Complex numbers Make math easy

Leonhard Euler (1707- 1783) 𝑒 𝑖𝜃 =𝑐𝑜𝑠𝜃+𝑖 𝑠𝑖𝑛𝜃

Trig vs. complex exponential

…and now your joke(s) of the day

Merry Go-Round 1) Plot the complex number z= 2 𝑒 𝑖 2𝜋 𝑡 for values of t = 0, 1/6, 1/3, 1/2, 2/3, 5/6, 1, 7/6, 4/3, 3/2, 5/3, 11/6, 2. 2) Do the same as above but for the complex number z=4 𝑒 𝑖 2𝜋 2 𝑡 3) Do the same as above, but for the complex number z=4 𝑒 𝑖 (2𝜋 2 𝑡+ 𝜋 2 ) 4) Plot the projection onto the real axis for your answers in 2 and 3.

Phasor = Rotating vector in complex plane

Phasor = Magnitude and Phase over time sweeps out sinusoid https://en.wikipedia.org/wiki/Phasor#/media/File:Unfasor.gif

Traveling Waves Angular frequency w = 2pf Wavenumber k = 2p/l Animation credit: Dan Russel, Penn St. : http://www.acs.psu.edu/drussell/Demos/wave-x-t/wave-x-t.html Angular frequency w = 2pf How wave varies with time Wavenumber k = 2p/l How wave varies in space

Describing a cosine wave with phasors 5 4.33 2.5

How about a sine wave?

Phasors! w Rotates in time coswt -coswt sinwt -sinwt w Another great animation at: http://resonanceswavesandfields.blogspot.com/2007/08/phasors.html Rotates in time But we just read off the magnitude and phase information

Applications: electronics I’m all about the bass

Low Pass Filter Example: Vout and Vin

Phasors! Rotates in time coswt -coswt sinwt -sinwt Another great animation at: http://resonanceswavesandfields.blogspot.com/2007/08/phasors.html Rotates in time But we just read off the magnitude and phase information

Application: Mechanical Vibrations

Optics: spectroscopy and lasers Fabry-Perot Interferometer Lasers

Inside the interferometer L

Quantum physics Schrodinger Equation Particle in a Box

Image sources Euler postage stamp: http://www.kidzsearch.com/wiki/File:Euler-10_Swiss_Franc_banknote_(front).jpg Electronics: http://drstienecker.com/tech-261-material/16-resistorinductorcapacitor-circuits-chapter-16/ Mechanical Vibrations: http://aviation.trendolizer.com/2015/07/faa-to-issue-airworthiness-directive-for-boeing-747-8-wing-flutter.html Kenmore washer (google image search) Fabry-Perot Interferometer: http://www.phy.davidson.edu/stuhome/cabell_f/diffractionfinal/pages/fabry.htm http://web.mit.edu/2.710/Fall06/2.710-wk8-a-ho.pdf