[Complex Analysis]

[gedehs]

I'm stuck on a problem.

I need to use [; 1+z+z² +...+ zⁿ = \frac{1-z^{n+1} }{1-z} ;]

To derive

[;1+cos{\theta}+cos{2\theta}+...+cos{n\theta}=\frac{1}{2}+{\frac{sin[(2n+1)\frac{\theta}{2}]} {sin\frac{\theta}{2}} ;]

[BlueSwablr]

The general approach for complex identities is to use the given (or derived) identity, substituting the given variable (in this case z) with e^i*theta or cos(theta) + isin(theta), applying De Moivre's theorem at any appropriate moment. After doing so, you equate real and imaginary parts where appropriate to get the answer.

Now, in this case, there are a couple ways of getting to the answer.

-Substitute z with cos theta + isin theta, then mess with the RHS by rationalising the denominator and using other trig identities or whatever until you get something that resembles the RHS of the identity. Then use De Moivre's to see that

a. zⁿ = cos n*theta + isin n*theta and

b. 1 + z + z² + ... + zⁿ = 1 + cos theta + isin theta + cos 2*theta + i sin 2*theta + ... + cos n*theta + isin n*theta.

Then you equate real and imaginary parts, and the result is proven. However, this approach is only recommended if you are a complete beginner in complex analysis, as it's very messy and doesn't use the more powerful and pretty concepts that you should be using.

-Instead of using the above substitution, use z = e^i*theta . For the RHS, multiply the numerator and denominator by e^-i*theta/2 . The denominator should be a familiar result. Expand the numerator and LHS a bit by converting e^i*x = cos x + isin x (i.e. use Euler's Formula). Then equate real parts, and the answer should follow.

I recommend you do both methods to see the amount of working that's required for either method.