[latexpage]

### Sources:

- http://mathworld.wolfram.com/FourierTransform.html
- http://www4.ncsu.edu/~franzen/public_html/CH736/math/ft/ft.html

### Recommended Videos:

::

### Fourier Series

Allows us to expand any periodic funciton on the range in terms of sinusoidal functions that are periodic on that interval

- Recall Euler’s formula:
- Euler’s identity gives (i.e., )
- This is a consequence of Euler’s formula:
- ⇒ let

- Since the sines and cosines can be combined into a complex exponential, we can use this equivalency to simplify into a single term:

where

### Review of sine and cosine:

#### Cosine Function

- Even function ⇒ ⇒ symmetric

#### Sine Function

- Odd function ⇒ 1 ⇒ antisymmetric

### The Fourier Transform

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

Let be a function of time and be a function of frequency

- Aside: where angular frequency and oscillation frequency
- Then, the FT of (if it exists) is …

#### Some common parameter choices

- Physics and Mathematica default:

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

- Classical physics: $a=-1, b=1$

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

### 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:
- Time () and frequency ():
- Position () and momentum ():

- Time () and frequency ():