(t)e^{i\omega t}dt} 

  • Pure mathematics and systems engineering: $a=1, b =-1$

G(\omega) = \sqrt { \frac { |-1| }{ (2\pi )^{ 1-1 } } } \int _{ -\infty }^{ \infty }{g(t)e^{i(-1)\omega t}dt} =  \sqrt { \frac { 1 }{ (2\pi )^0 } } \int _{ -\infty }^{ \infty }{g(t)e^{-i\omega t}dt} = \int _{ -\infty }^{ \infty }{g(t)e^{-i\omega t}dt}

G(\omega) =  \int _{ -\infty }^{ \infty }{g(t)e^{-i\omega t}dt}

  • Classical physics: $a=-1, b=1$

G(\omega) = \sqrt { \frac { |1| }{ (2\pi )^{ 1-(-1) } } } \int _{ -\infty }^{ \infty }{g(t)e^{i(1)\omega t}dt} =  \sqrt { \frac { 1 }{ (2\pi )^2 } } \int _{ -\infty }^{ \infty }{g(t)e^{i\omega t}dt} = \frac{1}{2\pi }\int _{ -\infty }^{ \infty }{g(t)e^{-i\omega t}dt}

G(\omega) = \frac{1}{2\pi }\int _{ -\infty }^{ \infty }{g(t)e^{-i\omega t}dt}

  • Signal processing: $a = 0, b = -2\pi $

G(\omega) = \sqrt { \frac { |-2\pi| }{ (2\pi )^{ 1-(0) } } } \int _{ -\infty }^{ \infty }{g(t)e^{i(-2\pi)\omega t}dt} = sqrt { \frac { 2\pi }{ 2\pi} } \int _{ -\infty }^{ \infty }{g(t)e^{-i2\pi \omega t}dt} = \int _{ -\infty }^{ \infty }{g(t)e^{-i2\pi \omega t}dt}

G(\omega) = \int _{ -\infty }^{ \infty }{g(t)e^{-i2\pi \omega t}dt}


Fourier Transform in Quantum Mechanics

Note: most of this material comes from the second link listed under my sources.

Conjugate pairs

  • A conjugate pair is a pair of variables that are related to one another via the FT.
  • Two conjugate pairs that exist in nature are:
    1. Time (t) and frequency (\upsilon):
      • G(\upsilon) = \mathcal{F} \{ g(t) \} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}{e^{-i\upsilon t}g(t)dt}
      • g(t) = \mathcal{F} \{ G(\upsilon) \} = \mathcal{F}^{-1} \{ g(t) \} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}{e^{i\upsilon t}G(\upsilon) d\upsilon }
    2. Position (x) and momentum (\rho):
      • \phi(\rho) = \mathcal{F} \{ \Psi(x) \} = \frac{1}{\sqrt{2\pi \hbar}} \int_{-\infty}^{\infty}{e^{-i\rho x/ \hbar} \Psi (x)dx}
      • \Psi(x) = \mathcal{F} \{ \phi (\rho) \} = \mathcal{F}^{-1} \{ \Phi(x) \} = \frac{1}{\sqrt{2\pi \hbar}} \int_{-\infty}^{\infty}{e^{i\rho x/ \hbar} \phi (\rho) d\rho}

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