# Application of the Biot-Savart Law to Parabolic Vortex Segments using Elliptic Integrals

## TL;DR

This paper develops explicit formulas using elliptic integrals to calculate the velocity induced by parabolic vortex segments, advancing aerodynamic modeling techniques.

## Contribution

It introduces a novel application of elliptic integrals and hypergeometric functions to derive explicit Biot-Savart law formulas for parabolic vortex segments.

## Key findings

- Formulas derived using Appell hypergeometric functions.
- Validation through numerical integration and perturbation expansion.
- Enhanced accuracy in vortex velocity calculations.

## Abstract

The Biot-Savart law is used in aerodynamic theory to calculate the velocity induced by curved vortex lines. Explicit formulas are developed, using multivariate Appell hypergeometric functions, for the velocity induced by a general parabolic vortex segment. The formulas are derived by constructing a particular pencil of elliptic curves whose period integrals provide the solution to the induced velocity. We use numerical integration and a perturbation expansion to evaluate the validity of our formulas.