I'm currently working through Raymond O. Wells' "Differential Analysis on Complex Manifolds" and I'm confused by example 2.13 in chapter 1.
In this example he is computing the global sections of the holomorphic line bundles on $P_1(C)$. I can follow the reasoning up until the last bit where he concludes what the space of sections is for each $k$. I don't see how this follows from the power series equality in the line above the conclusion?
Apologies if this is obvious and I'm just forgetting some basic results about complex analysis or something!
Here's a screenshot of the example with the preceding remark that he refers to within the example.
Edit: So I've used the advice from Andrew's comment to write a quick solution. For $k>0$, we can see that since $$ \sum_{n=0}^{\infty}a_nw^{-n} = w^k\sum_{n=0}^{\infty}b_nw^n,$$ then all the powers of $w$ on the left hand side are negative, whereas they are all positive powers on the right hand side. Thus since two Laurent series are equal only when they're corresponding coefficients are equal, then all the coefficients must be zero. Thus the section is zero on each chart and therefore defines a global section that must be zero everywhere. Hence $\Gamma(\mathbb{P}_1(\mathbb{C}), E^k) = 0$ when $k>0$.
When k=0 then the equality becomes $$ \sum_{n=0}^{\infty}a_nw^{-n} = \sum_{n=0}^{\infty}b_nw^n,$$ and so the only powers of $w$ that both sides have in common is the zero power. Therefore $a_0 = b_0$ and $a_j = 0, b_j = 0$ for all $j\neq 0$. Therefore the section $f$ is constant on each chart and agrees on the overlap so patches to a global constant section of $E^0$. Hence all sections of $E^0$ are constant and so $\Gamma(\mathbb{P}_1(\mathbb{C}), E^0) = \mathbb{C}$.
Finally, if $k<0$, then since $$ \sum_{n=0}^{\infty}a_nw^{-n} = w^k\sum_{n=0}^{\infty}b_nw^n,$$
we have that there will only be nonzero terms in the series when there are negative powers on the right hand side. Since $k < 0$, this is when $0 \leq n \leq -k$. So we can change the limits in the series to get $$ \sum_{n=0}^{-k}a_nw^{-n} = w^k\sum_{n=0}^{-k}b_nw^n.$$ So we can see that the sections are in fact polynomials. To see explicitly the degree $-k$ homogeneity, it is helpful to reintroduce the homogeneous coordinates $(z_0, z_1)$ via $w=\frac{z_0}{z_1}$ and rewrite the equality as $$ \sum_{n=0}^{-k}a_n(\frac{z_1}{z_0})^{n} = (\frac{z_0}{z_1})^k\sum_{n=0}^{-k}b_n(\frac{z_0}{z_1})^n.$$ We rearrange this to write it as $$ z_0^{-k}\sum_{n=0}^{-k}a_n(\frac{z_1}{z_0})^{n} = z_1^{-k}\sum_{n=0}^{-k}b_n(\frac{z_0}{z_1})^n.$$ Then under the change $(z_0, z_1) \mapsto (\lambda z_0, \lambda z_1)$ for some non-zero $\lambda$ we get $$ (\lambda z_0)^{-k}\sum_{n=0}^{-k}a_n(\frac{\lambda z_1}{\lambda z_0})^{n} = (\lambda z_1)^{-k}\sum_{n=0}^{-k}b_n(\frac{\lambda z_0}{\lambda z_1})^n.$$ This clearly simplifies to $$ \lambda^{-k}z_0^{-k}\sum_{n=0}^{-k}a_n(\frac{z_1}{z_0})^{n} = \lambda^{-k}z_1^{-k}\sum_{n=0}^{-k}b_n(\frac{z_0}{z_1})^n,$$ showing that when $k<0$ we have that $\Gamma(\mathbb{P}_1(\mathbb{C}), E^k)$ is the space of homogeneous polynomials of degree $-k$.
