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

Last 10 artitles

• 2025-04-05 16:11:46 - Simple Python Projects
• 2025-04-05 16:09:15 - Currency exchange rate
• 2025-03-28 08:41:03 - Privacy Policy - https://mycoding.uk/
• 2024-11-29 16:38:37 - Estimating super-sonic missile velocity in the atmosphere
• 2024-11-27 19:16:02 - YouTube manipulations
• 2024-10-31 19:17:21 - Hydrogen (H) electron density
• 2024-10-31 16:09:56 - Period 1 element
• 2024-10-31 12:46:28 - Atom electron clouds
• 2024-10-31 12:04:50 - Ephemerides
• 2024-10-19 14:46:49 - Polynomial fit with Numpy polyfit

9 popular artitles

• Python Panda (2231)
• Python NumPy (1625)
• Python: How to make absolute links in BeautifulSoup (1063)
• In-place and out-place Numpy functions (1008)
• Python: How to make video from images (977)
• Linear first order ODE (953)
• NumPy: What the difference between mgrid and meshgrid (905)
• Plotting of Interactive Electric field due to point charges with Matplotlib (803)
• Polynomial fit with Numpy polyfit (784)