I have been going over Courant’s textbook on Calculus (“Differential and Integral Calculus, Vol. 1”) and I have stumbled upon a problem I literally have no idea how to approach. It is located in the problem set of Appendix 2 to Chapter 1, and it states that I shall evaluate the expression $(3-3i)^{\frac{2}{3}}$. The problem is that (1) I have the understanding of how to evaluate integer powers of complex numbers, but have no idea how to deal with rational powers (the non integer ones in particular), (2) when I tried to solve it, I arrived at 3 distinct values for the expression; knowing that only one of them must be right (the Answers and Hints section says so) I plugged in those other 2 values into a calculator and found out that those did not satisfy the criterion of being equal to the desired expression. How do I even approach this problem and why am I getting some answers wrong? Is the latter in some connexion to the fact that raising a complex number to a power is not a bijective operation?

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    $\begingroup$ Formally speaking, you are right and there are 3 values. But Courant might use a convention with principal branch of Log, you should check the book. $\endgroup$ Oct 23 at 23:33

In this case, it is helpful to read the text and try to understand what the author is asking the reader to do. Near the bottom of page 40 of Courant's text, he writes

If we use this trigonometrical representation the multiplication of complex numbers takes a particularly simple form. For then \begin{align} c\cdot c' &= r(\cos\theta + i\sin\theta)\cdot r'(\cos\theta' + i\sin\theta')\\ &= rr'(\cos\theta\cos\theta' - \sin\theta\sin\theta') + i (\cos\theta\sin\theta' + \sin\theta\cos\theta').&&\text{[sic]} \end{align} ... We therefore multiply complex numbers by multiplying their absolute values and adding their amplitudes. ... It leads us at once to the relation $$ (\cos\theta + i \sin\theta)^n = \cos n\theta + i \sin n\theta,$$ which e.g. allows us to solve the equation $x^n = 1$ for positive integers $n$...

Note that there appears to be a typo in my copy of the text (there is a set of parentheses missing); I have reproduced the quoted text exactly here.

In problem 4(h) of the appendix to Chapter 1, Courant asks the reader to

Work out the following expressions, and state the modulus and amplitude of each of the numbers involved and of the answers:

(h) $(3-3i)^{2/3}$.

Presumably, the goal is to write $3-3i$ in its trigonometric (or polar) form, then apply the corollary to De Moivre's theorem, stated above. In this case,

$$ r = |3-3i| = 3\sqrt{2} \qquad\text{and}\qquad \theta = \arg(3-3i) = -\frac{\pi}{4}. $$

Therefore \begin{align} (3-3i)^{2/3} &= r^{2/3} \left[ \cos\!\left(\frac{2}{3} \theta\right) + i \sin\!\left( \frac{2}{3} \theta \right) \right] \\ &= (3\sqrt{2})^{2/3} \left[ \cos\!\left(-\frac{2}{3} \frac{\pi}{4}\right) + i \sin\!\left( -\frac{2}{3}\frac{\pi}{4} \right) \right] \\ &= (3\sqrt{2})^{2/3} \left[ \cos\!\left(-\frac{\pi}{6}\right) + i \sin\!\left( -\frac{\pi}{6} \right) \right]. \end{align}

One could, of course, simplify this further, but this seems to address the core of the exercise.


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