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Oct
28
comment How to make $\log x^a = a\log x$ work using multivalued complex approach
So if $f(1) = \{0, 1\}$, we established that $2f(1) = \{0, 1, 2\}$. What is $\frac{1}{2} f(1)$ equal to?
Oct
5
comment How to make $\log x^a = a\log x$ work using multivalued complex approach
I have a Python code that tests those identities here: theoretical-physics.net/dev/math/…
Oct
4
comment How to make $\log x^a = a\log x$ work using multivalued complex approach
For example, what if $a={1\over2}$? What are all the possible values for ${1\over2}\log z$?
Oct
3
comment How to make $\log x^a = a\log x$ work using multivalued complex approach
I think you are right! Do you have some references for how to do arithmetic with multivalued functions? E.g. what if "a" is not an integer (I know that's not the particular example I posted)?
Oct
3
comment How to make $\log x^a = a\log x$ work using multivalued complex approach
Just in case you can't see my edit, the final equation that you get, using your notation, is: $\log(k,x^a)=a\log(l, x)+2\pi i \left(k - la + \left\lfloor \pi-\Im (a\Log(x)) \over 2\pi \right\rfloor\right) ,k,l\in\mathbb{Z}$
Oct
3
comment How to make $\log x^a = a\log x$ work using multivalued complex approach
You need to be careful, it is not true for all $z$ that $\Log(e^z) = z$, but rather it is true for all $z$ that $\Log(e^z) = z + 2\pi i \left\lfloor \pi-\Im z \over 2\pi \right\rfloor$. I have proposed edits to your answer that fixes this problem (I don't know if you can see them or not). Either way, I gave you the bounty for your work, but it seems that the answer is that it is only possible if you use the floor function. @dafinguzman, yes, your winding number is precisely the floor function in all my equations.
Oct
3
comment How to make $\log x^a = a\log x$ work using multivalued complex approach
I think you made a mistake in equation (1), it should rather be $a\log(x) = a(\Log(x) + 2\pi i n) = a\Log(x) + 2 a\pi i n$, so then the corrected (1) and your (2) are not equal.
Oct
2
comment How to make $\log x^a = a\log x$ work using multivalued complex approach
Thanks for the answer. Yes, if you have to choose a specific branch, e.g. the principal branch with $\Log(z)$ as in your quote from the book, then the only way to have a formula valid for all $x$ and $a$ is to use the very first equation in my question: $\Log x^a = a\Log x + 2\pi i \left\lfloor \pi-\Im (a\Log x) \over 2\pi \right\rfloor$, which always holds. However, I thought that somehow you can make the "more natural" formula $\log x^a = a\log x$ work with multivalued $\log(z)$, perhaps by some convention how to evaluate it. Your answer seems to suggest that there is no such thing.
Oct
1
comment How to make $\log x^a = a\log x$ work using multivalued complex approach
I am interested in how to prove the relation $\log x^a = a \log x$ where $\log z$ is a multi-value analytic function. If you don't like $x=-1$, then use $x=1$. The result doesn't change, there are some values on LHS that are not included in the RHS.
Sep
30
comment How to make $\log x^a = a\log x$ work using multivalued complex approach
The problem is that there are are members of LHS (e.g. $n = 1$) that are not members of RHS (no matter what you choose for $m$). If that's the case, how can LHS be equal to RHS? Or to be specific, in what exact sense is it equal?
Sep
28
comment Principal Value of complex log
You made a mistake there, $\text{Log}^i\,(1-\sqrt3\,i) \neq e^{i\,\text{Log}\,(1-\sqrt3\,i)}$, but rather $\text{Log}^i\,(1-\sqrt3\,i) = e^{i\,\text{Log} \text{Log}\,(1-\sqrt3\,i)}$.
Nov
18
comment Derivative of big O symbol
$\sin(x^2)$ is infinitely differentiable at $x=0$ and the series is just a polynomial, so for it $O'(1) = O(1)$. You might be expanding around $x=\infty$, while I am asking about $x=0$. I clarified the question about this.
Jan
16
comment Is odd continuous function differentiable at $x=0$?
@N.S. Please do, I didn't want to put your solution into answers myself.
Jan
16
comment Solution of functional equation $f(x/f(x)) = 1/f(x)$?
Very nice! So if the function $f(x)$ is analytic, then you prove that $f(x)=1$. Assuming only that $f(x)$ is continuous, then the only possible other solutions are not analytic. That helps a lot.
Jan
16
comment Solution of functional equation $f(x/f(x)) = 1/f(x)$?
Right. So the domain of $g(x)$ must be an interval (e.g. not a union of two disjoint intervals). However, what if we only solve this on an interval [1, 10], let's say? I guess we would run into some contradictions with domains and ranges in the functional equation.
Jan
16
comment Solution of functional equation $f(x/f(x)) = 1/f(x)$?
Because $g(x)=1/x$ is also a solution of $g(g(x))$ and somehow it got eliminated. So something is not right. (Of course, it would get eliminated later anyway due to $g'(0)=1$, but that's not the point.)
Jan
16
comment Is odd continuous function differentiable at $x=0$?
N.S., you are right! $\frac{f(x)}{x} = \sin\frac{1}{x^2}$ which oscillates between -1 and 1 and so the limit does not exist. So this function is not differentiable at $x=0$. Thanks!
Sep
21
comment How to prove asymptotic limit of an incomplete Gamma function
I verified your steps, I think it's all correct. I am accepting your answer as it gives a simple proof.
Sep
21
comment How to prove asymptotic limit of an incomplete Gamma function
Thanks! As a matter of fact, I actually wanted to prove that $\gamma(z, x)\over\Gamma(z)$ goes to zero, but thought it'd be easier to do it with the upper incomplete gamma function.
Sep
21
comment How to prove asymptotic limit of an incomplete Gamma function
So you proved that $\gamma(z, x) < {x\over z-1-x}\Gamma(z)$ for $z-1 > x > 0$ and from that the limit follows immediately. Nice!