# On Lisbon integrals

## TL;DR

This paper introduces Lisbon Integrals, new complex integral transforms linked to polynomial root symmetric functions, and characterizes their associated differential systems, demonstrating their unique solutions and applications to exponential functions.

## Contribution

It defines Lisbon Integrals, determines their differential equations, and explores their role as integral transforms in complex analysis and algebraic geometry.

## Key findings

- Lisbon Integrals satisfy specific $\,\mathcal{D}$-modules.
- They are the unique global solutions to these differential systems.
- Application to exponential functions illustrates the transform's utility.

## Abstract

We introduce new complex analytic integral transforms, the Lisbon Integrals, which naturally arise in the study of the affine space $\mathbb{C}^k$ of unitary polynomials $P_s(z)$ where $s\in\mathbb{C}^k$ and $z\in \mathbb{C}$, $s_i$ identified to the $i-$th symmetric function of the roots of $P_s(z)$. We completely determine the $\mathcal{D}$-modules (or systems of partial differential equations) the Lisbon Integrals satisfy and prove that they are their unique global solutions. If we specify a holomorphic function $f$ in the $z$-variable, our construction induces an integral transform which associates a regular holonomic module quotient of the sub-holonomic module we computed. We illustrate this correspondence in the case of a $1$-parameter family of exponentials $f_t(z) = exp(t z)$ with $t$ a complex parameter.