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How Did Euler Derive His Complex Formula ?


Euler's formula:

It is one of the most prominent formula which illustrates the relationship between the complex exponential function and the trigonometric functions. It also provides a valid transformation between the Cartesian coordinates and the polar coordinates. Therefore, Euler's formula can be found in many mathematical branches, physics and engineering. 
      The mathematical representation for Euler's formula is:
 \( e^{i\theta}= cos(\theta) + isin(\theta) \)
Where \(e\) is the base of the natural logarithm, \(i\) is the imaginary unit and  \( \theta \in \mathbb{C} \) 
The notion \( e^{i \theta } \) where is called the unit complex number.

Proof of Euler's formula:

The Euler's formula derivation is based on Taylor series expansions of the exponential function \(e^{z}\) and the trigonometric functions \(sin(x)\) and \(cos(x)\) where \( z \in  \mathbb{C} \) , \(x \in\mathbb{R}\).

Beginning with the Taylor series expansion of the exponential function \(e^{z}\) to have:


Now, without loss of generality, let \(z=ix\) to have the following form:

\(e^{ix}=1+\frac{ix}{1!}+\frac{\left (ix\right )^{2}  }{2!}+\frac{\left (ix  \right )^{3}}{3!}+\frac{\left (ix  \right )^{4}}{4!}+\frac{\left (ix  \right )^{5}}{5!}+\frac{\left (ix  \right )^{6}}{6!}+\frac{\left (ix  \right )^{7}}{7!}+\frac{\left (ix  \right )^{8}}{8!}+...\)

Expand the powered numerators to have: 

\(e^{ix}=1+ix-\frac{x^{2}}{2!}-i \frac{x^{3}}{3!}+\frac{x^{4}}{4!}+i \frac{x^{5}}{5!}-\frac{x^{6}}{6!}-i \frac{x^{7}}{7!}+\frac{x^{8}}{8!}+... \)

Rearrange the right hand side terms to yield:

\(e^{ix}= \left (1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\frac{x^{8}}{8!} - ...  \right )+i\left   (x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+...  \right ) \)

\(e^{ix}= cos(x) + isin(x) \)

Moreover, we called the Euler's formula at \(z=\pi\); the Euler's Identity, which can be written in the form of \(e^{i \pi}+1=0\).

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