Power series, complex numbers

In summary, the conversation discussed finding the power series for exp(z) + exp(w*z) + exp(z*w^2) using the previously shown equation 1+w+w^2=0 and the complex number w=exp(2*Pi*i/3). The resulting power series was shown to be 3 + 0 +0 + 3*z^3/3! + 0 + 0 +3*z^6/6! etc, and it was determined that this was the desired form.
  • #1
gertrudethegr
6
0
We have already shown 1+ w+ w^2 =0

If w is the complex number exp(2*Pi*i/3) , find the power series for;
exp(z) +exp(w*z) + exp (z*w^2)

We have already shown 1+ w+ w^2 =0
 
Last edited:
Physics news on Phys.org
  • #2
If 1+w+w^2=0, can you tell me what are w^3, w^4, w^5 ...? Now do a power series expansion of the exponentials and combine like powers of z. Do you see a pattern?
 
  • #3
Thanks!

So w^3=1
w^4= w^1
w^5=w^2 etc

so
exp(z) +exp(w*z) + exp (z*w^2)= 3 + 0 +0 + 3*z^3/3! + 0 + 0 +3*z^6/6! etc,

But where do i go from here?
Thankyou for your time
 
  • #4
gertrudethegr said:
Thanks!

So w^3=1
w^4= w^1
w^5=w^2 etc

so
exp(z) +exp(w*z) + exp (z*w^2)= 3 + 0 +0 + 3*z^3/3! + 0 + 0 +3*z^6/6! etc,

But where do i go from here?
Thankyou for your time

You are done, aren't you? The problem asked for a simple form of the power series, and you just showed it to me.
 
  • #5
That is a valid point!

Thanks dick
 

1. What is a power series?

A power series is an infinite series of the form ∑n=0^∞ cn(x-a)n, where cn are constants, x is a variable, and a is a fixed point. It is a special type of Taylor series that represents a function as a sum of its derivatives evaluated at a specific point.

2. How do you find the radius of convergence of a power series?

The radius of convergence of a power series can be found by applying the ratio test. The ratio test states that if limn→∞|cn+1/cn| = L, then the series converges if L < 1 and diverges if L > 1. The radius of convergence is equal to 1/L.

3. What is the significance of complex numbers in power series?

Complex numbers are important in power series because they allow us to extend the domain of a function beyond the real numbers. This is especially useful in solving differential equations and other mathematical problems that involve complex variables.

4. Can a power series converge at its endpoints?

Yes, a power series can converge at one or both of its endpoints, depending on the function it represents. For example, the power series for ln(1+x) converges at x=-1, while the power series for sin(x) converges at both x=±1.

5. How are power series used in real-world applications?

Power series have a wide range of applications in fields such as physics, engineering, and economics. They are used to model and approximate various phenomena, such as electric circuits, chemical reactions, and economic growth. They are also used in numerical analysis to solve differential equations and other complex problems.

Similar threads

  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
592
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
169
  • Calculus and Beyond Homework Help
Replies
8
Views
557
  • Calculus and Beyond Homework Help
Replies
21
Views
754
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
441
  • Calculus and Beyond Homework Help
Replies
0
Views
418
  • Calculus and Beyond Homework Help
Replies
17
Views
825
Back
Top