Finding the Power Series for z^7

[fauboca]

How do I find the power series for $z^7$?

I can't remember.

 

[Mark44]

How do I find the power series for $z^7$?

I can't remember.

What would be the coefficient for the constant term?
What would be the coefficient for the z term?
What would be the coefficient for the z² term?
.
.
.

 

[Dick]

If you mean the power series expansion around a point that's not 0, say around z=1, then write z^7=((z-1)+1)^7 and expand that.

 

[fauboca]

If you mean the power series expansion around a point that's not 0, say around z=1, then write z^7=((z-1)+1)^7 and expand that.

Around zero. I have been looking through Rudin and Rosenlicht but I don't see an example of what I am looking for.

 

[Dick]

Around zero. I have been looking through Rudin and Rosenlicht but I don't see an example of what I am looking for.

Then what are you looking for? z^7 IS the power series for the function f(z)=z^7 around z=0.

 

[fauboca]

Then what are you looking for? z^7 IS the power series for the function f(z)=z^7 around z=0.

$$
f(z) = \frac{1}{2\pi i}\sum_{n = 0}^{\infty}\left[\int_0^{2\pi}\frac{u^7}{u^{n + 1}}du z^n\right] = z^7.
$$

I want to show the above equality. I know since u^7 is analytic in the unit disk, g will be the same as f. But is there a way to show that without stating this?

 

[Dick]

$$
f(z) = \frac{1}{2\pi i}\sum_{n = 0}^{\infty}\left[\int_0^{2\pi}\frac{u^7}{u^{n + 1}}du z^n\right] = z^7.
$$

I want to show the above equality. I know since u^7 is analytic in the unit disk, g will be the same as f. But is there a way to show that without stating this?

Well, the integral is zero unless n=7, isn't it? The integral of u^k is zero unless k=(-1). And I'm assuming you actually meant a contour integral around the origin, not u=0 to u=2pi.

 

[fauboca]

Well, the integral is zero unless n=7, isn't it? The integral of u^k is zero unless k=(-1).

u is a complex number and u^7 is analytic and continuous.

For all z inside of C (C the unit circle oriented counterclockwise),
$$f(z) = \frac{1}{2\pi i}\int_C \frac{g(u)}{u-z} du$$
where $g(u) = \bar{u}$ is a continuous function and $f$ is analytic in C. Describe $f$in C in terms of a power series.

$\displaystyle f(z) = \frac{1}{2\pi i}\int_C \frac{\bar{u}}{u-z} du$ I am confused with what I am supposed to do. I know it says describe $f$ in terms of a power series.

Only difference I am dealing with u^7 not the conjugate.

 

[Dick]

$\int_C \frac{u^7}{u^{n + 1}}du=\int_C u^{7 - n - 1}du$. That's zero unless the exponent is -1 which happens when n=7.

 

[fauboca]

$\int_C \frac{u^7}{u^{n + 1}}du=\int_C u^{7 - n - 1}du$. That's zero unless the exponent is -1 which happens when n=7.

Ok, I understand now, thanks.

 

[fauboca]

Ok so if the function was 1/u, we would have u^{-n-2}. This one would always be 0 then?

 

[Dick]

Ok so if the function was 1/u, we would have u^{-n-2}. This one would always be 0 then?

Sure, but f(u)=1/u doesn't have a power series expansion around 0 with only positive powers. You'd need a Laurent series instead of a power series to represent it.

 

[fauboca]

Sure, but f(u)=1/u doesn't have a power series expansion around 0 with only positive powers. You'd need a Laurent series instead of a power series to represent it.

If we haven't done Laurent Series yet, how should I handle it then?

 

[Dick]

If we haven't done Laurent Series yet, how should I handle it then?

You don't. f(z)=1/z doesn't have a power series expansion around z=0. f(z) has to be analytic at z=0 to have a power series expansion. 1/z isn't analytic at z=0.

 

[fauboca]

You don't. f(z)=1/z doesn't have a power series expansion around z=0. f(z) has to be analytic at z=0 to have a power series expansion. 1/z isn't analytic at z=0.

Evaluating the sum and the integral for it yields f(z) = 0. Is that correct to put down after evaluating f(z) since all the terms are 0?

 

[Dick]

Evaluating the sum and the integral for it yields f(z) = 0. Is that correct to put down after evaluating f(z) since all the terms are 0?

All of the terms in your series are zero, yes. But that still doesn't make f(z)=1/z=0. I'm not sure you are paying attention here.

 

[fauboca]

All of the terms in your series are zero, yes. But that still doesn't make f(z)=1/z=0. I'm not sure you are paying attention here.

I understand what you are saying but I am trying to solve for
$$
f(z) = \frac{1}{2\pi i}\sum_{n = 0}^{\infty}\left[\int_0^{2\pi}\frac{\frac{1}{u}}{u^{n + 1}}du z^n\right].
$$
Since Laurent series are out and all the terms are 0, what else could f(z) be?

 

[Dick]

I understand what you are saying but I am trying to solve for
$$
f(z) = \frac{1}{2\pi i}\sum_{n = 0}^{\infty}\left[\int_0^{2\pi}\frac{\frac{1}{u}}{u^{n + 1}}du z^n\right].
$$
Since Laurent series are out and all the terms are 0, what else could f(z) be?

I am trying to tell you that the series you are quoting is NOT valid for all functions f(z). f(z) has to be analytic at z=0 to apply that. f(z)=1/z is NOT analytic at z=0. I've already told you this.

 

[fauboca]

I am trying to tell you that the series you are quoting is NOT valid for all functions f(z). f(z) has to be analytic at z=0 to apply that. f(z)=1/z is NOT analytic at z=0. I've already told you this.

By the integral transform theorem, if you put a continuous function g(u) into f(z), you get out an analytic function. If g is analytic, you get the same function. So f(z) has to equal something.

 

[Dick]

By the integral transform theorem, if you put a continuous function g(u) into f(z), you get out an analytic function. If g is analytic, you get the same function. So f(z) has to equal something.

I'm not quite sure why this is so difficult. 1/u is continuous on the contour. And yes, you get an analytic function out. It's f(z)=0. Now you say "If g is analytic, you get the same function.". g(u)=1/u ISN'T analytic at u=0. So the function you get out f(z)=0, ISN'T the same as the function you put in f(z)=1/z.

 

[fauboca]

I'm not quite sure why this is so difficult. 1/u is continuous on the contour. And yes, you get an analytic function out. It's f(z)=0. Now you say "If g is analytic, you get the same function.". g(u)=1/u ISN'T analytic at u=0. So the function you get out f(z)=0, ISN'T the same as the function you put in f(z)=1/z.

I understand that. I was verifying that f(z) = 0 is correct. You kept saying you don't understand what I telling you.

 

[Dick]

I understand that. I was verifying that f(z) = 0 is correct. You kept saying you don't understand what I telling you.

Apologies if I'm misunderstanding. But I'm just saying 0 isn't the power series representation of 1/z because it doesn't have one. And f(z)=1/z is not equal to 0. That's all. So, ok. Yes f(z)=0. I somehow thought you were trying to represent 1/z as a power series. Sorry.

 

[fauboca]

Apologies if I'm misunderstanding. But I'm just saying 0 isn't the power series representation of 1/z because it doesn't have one. And f(z)=1/z is not equal to 0. That's all. So, ok. Yes f(z)=0. I somehow thought you were trying to represent 1/z as a power series. Sorry.

No problem. I know f(z) = 1/z\neq 0 but when I said f(z) I meant,

$$
f(z) = \frac{1}{2\pi i}\sum_{n = 0}^{\infty}\left[\int_0^{2\pi}\frac{\frac{1}{u}}{u^{n + 1}}du z^n\right].
$$

 

[Dick]

No problem. I know f(z) = 1/z\neq 0 but when I said f(z) I meant,

$$
f(z) = \frac{1}{2\pi i}\sum_{n = 0}^{\infty}\left[\int_0^{2\pi}\frac{\frac{1}{u}}{u^{n + 1}}du z^n\right].
$$

Yeah, of course it's 0. Sorry again for being a bit thick about what you were asking.