Question

Sketch the trajectory of the complex-valued function $\mathbf{z}\mathbf{=}{\mathbf{e}}^{\left(-0.1-2i\right)\mathbf{t}}$.

Short answer

The trajectory is in the inward spiral in the clockwise direction.

Step-by-step solution

Euler’s formula

The Euler’s formula is${\mathit{e}}^{\mathbf{i}\mathbf{t}}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{t}\mathbf{+}\mathit{i}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{t}$

Explanation of the solution

Consider the given complex valued function as follows:

$z=e^{\left(0.1-2i\right)t}$

Rewrite the function as follows:

$\begin{array}{l}Z=e^{(0.1+2i)}\\ Z=e^{-0.1t}{e}^{-2it}\end{array}$

By using Euler’s formula, the function can be written as follows:

$$\begin{gathered} z=e^{-0.1t}{e}^{-2it} \\ z=e^{-0.1t}\left(\cos \left(-2t\right)+i\sin \left(-2t\right)\right) \\ z=e^{-0.1t}\left(\cos \left(2t\right)-i\sin \left(2t\right)\right) \\ z=e^{-0.1t}\left(\cos \left(2t\right)-i\sin \left(2t\right)\right) \end{gathered}$$

Graphical representation of the solution

The trajectory of the function is sketched in figure 2 as follows:

Hence, the trajectory of $z=e^{-0.1t}\left(cos\right(2t)-isin(2t\left)\right)$is in the inward spiral in the clockwise direction.