holomorphic

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1: 21.7 Riemann Surfaces

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Figure 21.7.1: A basis of cycles for a genus 2 surface. Magnify

►On a Riemann surface of genus $g$, there are $g$ linearly independent holomorphic differentials ${\omega }_j$, $j=1,2,\mathrm{\dots },g$. If a local coordinate $z$ is chosen on the Riemann surface, then the local coordinate representation of these holomorphic differentials is given by …Note that for the purposes of integrating these holomorphic differentials, all cycles on the surface are a linear combination of the cycles $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$. … ►Define the holomorphic differential …

2: 1.9 Calculus of a Complex Variable

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Analyticity

►A function $f(z)$ is said to be analytic (holomorphic) at $z=z_0$ if it is complex differentiable in a neighborhood of $z_0$. …