The value of `i^(i)`, is

Options

A. `-pi/2`
B. `e^(-pi/2)`
C. `e^(pi/2)`
D. none of these

Explanation

To find the value of \( i^i \), we can follow these steps:

### Step 1: Define the expression
Let \( A = i^i \).

### Step 2: Take the natural logarithm
Taking the natural logarithm of both sides, we have:
\[ \ln A = \ln(i^i) \]
Using the property of logarithms, this simplifies to:
\[ \ln A = i \ln i \]

### Step 3: Find \( \ln i \)
To find \( \ln i \), we express \( i \) in exponential form. Recall that:
\[ i = e^{i\frac{\pi}{2}} \]
Thus,
\[ \ln i = \ln\left(e^{i\frac{\pi}{2}}\right) = i\frac{\pi}{2} \]

### Step 4: Substitute \( \ln i \) back into the equation
Now substituting \( \ln i \) back into our equation for \( \ln A \):
\[ \ln A = i \cdot i\frac{\pi}{2} = -\frac{\pi}{2} \]

### Step 5: Exponentiate to solve for \( A \)
To find \( A \), we exponentiate both sides:
\[ A = e^{-\frac{\pi}{2}} \]

### Conclusion
Thus, the value of \( i^i \) is:
\[ i^i = e^{-\frac{\pi}{2}} \]