Evaluating the Complex Expression $i^i$

We are asked to evaluate the mathematical expression $i^i$. This process of mathematical evaluation involves understanding exponentiation with complex numbers. The expression $i^i$ uses the imaginary unit $i$.

How to Evaluate $i^i$ Using Complex Exponentiation

To evaluate a complex number raised to a complex power, like $z^w$, we use the definition involving the complex logarithm:

$$z^w = e^{w \log z}$$

Here, our base is $z=i$ and our exponent is $w=i$.

Finding the Logarithm of $i$

The logarithm of a complex number $z$ is given by:

$$\log z = \ln|z| + i \arg(z)$$

For the imaginary unit $i$:

• The magnitude $|i|$ is 1. So, $\ln|i| = \ln(1) = 0$.
• The argument $\arg(i)$ is the angle from the positive real axis to the point representing $i$ in the complex plane. Using concepts related to Euler's formula, the principal value is $\frac{\pi}{2}$. The general value is $\frac{\pi}{2} + 2n\pi$, where $n$ is an integer ($\dots, -2, -1, 0, 1, 2, \dots$).

We can rewrite the general argument as $\frac{\pi + 4n\pi}{2} = \frac{(1+4n)\pi}{2}$.

So, the general value of the logarithm of $i$ is:

$$\log i = \ln(1) + i \left(\frac{\pi}{2} + 2n\pi\right)$$

$$\log i = 0 + i \frac{(1+4n)\pi}{2}$$

$$\log i = i \frac{(4n+1)\pi}{2}$$

This uses the concept of the general value of the logarithm, which is crucial for complex exponentiation like evaluating $i^i$. The connection between the polar form and exponential form via Euler's formula is fundamental here.

Calculating $i^i$

Now we can substitute $z=i$, $w=i$, and $\log i = i \frac{(4n+1)\pi}{2}$ into the formula $i^i = e^{i \log i}$:

$$i^i = e^{i \cdot \left(i \frac{(4n+1)\pi}{2}\right)}$$

$$i^i = e^{i^2 \frac{(4n+1)\pi}{2}}$$

Since $i^2 = -1$, we have:

$$i^i = e^{-1 \cdot \frac{(4n+1)\pi}{2}}$$

$$i^i = e^{-\frac{(4n+1)\pi}{2}}$$

This expression represents the general value of $i^i$. The specific value depends on the integer $n$. When $n=0$, we get the principal value:

$$i^i = e^{-\frac{(4(0)+1)\pi}{2}} = e^{-\frac{\pi}{2}}$$

This calculation involves essential concepts of complex numbers and exponentiation.

General Value vs Principal Value

The general value of $i^i$ is $e^{-\frac{(4n+1)\pi}{2}}$, where $n$ is any integer. The principal value is obtained by taking $n=0$, resulting in $e^{-\frac{\pi}{2}}$. Understanding the difference between the general value and the principal value is important in complex analysis.

Comparing the Result with Options

We found that the general value of the mathematical expression $i^i$ is $e^{-\frac{(4n+1)\pi}{2}}$.

Let's look at the options provided:

1. \(e^{\frac{\pi}{2}}\)
2. \(e \frac{-\pi}{2}\)
3. \(e^{(4 n+1)} \frac{\pi}{2}\)
4. \(e \frac{-(4 n+1)^\pi}{2}\)

Comparing our derived result $e^{-\frac{(4n+1)\pi}{2}}$ with the options, Option 4 text is \(e \frac{-(4 n+1)^\pi}{2}\), which literally represents $e \times \frac{-(4 n+1)^\pi}{2}$. This interpretation does not match our derived result $e^{-\frac{(4n+1)\pi}{2}}$. However, given that the calculation of $i^i$ leads to an exponentiation of $e$, and Option 4 contains $e$ and terms related to $n$ and $\pi$ divided by 2 in a mathematical expression, it is highly probable that Option 4 was intended to represent $e^{-\frac{(4n+1)\pi}{2}}$ despite the formatting in the provided text. Assuming the intent was to match the general value of $i^i$, Option 4, when correctly interpreted as $e^{-\frac{(4n+1)\pi}{2}}$, corresponds to our derived general value from the mathematical evaluation of $i^i$.