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Jan
24
comment Need an operator with given properties
@Cameron Williams as $+\infty$
Jan
24
comment Need an operator with given properties
@Cameron Williams zero to $x$ power.
Jan
24
comment Need an operator with given properties
@Cameron Williams the class is any functions defined in the neighbourhood of zero (not necessarily in zero itself), icluding values from affinely extended real line.
Jan
24
comment Need an operator with given properties
@Cameron Williams I need an operator that works primarily on analytic functions, but also desirably on functions of more generalized classes, including non-continuous in zero and distributions.
Jan
24
comment Differentiable only at $x=0$ and $f'(0)>0$
The example of David Mitra satisfies the first two conditions.
Jan
24
comment Differentiable only at $x=0$ and $f'(0)>0$
@user197137 the definition of derivative is $\lim_{h\to 0}\frac{f(x+h)-f(x)}h$. If it is positive, then there are infinitely many such h that $f(-h) < f(0)< f(h)$
Jan
24
comment Differentiable only at $x=0$ and $f'(0)>0$
@user197137 for the function to have positive derivative, such h should exist.
Jan
24
comment Numerical system that includes the limit targets such as $0^+$, $0^-$, $1^+$ etc
@ajotatxe limit will be excessive in this case, in this numerical system the function can be evaluated directly: $\sin 0^+=0^+$. You can see it as the simplifyed system of hyperreals, with $0^+$ substituted for any positive infinitesmal.
Jan
22
comment Why we cannot ascribe values to behavior of functions at poles?
I also think whether anythink will break if we remove infinity. That is, prohibit integration with limits at infinity (only allowing integration to $\omega$).
Jan
17
comment Evaluate the double integral $\int _0^1\int _0^1\frac{x+i}{(1-ix y) \ln (x y)} \,dx\,dy$
Great! But I wonder how is it related to the Euler's $\gamma$ and $\ln \pi/4$. I thought the answer should be somewhat continuous with them. What if we change the $i$ to somewhat like $(-1)^k$ or $e^{i\pi k}$? How the answer will be k-dependent?
Jan
16
comment Why we cannot ascribe values to behavior of functions at poles?
Well. What do u think will happen if we postulate $a \omega =\omega$ but $\omega+a\ne \omega$, $\omega-\omega$ undefined?
Jan
12
comment What is $0^{i}$?
@soultrane there are multiple formulas, all give the same result, why you do not like it? I can explain with details. Wolfram specifics are not involved.
Jan
12
comment What is $0^{i}$?
@soultrane the number after "^" determines the derivative order. Win $0$ it gives sin, with $1$ it gives cosine, with -1 it gives -cosine etc. But it is off-topic here, I can explain with more details as a separate answer to your question.
Jan
12
comment What is $0^{i}$?
@hjhjhj57 I mention that the mean of this diverging series is $0$.
Jan
12
comment What is $0^{i}$?
@hjhjhj57 why u do not criticize Slade for the same?
Jan
12
comment What is $0^{i}$?
No, it is not wrong, it is widely accepted defginition. Prove it is wrong.
Jan
12
comment What is $0^{i}$?
"follows the convention $0^x=0$" - where u got this "convention" from? "and $0^0=0$ is questionable" - it is not questionable, it it plainly wrong, it is usually define to be 1 or left undefined.
Jan
12
comment What is $0^{i}$?
@soultrane i-th order derivative of sine is $i \sinh (\frac \pi 2-ix)$: tinyurl.com/kdmdkvf
Jan
12
comment What is $0^{i}$?
@hjhjhj57 the topic starter already indicated that the limit does not exist, so this comment does not add anything. He was asking what value can be assigned to it nevertheless. The mean value of the sequence is $0$ (see my ansswer).
Jan
12
comment What is $0^{i}$?
@hjhjhj57 yes, and what?