# Modular graph functions and asymptotic expansions of Poincar\'e series

## TL;DR

This paper analyzes $SL(2,\

## Contribution

It introduces a method to represent modular functions as Poincaré series to derive their asymptotic expansions, capturing both perturbative and non-perturbative effects.

## Key findings

- Derived asymptotic expansions for modular graph functions.
- Identified perturbative and instanton contributions in string theory corrections.
- Provided a unified approach to analyze invariant functions in string theory.

## Abstract

In this note we study $SL(2,\mathbb{Z})$-invariant functions such as modular graph functions or coefficient functions of higher derivative corrections in type IIB string theory. The functions solve inhomogeneous Laplace equations and we choose to represent them as Poincar\'e series. In this way we can combine different methods for asymptotic expansions and obtain the perturbative and non-perturbative contributions to their zero Fourier modes. In the case of the higher derivative corrections, these terms have an interpretation in terms of perturbative string loop effects and pairs of instantons/anti-instantons.