Understanding Euler's Formula through its Derivation

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Hello. I am going to tell you how to intuitively understand Euler's formula through its derivation.

I presume that you understand the concept of Taylor series expansion. If not, please visit the video or the text entitled "The Simplest Way to Understand the Concept of Taylor Series Expansion" before reading this text.

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Look at this beautiful equation. This is called Euler's formula.

$$\mathrm{e}^{i\theta} = \cos \theta + i \sin \theta \nonumber$$

It relates the three different mathematical concepts: exponential function, imaginary number and trigonometric functions. If you substitute $\theta = \pi$, then we get

$$\mathrm{e}^{i\pi} = -1 \nonumber$$

This one connects exponential function, imaginary number and pi.

The most important point of Euler's formula is that the term $\mathrm{e}^{i\theta}$ plays an important role in science and engineering as the "oscillation kernel."

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Ok. Let's derive Euler's formula by applying the Taylor series expansion at $\theta = 0$ to $\mathrm{e}^{i\theta}$. Recall Taylor series expansion at $\theta = 0$ is

$$f(\theta) = a_0 + a_1 \theta + a_2 \theta^2 + a_3 \theta^3 + \cdots \nonumber$$

where

$$a_n = \frac{1}{n!}f^{(n)}(0) \nonumber$$

Keep in mind that the derivative of exponential function is exponential function.

$$\frac{d}{d\theta} \mathrm{e}^{i\theta} = i \mathrm{e}^{i\theta} \nonumber$$

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Let's calculate the coefficients $a_0, a_1, a_2, a_3, \cdots$.

$$a_0 = \mathrm{e}^{i\theta}\vert_{\theta=0} = \mathrm{e}^{i0} = 1 \nonumber$$

$$a_1 = ( \mathrm{e}^{i\theta} )'\vert_{\theta=0} = i\mathrm{e}^{i0} = i \nonumber$$

$$a_2 = \frac{1}{2}( i\mathrm{e}^{i\theta} )'\vert_{\theta=0} = -\frac{1}{2}\mathrm{e}^{i0} = -\frac{1}{2} \nonumber$$

$$a_3 = \frac{1}{6}( -\mathrm{e}^{i\theta} )'\vert_{\theta=0} = -i\frac{1}{6}\mathrm{e}^{i0} = -i\frac{1}{6} \nonumber$$

$$a_4 = \frac{1}{24}( -i\mathrm{e}^{i\theta} )'\vert_{\theta=0} = \frac{1}{24}\mathrm{e}^{i0} = \frac{1}{24} \nonumber$$

$$a_5 = \frac{1}{120}( \mathrm{e}^{i\theta} )'\vert_{\theta=0} = i\frac{1}{120}\mathrm{e}^{i0} = i\frac{1}{120} \nonumber$$

$$\vdots \nonumber$$

So, the Taylor series expansion of $\mathrm{e}^{i\theta}$ at $\theta = 0$ is

$$\mathrm{e}^{i\theta} = 1 + i \theta - \frac{1}{2} \theta^2 - i \frac{1}{6} \theta^3 + \frac{1}{24} \theta^4 + i \frac{1}{120} \theta^5 + \cdots \nonumber$$

If this series expansion is split into the even and the odd terms, we get

$$\mathrm{e}^{i\theta} = \left( 1 - \frac{1}{2} \theta^2 + \frac{1}{24} \theta^4 \cdots \right) + i \left( \theta - \frac{1}{6} \theta^3 + \frac{1}{120} \theta^5 \cdots \right) \nonumber$$

Well, amazingly, the even or the real part corresponds to the Taylor series expansion of cosine function and the odd or the imaginary part corresponds to that of sine function!

So, we get Euler's formula.

$$\mathrm{e}^{i\theta} = \cos \theta + i \sin \theta\nonumber$$

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On the complex plane, $\theta$ can be considered as the angle formed by the horizontal axis and the line connecting a point on the unit circle to the origin.

If we assume $\theta$ evolves with respect to time with a constant speed, we can write

$$\theta = \omega t \nonumber$$

where $\omega$ is the speed called angular frequency and $t$ is time. Substituting this into Euler's equation gives

$$\mathrm{e}^{i \omega t} = \cos \omega t + i \sin \omega t \nonumber$$

In this case, the point turns around on the unit circle counterclockwise. This is one aspect of Euler's formula and the reason why it plays an important role as the oscillation kernel. It is known that an actual phenomenon can be obtained by taking the real part of this oscillation kernel $\mathrm{e}^{i \omega t}$. Why? This will be explained in the video and the text entitled "Oscillation Kernel". By taking the real part, we get the oscillation waveform.

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Next, let's consider what if the variable of the exponential function is not a purely imaginary number but a complex number $\phi + i\theta$.

$$\mathrm{e}^{i\theta} \quad \rightarrow \quad \mathrm{e}^{\phi + i\theta} \nonumber$$

This can be written as

$$\mathrm{e}^{\phi + i\theta} = \mathrm{e}^{\phi} \mathrm{e}^{i\theta} \nonumber$$

and substituting Euler's formula into this equation gives

$$\mathrm{e}^{\phi + i\theta} = \mathrm{e}^{\phi} \left( \cos \theta + i \sin \theta \right) \nonumber$$

In most physical and engineering applications, $\phi$ and $\theta$ are in proportion to time or distance. To understand this equation better, let's consider the case where $\phi$ and $\theta$ are in proportion to time $t$.

$$\begin{eqnarray} \phi & = & \alpha t \nonumber \\ \theta & = & \omega t \nonumber \end{eqnarray} \nonumber$$

Then, we get the time evolving version of Euler's formula.

$$\mathrm{e}^{(\alpha + i\omega)t} = \mathrm{e}^{\alpha t} \left( \cos \omega t + i \sin \omega t \right) \nonumber$$

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Depending on the value of $\alpha$, this equation represents different types of oscillation. When $\alpha = 0$, the $\mathrm{e}^{\alpha t}$ term vanishes, and so this equation is the same as the previous purely imaginary case. In this case, this equation represents a point turns around on the unit circle counterclockwise on the complex plane. By taking the real part, we get the permanent oscillation waveform.

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When $\alpha < 0$, this equation represents a damped oscillation. Starting from the point 1 on the complex plane, the point turns around counterclockwise with an exponentially shrinking radius and converges into the origin, so winding around the origin spirally. If you take the real part of this equation, you get a damped oscillation waveform. This represents the response of most physical and engineering systems which are stable.

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When $\alpha>0$, on the other hand, the equation represents a divergent oscillation. Starting from the point 1 on the complex plane, the point turns around counterclockwise with an exponentially growing radius and diverges to infinity. Taking the real part of this equation gives you a divergent oscillation waveform. This represents the response of unstable physical and engineering systems.

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It is interesting to note that $\alpha$ is a criterion which determines whether the system of which response is described by the time evolving version of Euler's formula is stable or not.

So, the oscillation kernel $\mathrm{e}^{(\alpha + i\omega)t}$ represents a permanent, damped or divergent oscillation. $\alpha$ determines its stability, and $\omega$ determines its oscillation frequency. Then, we may write

$$s = \alpha + i\omega \nonumber$$

and $s$ is called the complex frequency in engineering. Actually, this is the operator of Laplace transform. If you are interested also in Laplace transform, see the video or text entitled "Understanding Laplace Transform in Terms of Complex Frequency."

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I hope you have reached real understanding of Euler's formula. Thank you.