# Riemann-Roch Theorem

## Theorem

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## Proof

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## Motivation

The **Riemann-Roch Theorem** gives a formula which is used to calculate the exact number of independent holomorphic functions that can be defined on a given Riemann surface.

This formula involves the number of independent differential forms of special kinds.

Some of these are equal to topological invariants of the Riemann surface.

These are, in general, easy to calculate.

## Also see

## Source of Name

This entry was named for Georg Friedrich Bernhard Riemann and Gustav Roch.

## Historical Note

The **Riemann-Roch theorem** was stated and proved in the mid-$19$th century by Bernhard Riemann and his student at the time Gustav Roch.

Bernhard Riemann supposedly achieved his version by considering configurations of closed spaces and imagining experiments performed on them.

It has been used in recent times in the study of codes.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Riemann-Roch theorem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Riemann-Roch theorem**