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Using complex exponentials to prove 1+acostheta

  • #1

Homework Statement


Use complex exponentials to prove 1 + acos(theta) + a^2cos(2theta) + a^3cos(3theta)... = (1 - acos(theta))/(1 - 2acos(theta) + a^2)


Homework Equations



euler's e^itheta/2 +e^-itheta/2=2cos(2theta)

The Attempt at a Solution



a^(n)cos(ntheta) = e^nitheta = e^-nitheta

from there i got the series

(a^n(e^itheta)^n)/2

now from here I think I setup the summation formula but this is where I get stuck. Any help is greatly apprecited.
 

Answers and Replies

  • #2
6,054
390
Is |a| < 1?
 
  • #3
Yes, a is a real constant and |a| < 1. sorry about that
 
  • #4
6,054
390
[tex]
a^n \cos n\theta = a^n\frac {e^{in\theta} + e^{-in\theta}} {2}
= \frac {a^ne^{in\theta} + a^ne^{-in\theta}} {2}
= \frac {p^n + q^n} {2}
\\ p = (\ln a)e^{i\theta}, \ |p| < 1
\\ q = (\ln a)e^{-i\theta}, \ |q| < 1
[/tex]

What is the sum of [itex]p^n[/itex] and [itex]q^n[/itex]?
 
  • #5
That makes sense. That will then give me a real and an imaginary result of which I take the real I believe. Also, I apologize for the typo on the first post.
 
  • #6
6,054
390
Well, you can take the real part, but the sum is real anyway.
 

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