# Definition:Abelian Function

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## Definition

An **Abelian function** is an inverse function of an Abelian integral.

## Source of Name

This entry was named for Niels Henrik Abel.

## Historical Note

The foundations for the theory of Abelian functions were laid in Abel's paper *Mémoire sur une Propriété Générale d'une Classe Très-Étendue de Fonctions Transcendantes*.

Bernhard Riemann studied Abelian functions by applying Dirichlet's Principle for Harmonic Functions combined with geometric reasoning.

The work on extending our understanding of them continued with Karl Weierstrass, working alone with no contact with other mathematicians.

He took up their study as his life's work.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($\text {1815}$ – $\text {1897}$)