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Mar
28
comment Given this operator what is inverse operator?
@Martin R 4 well the limit is added for some functions that have t in the denomenator (like 1/x). The sum should be computerd in closed form then take limit. The limit is not necessary for other functions.
Mar
28
comment Given this operator what is inverse operator?
@Olivier Oloa actually, $$\Delta_{sym}[f(x)]=(\Delta_{full}[f(x)]+\Delta_{full}[f(x-\varepsilon)])/2$$, but what does it help?
Mar
26
comment Why hyperreal numbers are built so complicatedly?
Differentiability: $$(f(x+\varepsilon)-f(x-\varepsilon))/(2\varepsilon)$$ What needs more definition with it?
Mar
26
comment Why hyperreal numbers are built so complicatedly?
Well what questions still remain unanswered with the definition from the question?
Mar
26
comment Has anybody ever considered “full derivative”?
Does it mean time scales?
Mar
26
comment Has anybody ever considered “full derivative”?
@Mark S. what a problem in defining $\sin \varepsilon$? It just can be represented as a series or in closed form... Where the problem is?
Mar
25
comment Has anybody ever considered “full derivative”?
@Kevin Carlson is this field truly hyperreal?
Mar
25
comment Has anybody ever considered “full derivative”?
@Mark S. so basically if to add to rationals $\varepsilon$ as in this post we get Levi-Civita field?
Mar
25
comment Why hyperreal numbers are built so complicatedly?
What questions about calculus remain unanswered?
Mar
25
comment Has anybody ever considered “full derivative”?
@Mark S. $No(\omega)$ is a hyperreal system, subfield of surreals ohio.edu/people/ehrlich/Unification.pdf
Mar
25
comment Has anybody ever considered “full derivative”?
@Hayden I am about introducing one element, $\varepsilon$, similarly to how complex $i$ introduced. One can argue such system would be undefinable, but then complex numbers are also undefinable because $i$ is indistinguishable from $-i$. By the way, in the context of surreals, hyperreal numbers can be considered a subfield $No(\omega)$. Then this $\omega$ has definite meaning: it is considered equal to the first infinite ordinal.
Mar
25
comment General form for series coefficient of Taylor series expansion of $(x+1)^{1/x}$
@mickep around zero
Mar
25
comment Has anybody ever considered “full derivative”?
@Hayden in usual non-standard analysis they usually do not introduce distinguished elements $\varepsilon$ and $\omega$, thus they cannot uniquely define full derivative.
Mar
25
comment Has anybody ever considered “full derivative”?
@kjetil b halvorsen this is not q-derivative, totally different thing.
Mar
25
comment Why hyperreal numbers are built so complicatedly?
Thanks for pointing out the Levi-Civita field
Mar
23
comment Who discovered the first explicit formula for the n-th prime?
It is not elementary functions.
Mar
17
comment Why hyperreal numbers are built so complicatedly?
Why is the downvote?
Mar
15
comment What function satisfies the following equation?
What if additional condition is imposed, that isfunction $f(x+\pi/4)e^{-x}\Gamma(x/\pi +1/4)$ is even?
Mar
10
comment What function satisfies the following equation?
This is Great!!! Thank you very much!
Mar
10
comment What exactly *is* the Riemann zeta function?
The latest identity is wrong :-(