Wikipedia:Reference desk/Archives/Mathematics/2011 December 26

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

December 26

Confused about complex exponentiation

The article on exponentiation says that, if a, b, and c are complex numbers, then in general, $(a^{b})^{c}\neq a^{bc}$. But there are obvious times when the rule does work, like when computing exponentials: eg $i^{i}=(e^{i{\frac {\pi }{2}}})^{i}=e^{(i)(i{\frac {\pi }{2}})}=e^{-{\frac {\pi }{2}}}$. So my questions are, 1) when does the rule work?, and 2) when it does work, why does it work? Is it an axiom, or can it be proven? And final question: does $a^{b+c}=a^{b}a^{c}$ always? Thanks. 65.92.7.9 (talk) 06:09, 26 December 2011 (UTC)

See Exponentiation#Computing complex powers about your 'obvious times when the rule does work'. The problem is that complex exponentiation doesn't have a unique value. It has a generally agreed principal value and that's about it. For the second part the principal value is the same and you don't get any different set of values from either side so in that sense they are equal. Dmcq (talk) 08:31, 26 December 2011 (UTC)

I was assuming the principle value was taken during exponentiation. 65.92.7.9 (talk) 08:48, 26 December 2011 (UTC)

If c is irrational then all is rather simple: The rule $(a^{b})^{c}=a^{bc}$ works, if and only if $-\pi <Im(b\ln(a))\leq \pi$. However, if c is rational, then that's more complicated: The rule $(a^{b})^{c}=a^{bc}$ works, if and only if there is an integer k such that $Im(cb\ln(a))\mod 2\pi =c(Im(b\ln(a))\mod 2\pi )+2k\pi$.

As you see, it's not recommended to use the rule $(a^{b})^{c}=a^{bc}$ in the complex plane, even when referring to the principle values only. However, when referring to the principle values only - you can use the other rule: $a^{b+c}=a^{b}a^{c}$, which directly derives from the very definition of the power function: $a^{1}=a$, and $a^{n}=a^{n-1}*a$, as well as from some trivial holomorphic considerations.

Hope this helps. 87.68.254.218 (talk) 14:14, 26 December 2011 (UTC)

I don't think the article is making the claim that (a^b)^c = a^bc for arbitrary complex numbers. If it was then it would be wrong and I'd fix it but I've gone through the article several times and can't find what the OP is referring to.--RDBury (talk) 13:51, 26 December 2011 (UTC)

The OP hasn't claimed what you think they claimed. On the contrary: they claimed: "The article on exponentiation says that, if a, b, and c are complex numbers, then in general, $(a^{b})^{c}\neq a^{bc}$. But there are obvious times when the rule does work".

I think they just replaced (not on purpose) the sign ≠ by the sign /=. 87.68.254.218 (talk) 14:14, 26 December 2011 (UTC)

You're right, I was misreading it.--RDBury (talk) 19:45, 26 December 2011 (UTC)

Probably you misread because of improper 'not equal' notation. I just corrected that. --CiaPan (talk) 17:13, 29 December 2011 (UTC)