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Nov
19
comment Is there a continuous function on R such that $f(f(x))=e^{-x}$?
This is only if f required to be real-valued. If it is allowed being complex-valued (but continuous and defined on R), it can exist.
Nov
19
comment Is there exact formula that returns minimal period of a periodic function?
@Travis yes, infimum of all positive periods. Ideally. But if it worked only for non-constant analytic functions, it would be ok.
Nov
19
comment Is there exact formula that returns minimal period of a periodic function?
@GEdgar I want a formula(expression) representation, that would give it for arbitrary analytic function. Like a limit, series, integral, sum etc.
Nov
19
comment Is there exact formula that returns minimal period of a periodic function?
@Travis for such functions the formula ideally would return $0$ like for constants. But I would be satisfied if the formula worked for only analytic functions.
Nov
19
comment Is this similarity just a coincidence?
@alex.jordan digamma is indefinite sum of 1/x if u prefer en.wikipedia.org/wiki/…
Nov
17
comment How to take this integral?
@Ron Gordon typo due to a bug in Mathematica! It puts the argument before the function when copying!
Nov
9
comment Ascribing values to Gamma of negative integers
@Claude Leibovici Numerator for non-Eurer-Masceroni part is Wolstenholme numbers (A001008), denomenator is A102928 multiplyed by factorial.
Oct
31
comment Ascribing values to Gamma of negative integers
@Lucian This InverseFourierTransform[FourierTransform[f[t], t, w] (-I w)^x, w, 1] gives Cos[Pi x]Gamma[1+x] in Mathematica. Now try 0, -1, -2, -3 for x and find Gamma[1+x] from here.
Oct
31
comment Ascribing values to Gamma of negative integers
@Lucian where they need it to be infinite?
Oct
31
comment Ascribing values to Gamma of negative integers
@Lucian "require those values to be infinite" - what applications?
Oct
31
comment Ascribing values to Gamma of negative integers
@Claude Leibovici $$-\frac{((1/f))'')^2}{2(1/f)'}$$
Oct
31
comment How to express this operator through derivetives?
$$-2\frac {-(f(x))^2-f(x)f''(x)h^2+(f'(x))^2h^2}{2f(x)+f''(x)h^2}$$
Oct
31
comment How to express this operator through derivetives?
Is there a common name for this operator (from your answer)? It seems it is constant for logarithm.
Oct
31
comment How to express this operator through derivetives?
Can u please write the final result with $f(x)\ne0$ if it is not that difficult?
Oct
31
comment How to express this operator through derivetives?
how would it look like? Sorry, I do not see where u assumed this.
Oct
31
comment How to express this operator through derivetives?
Works! Except in non-pole points it does not give the original function, but this is even better. Also often need to use lomit on the quotinent.
Oct
31
comment Ascribing values to Gamma of negative integers
@Claude Leibovici or, without using complex numbers: Limit[(Gamma[-n + x] + Gamma[-n - x])/2, x -> 0]
Oct
31
comment Ascribing values to Gamma of negative integers
@Claude Leibovici It gives Gamma[-n] for all other real values except negative integers.
Oct
31
comment Ascribing values to Gamma of negative integers
@Claude Leibovici Mathematica code: Limit[Re[Gamma[-n + I x]], x -> 0]
Oct
31
comment Ascribing values to Gamma of negative integers
@Henry For instance it gives $(1/t)^{-1}=\log|x|+\gamma$ (which is by the way, consistent with discrete integral of $1/t$ which is $\psi(t)$ and asymptotically approaches $\log|x|+\gamma$ at $x\to+\infty$