Announcements 10/6/10 Prayer Exam goes until Saturday a. a.Correction to syllabus: on Saturdays, the Testing Center gives out last exam at 3 pm, closes at 4 pm. Homework survey—survey closes tonight. Please respond this afternoon/evening if you haven’t already. Taylor’s Series review: a. a.cos(x) = 1 – x 2 /2! + x 4 /4! – x 6 /6! + … b. b.sin(x) = x – x 3 /3! + x 5 /5! – x 7 /7! + … c. c.e x = 1 + x + x 2 /2! + x 3 /3! + x 4 /4! + …

Reminder What is  ? What is k? How do they relate to the velocity? Relationship between  and T Relationship between k and Consistency check:  /k = ?

Reading Quiz What’s the complex conjugate of: a. a. b. b. c. c. d. d.

Complex Numbers – A Summary What is “i”? What is “-i”? The complex plane Complex conjugate a. a.Graphically, complex conjugate = ? Polar vs. rectangular coordinates a. a.Angle notation, “A  ” Euler’s equation…proof that e i  = cos  + isin  a. a.  must be in radians b. b.Where is 10e i(  /6) located on complex plane? What is the square root of 1… 1 or -1?

Complex Numbers, cont. Adding a. a.…on complex plane, graphically? Multiplying a. a.…on complex plane, graphically? b. b.How many solutions are there to x 2 =1? c. c.What are the solutions to x 5 =1? (x  x  x  x  x=1) Subtracting and dividing a. a.…on complex plane, graphically?

Polar/rectangular conversion Warning about rectangular-to-polar conversion: tan -1 (-1/2) = ? a. a.Do you mean to find the angle for (2,-1) or (-2,1)? Always draw a picture!!

Using complex numbers to add sines/cosines Fact: when you add two sines or cosines having the same frequency (with possibly different amplitudes and phases), you get a sine wave with the same frequency! (but a still-different amplitude and phase) a. a.“Proof” with Mathematica… (class make up numbers) Worked problem: how do you find mathematically what the amplitude and phase are? Another worked problem?

Using complex numbers to solve equations Simple Harmonic Oscillator (ex.: Newton 2 nd Law for mass on spring) Guess a solution like what it means, really: (and take Re{ … } of each side)

Complex numbers & traveling waves Traveling wave: A cos(kx –  t +  ) Write as: Often: …or – – where “A-tilde” = a complex number, the phase of which represents the phase of the wave – – often the tilde is even left off