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Oct
14
comment How to create a reliable natural integral operator?
@Daniel Lichtblau what if Fourier integral in taken over whole complex plane?
Oct
13
comment How to create a reliable natural integral operator?
@Daniel Lichtblau well, exponent HAS a formal Fourier transform, $$\mathcal{F}[e^{ax}]=\delta(x-\frac a{2\pi i})$$ So I wonder, if it is possible to make Mathematica to know it?
Oct
13
comment How to create a reliable natural integral operator?
@Daniel Lichtblau that's why my incidential discovery of the Fourier formula is so valuable (it can find natural integral even for Dirac Delta and sgn(x)) math.stackexchange.com/a/906379/2513 I wonder whether it is easier to polish the Fourier formula. For instance, why it works for sine and cosine but does not for hyperbolic functions? Is there a way to fix it?
Oct
13
comment How to create a reliable natural integral operator?
@Daniel Lichtblau no, it won't, 1st derivative is non-zero, all others are zero. Newton series will not converge for such function. That's why Fourier method is used. But the question is how to obtain the functions in closed form.
Oct
13
comment How to create a reliable natural integral operator?
@Daniel Lichtblau the second formula diverges for f(x)=x. You can see some functions for which it works in my reply to Dr. Wolfgang Hintze above.
Oct
13
revised How to create a reliable natural integral operator?
added 271 characters in body
Oct
13
comment Oddify an even function and vice versa?
@Kuba this works: f[x_]:=Sinh[x] Manipulate[Plot[Evaluate[{Sum[(x)^(2 n)/((2 n)!) Sum[Binomial[n-1/2,m] Sum[Binomial[m,k] (-1)^(m-k)Function[s,Evaluate[D[f[s],{s,2 k+1}]]][0],{k,0,m}],{m,0,r}],{n,0,r}],f[x]}],{x,-2,2},AspectRatio->Automatic],{{‌​r,25},1,50,2}]
Oct
13
comment Oddify an even function and vice versa?
@Kuba also it should be (2n)!, not 2 n!
Oct
13
comment Oddify an even function and vice versa?
@Kuba it also seems your formula has error. Why do u evaluate the derivative at x rather than 0?
Oct
13
comment Oddify an even function and vice versa?
@Kuba for the first pair of functions it will diverge, one needs to modify the formula so it to work. But essentially the principle is the same.
Oct
13
revised Oddify an even function and vice versa?
edited title
Oct
13
comment How to create a reliable natural integral operator?
@Dr. Wolfgang Hintze yes, it is possible to simplify it this way. One can also use $$f^{(-1)}(x)=\lim_{n\to\infty}\binom {-1}n\sum_{k=0}^n\frac{s-n}{s-k}\binom nk(-1)^{n-k}f^{(k)}(x)$$ or $$f^{(-1)}(x)=\lim_{n\to\infty}\frac{\sum_{k=0}^{n} \frac{(-1)^k f^{(k)}(x)}{(-1-k)k!(n-k)!}}{\sum_{k=0}^{n} \frac{(-1)^k }{(-1-k) k!(n-k)!}}$$ with a benefit of the leter formula in that it can be adapted to cases when particular derivative in a point becomes infinite.
Oct
12
comment How to create a reliable natural integral operator?
@Daniel Lichtblau is there a way to implement the second definition in Mathematica in such a way that it would produce results in a closed form (like the first formula does)?
Oct
12
comment Oddify an even function and vice versa?
@JohnD No, I am interested in those operators I posted in the question. I am not interested in piecewise extensions. Among other properties these operators have a property that they asymoptotically approach each other as $x\to+\infty$ and their sum goes to zero as $x\to-\infty$. Also the value of the even function in $x=0$ is equal to the derivative of the odd function.
Oct
12
comment How to create a reliable natural integral operator?
@Daniel Lichtblau Look here for a hint how the definitions may be related: mathoverflow.net/a/179218/10059
Oct
12
comment Oddify an even function and vice versa?
@Kuba I only can construct these operators in numerical form in Mathematica, I have no idea how to make such operator that would give results in closed form.
Oct
12
revised Oddify an even function and vice versa?
added 81 characters in body
Oct
12
asked Oddify an even function and vice versa?
Oct
12
comment How to create a reliable natural integral operator?
@Jens the question includes the definition, so what is unclear?
Oct
12
comment How to create a reliable natural integral operator?
@Dr. Wolfgang Hintze $f(x)=2^x$, $f(x)=e^x$, $f(x)=e^{x \cos a} \sin(x \sin a), 0\le a\le\pi/3$, $f(x)=x^2 e^x$, $f(x)=P(x) e^x$.