Expansions of Functions Analytic at Infinity

In summary: yieldsf(z) = 1 - \frac{2}{z+1}\sum_{n=0}^{+\infty} \frac{a_{n}}{z^{n}} converging outside some disk with radius 1/radius of the g disk.
  • #1
ferret123
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Homework Statement



Prove that if f(z) is analytic at infinity, then it has expansion of the form

[itex]f(z) = \sum_{n=0}^{+\infty} \frac{a_{n}}{z^{n}}[/itex]

converging outside some disk.2. The attempt at a solution

I know that for f(z) to be analytic at infinity we want to consider the composite function g(w) = f(\frac{1}{w}), then if g is analytic at 0 f will be analytic at infinity.

I've been attempting to Taylor expand g(w) around the point w=0 and then substituting the appropriate f terms in. However I'm not sure if this is a valid method as I can't see much pattern in the derivatives.

So far I have

[itex]g(w) = g(0) + g'(0)w + \frac{g''(0)w^{2}}{2} + \frac{g'''(0)w^{3}}{3!} + ...[/itex]

as well as

[itex]g'(w) = \frac{-f'(\frac{1}{w})}{z^{2}}[/itex]
[itex]g''(w) = \frac{2f'(\frac{1}{w})}{w^{3}} + \frac{f''(\frac{1}{w})}{w^{4}}[/itex]
[itex]g'''(w) = \frac{-6f'(\frac{1}{w})}{w^{4}} - \frac{6f''(\frac{1}{w})}{w^{5}} - \frac{f'''(\frac{1}{w})}{w^{6}}[/itex]

meaning I'm just gaining more of each derivative each time and I'm not sure if this is the right way to go about this. Any help or tips would be much appreciated.
 
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  • #2
ferret123 said:

Homework Statement



Prove that if f(z) is analytic at infinity, then it has expansion of the form

[itex]f(z) = \sum_{n=0}^{+\infty} \frac{a_{n}}{z^{n}}[/itex]

converging outside some disk.2. The attempt at a solution

I know that for f(z) to be analytic at infinity we want to consider the composite function g(w) = f(\frac{1}{w}), then if g is analytic at 0 f will be analytic at infinity.

What you need here is that if [itex]f[/itex] is analytic at infinity then [itex]g[/itex] is analytic at zero.

I've been attempting to Taylor expand g(w) around the point w=0 and then substituting the appropriate f terms in.

You don't need to do that. It is enough to note that if [itex]g[/itex] is analytic at zero then it has a power series expansion
[tex]
g(w) = \sum_{n=0}^\infty a_n w^n
[/tex]
which converges inside some disc.
 
  • #3
Ok I've really over complicated it then, thanks! So then when I change back to [itex]f[/itex] I get the desired result analytic outside of a disk with radius 1/radius of the [itex]g[/itex] disk.

The next part of the question is looking for an expansion of this form for for [itex]\frac{z-1}{z+1} \& \frac{z^{2}}{z^{2} - 1}[/itex]. I think for the first of these I can re-express as [itex]1 - \frac{2}{z+1}[/itex] and then expand this?
 

FAQ: Expansions of Functions Analytic at Infinity

1. What does it mean for a function to be analytic at infinity?

A function is considered analytic at infinity if it has a power series expansion that converges for all values of x, including x approaching infinity. This means that the function must be well-behaved and have no singularities at infinity.

2. How do you determine if a function is analytic at infinity?

One way to determine if a function is analytic at infinity is to use the criteria of the Cauchy-Hadamard theorem, which states that if the limit of the absolute value of the function's coefficients in its power series expansion is equal to zero, then the function is analytic at infinity. Alternatively, you can also check if the function has a Laurent series expansion that converges at infinity.

3. Can a function be analytic at infinity but not at other points?

Yes, it is possible for a function to be analytic at infinity but not at other points. For example, the function f(x) = 1/x is analytic at infinity, but has a singularity at x = 0. This is because the function's power series expansion only converges for x approaching infinity, but not at x = 0.

4. What are some examples of functions that are analytic at infinity?

Some common examples of functions that are analytic at infinity include polynomials, exponential functions, trigonometric functions, and rational functions. These functions have power series expansions that converge for all values of x, including x approaching infinity.

5. How is the concept of analyticity at infinity useful in mathematics and science?

Analyticity at infinity is a useful concept in complex analysis, as it allows for the study of functions on an extended complex plane. It is also important in the study of differential equations and their solutions, as well as in the analysis of asymptotic behavior of functions. In science, it can be used to model and analyze physical phenomena, such as the behavior of electromagnetic fields at infinity.

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