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Sep
11
comment How to solve the following delay differential equation?
what is $g_1$?.
Sep
11
asked How to solve the following delay differential equation?
Sep
2
revised What does this expression give?
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Sep
2
comment What does this expression give?
@Mhenni Benghorbal no
Sep
1
revised What does this expression give?
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Sep
1
revised What does this expression give?
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Sep
1
asked What does this expression give?
Aug
27
comment Is there an easier way to find the “natural” integration constant?
"When h>0, it does not converge, but the most logical value would be extending it using it the geometric series formula. When h<0, the series converges, and the h cancels the negative." - so, you have just "discovered" that $-\int_{-\infty}^x e^x$ converges, while $-\int_{x}^{\infty} e^x$ does not. Very novel!
Aug
27
comment Is there an easier way to find the “natural” integration constant?
Riemann sum has infinity on top of the sum if the integral has infinity on top of the sum. If $h$ is negative, the integral will be $-\int_{-\infty}^x f(x)$. The limit without sign before $h$ will exist if and only if $-\int_{-\infty}^x f(x)=-\int_{x}^{\infty} f(x)$. This is not the case for any of the functions you mentioned, neither for sin or exponent. And for sin and cos neither of these integrals exists. So this suggestion is total rubbish.
Aug
27
comment Is there an easier way to find the “natural” integration constant?
How you got positive value for $e^x$ by summing all-positive terms and taking it with negative sign? For exponent all the terms under sum are positive.
Aug
27
comment Is there an easier way to find the “natural” integration constant?
And look also that your definition just does not depend on the value of the function at negative $x$, completely.
Aug
27
comment Is there an easier way to find the “natural” integration constant?
How you got the results for sine and exp? Look at the formula at the bottom of the first page here: mathcentre.ac.uk/resources/… or at the formula (1) here: www3.ul.ie/~mlc/support/Loughborough%20website/chap15/15_1.pdf It is Riemann sum (en.wikipedia.org/wiki/Riemann_sum) for $-\int_x^\infty f(t)dt$
Aug
27
comment Is there an easier way to find the “natural” integration constant?
Err, by the way, your formula looks like the integral sum (definition of definite integral) for $-\int_x^\infty f(t) dt$. This should give 1 in $x=0$ only for $-e^{-x}$ and actually far from "natural integral". Am I missing something?
Aug
23
accepted Is there an easier way to find the “natural” integration constant?
Aug
23
comment Is there an easier way to find the “natural” integration constant?
@Rahul for what it does not converge?
Aug
23
comment Is there an easier way to find the “natural” integration constant?
@Rahul and for n=0 the value also can be estimated as zero because $\delta(x)/x$ is an odd function whose integral from $-\infty$ to $\infty$ should be zero. This coincides with the notion that natural integral of an even function should be always zero in x=0 because it is odd.
Aug
23
comment Is there an easier way to find the “natural” integration constant?
@Rahul also it diverges for odd negative n (which is understandable given that the integrals have a pole in zero) and converges only in generalized sence when n is negetive and even (then the pole in zero has opposite signs). But shifting these functions one still establish the constant, say, in x=1.
Aug
23
comment Is there an easier way to find the “natural” integration constant?
@Rahul it seems it does not. But I have just implemented the formula in Mathematica, and this is just what I obtained, it uses its own tables of Fourier transforms (the FurierTransform function more often converges than just integral, although gives the same result). It extensively uses Dirac Delta function.
Aug
23
comment Is there an easier way to find the “natural” integration constant?
@Rahul it seems the other method I found, resolves the question of polinomials! Only for $f(x)=const$ the expansion diverges, for $f(x)=x^n$ where n is integer, the expansion always gives $0$.
Aug
22
revised Is there an easier way to find the “natural” integration constant?
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