- #1
Bashyboy
- 1,421
- 5
Homework Statement
Is ##z^{1/4}## multi-valued or single-valued? How many possible values can it have in general?
Homework Equations
##z## raised to some complex power ##c## is defined as ##z^c = e^{c \log z}##.
The Attempt at a Solution
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z^{\frac{1}{4}} = e^{\frac{1}{4} \log z} \iff
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z^{\frac{1}{4}} = e^{\frac{1}{4} (\ln|z| + i \arg z)} \iff
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z^{\frac{1}{4}} = e^{\ln |z|^{1/4} + \frac{i}{4} (\theta + 2 n \pi)} \iff
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z^{\frac{1}{4}} = \sqrt[4]{|z|} e^{i(\frac{\theta}{4} + \frac{\pi}{2} n)}
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I would say, "yes, it is multi-valued, as each value of ##n## gives you a distinct solution." However, in general, I am unsure of how many solutions there are. For a given ##z##, doesn't ##\sqrt[4]{|z|}## give us four distinct roots; and if we let ##n## be 0, 1, 2, 3, would we get four distinct solutions (n=4 and beyond would give us repetitive solutions)? Furthermore, how do I know the solutions I get from ##\sqrt[4]{|z|}## correspond to the ones gotten by enumerating values of ##n##?