Transcript complex numbers

Announcements 10/6/10
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Prayer
Exam goes until Saturday
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. cos(x) = 1 – x2/2! + x4/4! – x6/6! + …
b. sin(x) = x – x3/3! + x5/5! – x7/7! + …
c. ex = 1 + x + x2/2! + x3/3! + x4/4! + …
Reminder
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What is w?
What is k?
How do they relate to the velocity?
Relationship between w and T
Relationship between k and l
Consistency check: w/k = ?
Reading Quiz
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1  3i
What’s the complex conjugate of:
4  5i
a.
b.
c.
d.
1  3i
4  5i
1  3i
4  5i
1  3i
4  5i
1  3i
4  5i
Complex Numbers – A Summary
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What is the square
What is “i”?
root of 1… 1 or -1?
What is “-i”?
The complex plane
Complex conjugate
a. Graphically, complex conjugate = ?
Polar vs. rectangular coordinates
a. Angle notation, “Aq”
Euler’s equation…proof that eiq = cosq + isinq
a. q must be in radians
b. Where is 10ei(p/6) located on complex plane?
Complex Numbers, cont.
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Adding
a. …on complex plane, graphically?
Multiplying
a. …on complex plane, graphically?
b. How many solutions are there to x2=1?
c. What are the solutions to x5=1? (xxxxx=1)
Subtracting and dividing
a. …on complex plane, graphically?
Polar/rectangular conversion
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Warning about rectangular-to-polar conversion:
tan-1(-1/2) = ?
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
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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. “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
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Simple Harmonic Oscillator
(ex.: Newton 2nd Law for mass on spring)
2
d x
k
 x
2
m
dt
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Guess a solution like x(t )  Ae
iwt  
what it means, really: x(t )  A cos(wt   )
(and take Re{ … } of each side)
Complex numbers & traveling waves
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Traveling wave: A cos(kx – wt + )
Write as: f (t )  Ae
i kx wt  
i i kx wt 
Often: f (t )  Ae e
…or f (t )  Ae
i kx wt 
– where “A-tilde” = a complex number, the phase
of which represents the phase of the wave
– often the tilde is even left off