The Exposition and Proof of Euler’s Formula

In the 18th century, Euler’s formula (EF) was discovered by Leonhard Euler, and it has since been recognized as one of the most famous and beautiful equations in the mathematical world, occupying an extremely important place in numerous fields. The theory of complex functions has been significantly enriched by this formula, as it extends the domain of trigonometric functions to complex numbers and establishes a relationship between trigonometric functions and exponential functions. This paper explores the extended applications of Euler’s formula in complex numbers, geometry, topology, and graph theory. The proof of Euler’s formula in its algebraic form is achieved through the utilization of the Maclaurin series and the method of separation of variables, starting with the derivatives of functions. Additionally, the implications of the formula are investigated, with related proofs and applications being examined in plane geometry, graph theory, and physics. It is highlighted that Euler’s formula not only occupies a significant position in mathematics but has also been a driving force behind the continuous development and innovation of modern technology.