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Apr
4
comment Does it make sense to learn any other language except English, being a mathematician?
@Howard Langtone consider Gelfond, Calculus of finite differences (1959). inis.jinr.ru/sl/vol1/UH/_Ready/Mathematics/… It has been translated to English only in 1971 in India. It is the only book where I found criteria of possibility to represent an analytic function as Newton series.
Apr
1
comment Has anybody ever considered “full derivative”?
@columbus8myhw this is not closed form...
Apr
1
comment Has anybody ever considered “full derivative”?
@columbus8myhw it is not equal to e.
Apr
1
comment Has anybody ever considered “full derivative”?
@columbus8myhw I think e can be expressed in closed form by modifying the formula.
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@Martin R as $\log |a|$
Mar
28
comment Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$
@Timbuc en.wikipedia.org/wiki/Non-standard_analysis
Mar
28
comment Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$
@abel Cosine integral mathworld.wolfram.com/CosineIntegral.html
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@Arpan Banerjee we get infinity in both numerator and denomenator with this rule. If u know how to apply it properly, make an answer please.
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@Arpan Banerjee logarithm is not differentiable at 0, L'Hopital's rule is not applicable
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@Arpan Banerjee counter-example: $x=1$, $\varepsilon$=2
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@math also if $x-\varepsilon$ <0 the seond factor also becomes infinite, and I am exactly interested in the case $|x|<\varepsilon$...
Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@math are u sure? What if a tends to zero from below? then the second factor in the numerator becomes infinite...
Mar
28
comment Given this operator what is inverse operator?
@Martin R regarding sums to non-integer limits, look here: en.wikipedia.org/wiki/Indefinite_sum anyway, I found what I was looking for.
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?