Help Solving Series Related to z = rexp(ix)

  • Thread starter Thread starter bodensee9
  • Start date Start date
  • Tags Tags
    Series
bodensee9
Messages
166
Reaction score
0

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

 
Physics news on Phys.org
Written in exponential form it's a geometric series. Sum it.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
View full post »

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

Deduce orthogonality relations for sine and cosine w/ Euler's Formula
Replies
1
Views
1K
Solve integral by finding Fourier series of complex function
Replies
3
Views
1K
Calculating Laurent series of complex functions
Replies
3
Views
1K
Link between Z-transform and Taylor series expansion
Replies
2
Views
2K
On pointwise convergence of Fourier series
Replies
3
Views
2K
How to show that these sequences are summable?
Replies
1
Views
1K
Learning to use the Cauchy criterion for infinite series
Replies
7
Views
2K
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
Replies
1
Views
1K
Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
Replies
1
Views
1K
A problem with the convergence of a series
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top