How do I find real and imaginary parts of complex powers like i^i and (1+i)^(1−i)?

In one line

Use \(a^b=e^{b\log a}\): find \(\log a\) for the complex base, multiply by the exponent, separate \(e^{u+iv}=e^u(\cos v+i\sin v)\).

Explained simply

Your calculator cannot do \(i^i\) directly, but it can do \(e^{\text{something}}\). The trick is the same as for real numbers, where \(2^3=e^{3\log2}\). Convert any power into an exponential using the logarithm, and then Euler's formula splits it into real and imaginary parts.

Technically, for complex \(a\ne0\) and \(b\): \(a^b=e^{b\log a}\). Since \(\log a\) has many values, so can \(a^b\); the principal value uses the principal log.

The surprising result of this topic is that \(i^i\) is a real number. Many students find this hard to believe, but it follows directly from \(i=e^{i\pi/2}\): raising it to the power \(i\) gives \(e^{i\cdot i\pi/2}=e^{-\pi/2}\). Once you trust the rule \(a^b=e^{b\log a}\), every such power becomes a routine calculation in three short steps.

How it works

1. Write \(\log a=\log r+i(\theta+2n\pi)\) (use \(n=0\) for principal value).
2. Multiply by the exponent \(b=c+id\) and simplify to \(u+iv\).
3. Then \(a^b=e^{u}(\cos v+i\sin v)\): real part \(e^u\cos v\), imaginary part \(e^u\sin v\).
4. For "prove that" questions, take log of both sides first and compare real and imaginary parts.

Key formula

\(a^b=e^{b\log a}\), \(\ e^{u+iv}=e^u(\cos v+i\sin v)\), where \(\log a=\log|a|+i(\arg a+2n\pi)\).

Quick example

(a) \(i^i\): \(\log i=i\left(\dfrac\pi2+2n\pi\right)\). So \(i^i=e^{i\cdot i(\pi/2+2n\pi)}=e^{-(\pi/2+2n\pi)}\). All values are real; the principal value is \(e^{-\pi/2}\approx0.208\).

(b) \((1+i)^{1-i}\), principal value: \(\log(1+i)=\dfrac12\log2+\dfrac{i\pi}4\).

\((1-i)\left(\dfrac12\log2+\dfrac{i\pi}4\right)=\left(\dfrac12\log2+\dfrac\pi4\right)+i\left(\dfrac\pi4-\dfrac12\log2\right)\).

So \((1+i)^{1-i}=\sqrt2\,e^{\pi/4}\left[\cos\left(\dfrac\pi4-\dfrac12\log2\right)+i\sin\left(\dfrac\pi4-\dfrac12\log2\right)\right]\).

Common mistakes

• Forgetting \(i\cdot i=-1\) when multiplying, giving \(i^i\) as complex instead of real.
• Dropping the \(2n\pi\) when the question asks for "all values" or "general value".
• Writing \(e^{u+iv}=e^u+e^{iv}\); it is a product, \(e^u\cdot e^{iv}\).

How MU asks it

Frequent 5/6-mark parts: "Prove that \(i^i\) is wholly real and find its principal value", "Find the real part of \((1+i)^{i}\)", or "If \(i^{\alpha+i\beta}=\alpha+i\beta\), prove \(\alpha^2+\beta^2=e^{-(4n+1)\pi\beta}\)". For full marks write \(a^b=e^{b\log a}\), the general log, the multiplication, and the final \(e^u(\cos v+i\sin v)\) form.