Q9P - Mathematical Methods in Physical Sciences

Question

Evaluate each of the following in x+iy  form, and compare with a computer solution ${\left(-1\right)}^{\mathbf{i}}$.

Step-by-Step Solution

Answer

The x+iy form of the given equation ${\left(-1\right)}^i is e^{\left[-\left(\pi +2n\pi \right)\right]}$

Step 1: Given Information.

The given expression is ${\left(-1\right)}^i$.

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, $w={\left(-1\right)}^i.$ .

 

Write the given number in polar form.

$w=e^{\left[In{\left(-1\right)}^i\right]}$

 

Use the logarithm property.

$$\begin{gathered} w=e^{\left[i In\left(-1\right)\right]} \\ w=e^{\left[i In\left(r{e}^{i\theta }\right)\right]} \\ w=e^{\left[i In\left(r\right)-\left(\theta +2n\pi \right)\right]} \\ w=e^{\left[i In \left(1\right)-\left(\pi +2n\pi \right)\right]} \\ w=e^{\left[-\left(\pi +2n\pi \right)\right]} \end{gathered}$$

Where $n=0,\pm 1,\pm 2,\pm 3,...$

 

Therefore, x+iy form of the given equation ${\left(-1\right)}^i is e^{\left[-\left(\pi +2n\pi \right)\right]}$.