My Coding > Mathematics > Complex numbers > Estimate (-1.2)^x

Estimate (-1.2)^x

Actually, it is a big confusion can take place in the understanding of the function \(f(x) = (-1.2)^x\). It is important to understand, that \((-1.2)^x \neq -1.2^x \)!!! This is because in the first case, we take the fractional power from the negative value and in the second case we take a fraction power from the positive value and then change in sign. This is a crucial difference and it is very important to understand it.

Let's calculate our proper function without this very silly, but common error.

First of all, let's apply the property of power to a product of two numbers:

\[(-1.2)^x = 1.2^x\times (-1)^x\]

It is trivial to calculate \(1.2^x\) and therefore, our main interest will be targeted towards \((-1)^x\).

\[ -1 = e^{i(\pi + 2\pi n)} \\ (-1)^x = e^{i(\pi + 2\pi n)\times x} = e^{i(\pi\times x + 2\pi n\times x)} = \\ = \cos{(\pi\times x + 2\pi n\times x)} + i\sin{(\pi\times x + 2\pi n\times x)} \\ (-1.2)^x = \begin{cases} Re = 1.2^x\times\cos{(\pi\times x + 2\pi n\times x)} \\Im = 1.2^x\times\sin{(\pi\times x + 2\pi n\times x)} \end{cases} \\ n = 0, \pm 1, \pm 2, \pm 3, \cdot\]

As you can see, it is a complicated function and it significantly depends on the value of \(n\). It is much easier to plot this function graphically with Python.

Watch the video, where I explain how to write this Python code to plot this function and do a simple analysis.

For more detailed explanation, please go to watch our video:


Published: 2023-05-11 03:22:56
Updated: 2023-05-11 03:25:11

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