# A sublime proof of Euler’s formula

A proof of Euler’s formula using the Leibniz product rule.

Aidan Rocke https://github.com/AidanRocke
10-21-2022

The most remarkable formula in mathematics.-Feynman

The complex exponential may be defined as follows:

$$$\text{exp}: \mathbb{C} \rightarrow \mathbb{C} \\ z \mapsto \sum_{n \geq 0} \frac{z^n}{n!} \tag{1}$$$

Using the Cauchy Product and Mertens’ theorem, this implies that:

$$$\forall a, b \in \mathbb{C}, \text{exp}(a) \cdot \text{exp}(b) = \text{exp}(a + b) \tag{2}$$$

Now, let’s consider the function:

$$$\forall t \in \mathbb{R}, f(t) = e^{-it} \cdot (\cos t +i\sin t ) \tag{3}$$$

By the Leibniz product rule:

$$$\forall t \in \mathbb{R}, f'(t) = -i \cdot e^{-it} \cdot (\cos t +i\sin t ) + e^{-it} \cdot (i \cos t -\sin t ) = 0 \tag{4}$$$

Thus, $$f$$ is constant everywhere.

Now, given that:

$$$f(0)=1 \implies \forall t \in \mathbb{R}, f(t)=1 \tag{5}$$$

we may conclude that:

$$$\forall t \in \mathbb{R}, e^{it} = \cos t + i \sin t \tag{6}$$$

QED.

### Citation

For attribution, please cite this work as

Rocke (2022, Oct. 21). Kepler Lounge: A sublime proof of Euler's formula. Retrieved from keplerlounge.com

BibTeX citation

@misc{rocke2022a,
author = {Rocke, Aidan},
title = {Kepler Lounge: A sublime proof of Euler's formula},
url = {keplerlounge.com},
year = {2022}
}