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Complex numbers definition

Conversion of a complex number

The conversion of a complex number from trigonometric form to exponential form can be proven using the Taylor series expansion of the exponential function. Starting with the exponential form:

\[\begin{equation*}z=re^{i\theta}\end{equation*}\]

Taking the Taylor series expansion of \(e^{ix}\) around \(x = 0\), we get:

\[\begin{equation*}e^{ix}=1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}-\frac{x^6}{6!}-i\frac{x^7}{7!}+\cdots\end{equation*}\]

Substituting \(x = \theta\) into this expression, we get:

\[\begin{equation*}e^{i\theta}=1+i\theta-\frac{\theta^2}{2!}-i\frac{\theta^3}{3!}+\frac{\theta^4}{4!}+i\frac{\theta^5}{5!}-\frac{\theta^6}{6!}-i\frac{\theta^7}{7!}+\cdots\end{equation*}\]

Multiplying both sides by \(r\), we get:

\[\begin{equation*}re^{i\theta}=r\left(1+i\theta-\frac{\theta^2}{2!}-i\frac{\theta^3}{3!}+\frac{\theta^4}{4!}+i\frac{\theta^5}{5!}-\frac{\theta^6}{6!}-i\frac{\theta^7}{7!}+\cdots\right)\end{equation*}\]

Simplifying this expression, we get:

\[\begin{equation*}re^{i\theta}=r\cos(\theta)+ir\sin(\theta)\end{equation*}\]

which is the trigonometric form of the complex number. Therefore, we have shown that the conversion from exponential form to trigonometric form is valid using the Taylor series expansion of the exponential function.


Published: 2023-05-08 08:02:14
Updated: 2023-05-08 08:20:14

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