Question

Find the principal value of \(i^{i}\). Rewrite the base, \(i\), as an exponential first.

Short answer

\(i^{i}\) is equal to \(e^{-\frac{\pi}{2}}\) or approximately 0.20788 when calculated to 5 decimal places.

Step-by-step solution

Represent \(i\) in polar form

We know that \(i\) is equal to \(e^{\frac{\pi}{2}i}\) when represented in polar form. This is because \(i\) lies on the imaginary axis and the argument (the angle it makes with the positive real axis, measured in the counter-clockwise direction) is \(\frac{\pi}{2}\) (or 90 degrees).

Substitute \(i\) with its Exponential Equivalent in the Expression \(i^{i}\)

Replace \(i\) with \(e^{\frac{\pi}{2}i}\) in the equation \(i^{i}\). This gives us \((e^{\frac{\pi}{2}i})^{i}\). Then using the power rule of exponents, we get \(e^{-\frac{\pi}{2}}\).

Evaluate the expression

We evaluate \(e^{-\frac{\pi}{2}}\). This is done by calculating the value using the exponential function in a calculator and rounding to some decimal places or by leaving it as is. The principal value is found to be approximately 0.20788, but the principal value in terms of \(e\) is more exact and would be \(e^{-\frac{\pi}{2}}\)

Key concepts

Exponentiation of Complex Numbers

The concept of exponentiation involves raising a number to a power. When dealing with complex numbers, this requires some adjustments.
To find the exponentiation of a complex number like in the exercise, we use polar coordinates. The number is first expressed in exponential form as a base raised to a power which is itself a complex number.
For instance, for the complex number \(i\), we express it as \(e^{i\theta}\), where \(\theta\) represents the angle in the polar form. In the step-by-step solution, this is seen when \(i^i\) is transformed into \((e^{\frac{\pi}{2}i})^i\).
Using exponentiation rules, this translates to \(e^{i\cdot\frac{\pi}{2}i}\), simplifying ultimately to \(e^{-\frac{\pi}{2}}\).

• This process involves converting the complex base to its exponential form.
• Then applying properties of exponents to combine powers efficiently.

Polar Form

Polar form is a way of expressing complex numbers. Complex numbers can be split into two components: real and imaginary. In polar notation, these are represented using a magnitude and an angle.
To work with complex exponentiation, converting numbers to polar form is key.
For example, the imaginary unit \(i\) is shown in polar form as \(e^{\frac{\pi}{2}i}\). Here, the angle \(\frac{\pi}{2}\) (or 90 degrees) reflects \(i\)'s position on the complex plane, lying on the imaginary axis.

• Polar form incorporates both the radius and the angle the number makes with the positive real axis.
• This facilitates operations like multiplication, division, and exponentiation of complex numbers.

Converting to polar form simplifies the handling of powers and roots, since it transforms the problem into manageable multiplication and addition tasks.

Principal Value

The principal value of a complex number solution is the most commonly accepted one, often seen as the primary angle or result.
In exponential expressions like \(i^i\), achieving the principal value involves evaluating the expression within its main branch of results.
In step 3 of the solution, \(e^{-\frac{\pi}{2}}\) is named the principal value. It is regarded as the primary and simplest result derived from the process.

• The principal value ensures the complex power solutions are in a standardized form.
• While there can be multiple results due to periodicity in complex numbers, the principal value is often used for clarity and precision.

This approach aids consistency and reduces ambiguity in complex calculations.