Question

Evaluate each of the following in x+iyform, and compare with a computer solution ${\mathbf{2}}^{\mathbf{i}}$.

Short answer

The x+iy form of the given equation$2^{i}is,e^{\left(-2n\pi \right)}\left[0.77+0.64i\right]$

Step-by-step solution

Given Information.

The given expression is,$2^i$ .

Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of x+iy in which x is the real part and y is the imaginary part.

Convert in rectangular form.

Consider,$w=2^i$ .

Write the given number in polar form.

$w=e^{\left(iIn\left(2\right)\right)}$

Use the logarithm property.

$$\begin{gathered} w=e^{\left(iIn\left(2\right)\right)} \\ w=e^{\left(i\left[In\left(2\right)+2n\pi i\right]\right)} \\ w=e^{\left(iIn\left(2\right)-2n\pi \right)} \\ w=e^{\left(iIn\left(2\right)\right).exp\left(-2n\pi \right)} \\ w=e^{\left(-2n\pi \right)}\left[\cos \left(In2\right)+i\sin \left(In2\right)\right] \end{gathered}$$

Where$n=0,\pm 1,\pm 2,\pm 3,....$

$w=\left[0.77+0.64i\right]\left[1+e^{-2\pi },e^{\left(-4\pi \right)},....\right]$

Therefore, x+iy form of the equation$2^{i}is,e^{\left(-2n\pi \right)}\left[0.77+0.64i\right].$