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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
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answered Is there an easier way to find the “natural” integration constant?
Aug
22
comment Is there an easier way to find the “natural” integration constant?
@Rahul on the other hand, by decomposing the trigonometric functions into exponentials and applying the rule derived above and generalised to all exponential bases, one can see that natural integral of $\cos x$ is $\sin x$ and natural integral of $\sin x$ is $-\cos x$, that is $-1$ in zero. But applying the series directly to sine and cosine would not work, it will diverge. Thus I am seeking a method of finding the same value but which would work for say, sine and cosine directly.
Aug
22
comment Is there an easier way to find the “natural” integration constant?
@Rahul it is quite expected to diverge for polynomials, the natural constant tends to infinity in this case. One can see monomials (power functions) as integrals of constant function $f(x)=1$, that is of exponent $1^x$. Following the above rule, its consecutive derivatives would be $(\ln 1)^x$ that is $0^x$. This is $0$ for positive derivatives, $1$ for the zeroth one and tends to infinity for integrals (that is the monomials) so they just have no natural integration constant other than infinity. I know this looks weird given that polynomials are so widespread, but it is how it is.
Aug
22
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22
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19
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