= Sections of the degree-one line bundle on the projective line
{title2=$H^0(\mathbb{CP}^1,\mathcal O(1))\cong\mathbb C^2$}
A global <holomorphic section> of $\mathcal O(1)$ on $\mathbb{CP}^1$ is a linear homogeneous polynomial in the two projective coordinates. On one patch its coefficient is an entire <function> $p(\lambda)$; regularity on the other patch says $p(\lambda)/\lambda$ is regular at infinity. Hence $p$ has at most linear growth. <Cauchy estimates> make its second and higher <derivatives> zero, so $p=a+b\lambda$. The same argument entrywise applies to matrix-valued sections and makes the logarithmic <derivative> in the <Penrose-Ward correspondence> linear in the primed <two-component spinor>.
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