Help Solving Series Related to z = rexp(ix)

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In summary, the function z = rexp(ix) has a series representation given by Σ<sub>n=0</sub> (irexp(ix))<sup>n</sup>/n!, with an infinite radius of convergence. This series is a special case of the Taylor series for e<sup>ix</sup>, and can be used to solve differential equations using the method of Frobenius. It can also be used to evaluate the function for non-real values of z.
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Homework Statement



Hello, I am wondering if someone can help with the following. I am supposed to show that

Series from n = 1 to inf r^n*cos(nx) = rcosx -r^2/(1-2*rcosx + r^2). I am supposed to relate it to the fact that z = rexp(ix). I know that this expression is the real part of z^n or r^n*exp(inx). But I'm not sure what to do after that?

Any hints would be greatly welcome! Thanks!

Homework Equations





The Attempt at a Solution

 
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Written in exponential form it's a geometric series. Sum it.
 

FAQ: Help Solving Series Related to z = rexp(ix)

What is the series representation of the function z = rexp(ix)?

The series representation of the function z = rexp(ix) is given by Σn=0 (irexp(ix))n/n!

What is the radius of convergence for the series representation of z = rexp(ix)?

The radius of convergence for the series representation of z = rexp(ix) is infinite, meaning the series converges for all values of z.

What is the relationship between the series representation of z = rexp(ix) and the Taylor series for eix?

The series representation of z = rexp(ix) is a special case of the Taylor series for eix. In fact, the only difference between the two is the presence of the imaginary unit i in the series for z = rexp(ix).

How can the series representation of z = rexp(ix) be used to solve differential equations?

The series representation of z = rexp(ix) can be used to solve differential equations by substituting it into the equation and solving for the coefficients of the series. This technique is known as the method of Frobenius.

Can the series representation of z = rexp(ix) be used to evaluate the function for non-real values of z?

Yes, the series representation of z = rexp(ix) can be used to evaluate the function for non-real values of z. This is because the series converges for all values of z, including non-real values.

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