general formula for
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deceptie3j
Answered question
2022-06-05
general formula for when is negative I'm looking for a general formula for solving a problem of the form when . It seems like the formula is , but I would like to know how this is derived, or a proof for it. I got this this from looking at the answer of wolfram alpha, so this formula may not be correct.
Answer & Explanation
iceniessyoy
Beginner2022-06-06Added 27 answers
The complex logarithm function is defined as
with the usual, real logarithm. But, as we know, the argument of a complex number is defined only up to an integer multiple of , so the above definition in fact gives us infinite possibilities. Another fact, way more advanced, is that is an acute problem here, and every time we "go around" the origin we add (substract) a multiple of to the number's argument, so in order to have a more or less nice logarithm function we must choose a complex "branch" for the function, and in this case this means to choose a chunck of the complex plane that we're going to "throw away" so that in the remaining domain the function's well and nicely defined. The main branch here is the non-negative real axis , and in the function is well defined, continuous and all. Please do observe then that , for any real