by Bruna Crisóstomo de Oliveira Möckli
Friday 8 September 2017
Ceiling lamp
Ceiling lamp: Raw image. Not edited. Contrast. Light pattern forming e penumbra. Perfect circular symmetry of the lamp. White hole.
Elegant proof of the Euler equation
The Euler equation is one of the most elegant equations in mathematics. It reads
$$e^{i\varphi} = \cos\varphi +i\sin\varphi.$$
Its demonstration usually involves the Taylor expansions of the trigonometric functions. Here I reproduce a simple and straightforward demonstration that I have found on a math forum.
Consider the the function
$$f(\varphi)=e^{-i\varphi}\left(\cos\varphi+i\sin\varphi\right),$$
with $\varphi\in\mathbb{R}$. It is easy to verify that $f^\prime(\varphi)=0$ for all $\varphi\in\mathbb{R}$. For this reason $f(\varphi)$ is a constant function. Since $f(0)=1$, one must have that
$$e^{i\varphi} = \cos\varphi +i\sin\varphi.$$
If you know another neat way of demonstrating the Euler equation, please comment below.
$$e^{i\varphi} = \cos\varphi +i\sin\varphi.$$
Its demonstration usually involves the Taylor expansions of the trigonometric functions. Here I reproduce a simple and straightforward demonstration that I have found on a math forum.
Consider the the function
$$f(\varphi)=e^{-i\varphi}\left(\cos\varphi+i\sin\varphi\right),$$
with $\varphi\in\mathbb{R}$. It is easy to verify that $f^\prime(\varphi)=0$ for all $\varphi\in\mathbb{R}$. For this reason $f(\varphi)$ is a constant function. Since $f(0)=1$, one must have that
$$e^{i\varphi} = \cos\varphi +i\sin\varphi.$$
If you know another neat way of demonstrating the Euler equation, please comment below.
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