A finer point of series and summation.

[Anonymous No.6458233]

Let [ math] f(z) [ /math ] have a Laurent series expansion of the form <span class="math">\sum_{j=0}^{\infty}a_jz^j[\math] in <span class="math"> 0<|z|<\rho.

Would it be fair to say that this is also a power series? Can a series be both a Laurent series and a power series? I've been arguing with my prof about this for a few days. I claimed it was both in a proof and told me I couldn't and gave me no marks for it.

Similarly, if I define a polynomial p(x) = 1[\math], would it be fair to say that p(x) is both a polynomial and a number? or is it merely equal to a number?[/spoiler][/spoiler]

[Anonymous No.6458236]

Let [ math] f(z) [ /math ] have a Laurent series expansion of the form <span class="math"> \sum_{j=0}^{\infty}a_jz^j[/spoiler] in <span class="math"> 0<|z|<\rho[/spoiler].

Would it be fair to say that this is also a power series? Can a series be both a Laurent series and a power series? I've been arguing with my prof about this for a few days. I claimed it was both in a proof and told me I couldn't and gave me no marks for it.

Similarly, if I define a polynomial <span class="math">p(x) = 1[/spoiler], would it be fair to say that p(x) is both a polynomial and a number? or is it merely equal to a number?

[Anonymous No.6458239]

>>6458233
>>6458236
On a related note: how did I fuck this up?

[Anonymous No.6458252]

Yes, all power series are Laurent series, but I fail to see how this is anything more than semantics.

[Anonymous No.6458343]

>>6458239
You forgot some math tags and have a few slashes going the wrong direction.

[Anonymous No.6458607]

>>6458252
I spoke with him and we concluded that he's right. The Laurent series expansion isn't necessarily a power series.

Apparently a power series is defined as a series of the form <span class="math">\sum_{j=0}^\infty a_j(z-z_0)^j[/spoiler] which converges on some disk <span class="math"> |z-z_0| < R[\math].

we don't know if the Laurent series expansion converges on <span class="math">|z| < \rho[/spoiler], so we cannot conclude it's a power series. We could however define a power series <span class="math"> g(z) = \sum_{j=0}^\infty a_j(z-z_0)^j[\math] to be a power series and state that g(z) = f(z) = \sum_{j=0}^\infty a_j(z-z_0)^j on the punctured disk 0<|z|<\rho[\math][/spoiler][/spoiler]

[Anonymous No.6458615]

>>6458607
Well DUH, not all Laurent series are power series...
Only the ones for which <span class="math">a_n = 0 \forall n<0[/spoiler].
How is this not obvious?

also, if we're talking about formal power/Laurent series, the series doesn't necessarily have to converge at all.

[Anonymous No.6458708]

>>6458615
Even then, it's not necessarily true. Laurent series converge on an annulus, whereas power series converge on a disk. If we define a Laurent series that looks like a power series we still cannot claim it's a power series because we're not certain that it converges on a disk. All we know is that it converges on the annulus.

This is for a complex variables course.