1. Complex numbers
Make math easy

2. Leonhard Euler ( )
𝑒 𝑖𝜃 =𝑐𝑜𝑠𝜃+𝑖 𝑠𝑖𝑛𝜃

3. Trig vs. complex exponential

4. …and now your joke(s) of the day

5. 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.

6. Phasor = Rotating vector in complex plane

7. Phasor = Magnitude and Phase over time sweeps out sinusoid

8. Traveling Waves Angular frequency w = 2pf Wavenumber k = 2p/l
Animation credit: Dan Russel, Penn St. :
Angular frequency w = 2pf
How wave varies with time
Wavenumber k = 2p/l
How wave varies in space

9. Describing a cosine wave with phasors5
4.33
2.5

10. How about a sine wave?

11. Phasors! w Rotates in time
coswt
-coswt
sinwt
-sinwt w
Another great animation at:
Rotates in time
But we just read off the magnitude and phase information

12. Applications: electronics
I’m all about the bass

13. Low Pass Filter Example: Vout and Vin

14. Phasors! Rotates in time
coswt
-coswt
sinwt
-sinwt
Another great animation at:
Rotates in time
But we just read off the magnitude and phase information

15. Application: Mechanical Vibrations

16. Optics: spectroscopy and lasers
Fabry-Perot Interferometer
Lasers

17. Inside the interferometerL

18. Quantum physics
Schrodinger Equation
Particle in a Box

19. Image sources
Euler postage stamp:
Electronics:
Mechanical Vibrations:
Kenmore washer (google image search)
Fabry-Perot Interferometer: