Q11P - Mathematical Methods in Physical Sciences

Question

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

Step-by-Step Solution

Answer

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

Step 1: Given Information.

The given expression is, $2^i$ .

Step 2: 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.

Step 3: Convert in rectangular form.

Consider, $w=2^i$ .

 

Write the given number in polar form.

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

 

Use the logarithm property.

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

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 $2^i is, e^{\left(-2n\pi \right)}\left[0.77+0.64i\right].$