Power series expansion for Log z

In summary, the task is to find the power series expansion of Log z about the point z = i-2 and to show that the radius of convergence of the series is R = √5. The given Log z equation is modified to (z-3-i) - (1/2)(z-3-i)^2 + (1/3)(z-3-i)^3 -... and it is asked to find the value of a_k for the series. It is also discussed that the function may not be analytically extended on any open disk containing the origin, but it is analytic on the disk of radius √5 centered at i-2.
  • #1
smanalysis
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Homework Statement


Find the power series expansion of Log z about the point z = i-2. Show that the radius of convergence of the series is R = [tex]\sqrt{5}[/tex].


Homework Equations


None


The Attempt at a Solution


I know that Log z = (z-1) - (1/2)(z-1)^2 + (1/3)(z-1)^3 -...
So wouldn't this become (z-3-i) - (1/2)(z-3-i)^2 + (1/3)(z-3-i)^3 -... about the point z = i-2? What would a_k be for the series? Also, since values of Log z are unbounded in any neighborhood of the origin, I would think that this function cannot be extended analytically on any open disk containing the origin. But wouldn't it be analytic on the disk of radius |i-2| = |-2+i| = Sqrt((-2)^2 + 1^2) = Sqrt(5)? Then I would say that the radius of convergence would be Sqrt(5) for the disk centered at i-2, right? I would appreciate any suggestions, thanks in advance! :-)
 
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  • #2
"about the point z = i-2" means set z=ω-(i-2) and find log(z) as a power series in ω.
 

1. What is a power series expansion for Log z?

A power series expansion for Log z is a mathematical representation of the natural logarithm function, where the logarithm of a complex number z is expressed as an infinite sum of powers of z. It is written as Log z = ln|z| + i arg(z), where ln|z| represents the real part and i arg(z) represents the imaginary part.

2. How do you derive a power series expansion for Log z?

To derive a power series expansion for Log z, we use the formula for the Taylor series expansion of a function. In this case, we use the Taylor series expansion for the natural logarithm function, which is ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + ... By substituting z-1 for x, we can obtain the power series expansion for Log z.

3. What is the convergence radius of a power series expansion for Log z?

The convergence radius of a power series expansion for Log z is equal to the distance from the center of the series (in this case, z=1) to the nearest singularity of the function. In other words, it is the maximum distance from the center where the series will still converge and provide an accurate representation of the natural logarithm function.

4. How can a power series expansion for Log z be used to approximate the value of Log z?

A power series expansion for Log z can be used to approximate the value of Log z by truncating the series at a certain number of terms. The more terms we include in the series, the more accurate the approximation will be. This method is particularly useful for calculating logarithms of complex numbers, where it may be difficult to directly calculate the value using traditional methods.

5. Are there any limitations to using a power series expansion for Log z?

Yes, there are limitations to using a power series expansion for Log z. The series will only converge within its convergence radius, and outside of this radius, the approximation will become increasingly inaccurate. Additionally, the series may converge slowly or may not converge at all for certain values of z. Therefore, it is important to carefully consider the convergence radius and the number of terms used when using a power series expansion for Log z.

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