# On the representation of cylinder functions

## TL;DR

This paper introduces a new mixed integral-sum representation for cylinder functions valid for all complex orders and arguments, facilitating analysis and applications in complex analysis.

## Contribution

It provides a novel, general representation of cylinder functions applicable to unrestricted complex orders and variables, expanding their analytical framework.

## Key findings

- Derived a mixed integral-sum representation valid for all complex orders and arguments.
- Discussed limiting forms and related function representations.
- Explored applications of the new representation in complex analysis.

## Abstract

In this paper, we present a mixed-type integral-sum representation of the cylinder functions $\mathscr{C}_\mu(z)$, which holds for unrestricted complex values of the order $\mu$ and for any complex value of the variable $z$. Particular cases of these representations and some applications, which include the discussion of limiting forms and representations of related functions, are also discussed.