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revised Ascribing values to Gamma of negative integers
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comment Ascribing values to Gamma of negative integers
@Claude Leibovici or, without using complex numbers: Limit[(Gamma[-n + x] + Gamma[-n - x])/2, x -> 0]
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comment Ascribing values to Gamma of negative integers
@Claude Leibovici It gives Gamma[-n] for all other real values except negative integers.
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comment Ascribing values to Gamma of negative integers
@Claude Leibovici Mathematica code: Limit[Re[Gamma[-n + I x]], x -> 0]
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comment Ascribing values to Gamma of negative integers
@Henry For instance it gives $(1/t)^{-1}=\log|x|+\gamma$ (which is by the way, consistent with discrete integral of $1/t$ which is $\psi(t)$ and asymptotically approaches $\log|x|+\gamma$ at $x\to+\infty$
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comment Ascribing values to Gamma of negative integers
@Henry good point indeed... But on the other hand they are consistent with Fourier differintegral: $$(t^n)^{(s)}|_{t=1}=\frac{1}{2\pi}\int_{-\infty}^{+\infty} \frac{e^{- i \omega }}{(-i\omega)^s} \int_{-\infty}^{+\infty}t^n e^{i\omega t}dt \, d\omega=\Gamma(s+1)(-1)^s t^{n-s}$$
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revised Ascribing values to Gamma of negative integers
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revised Ascribing values to Gamma of negative integers
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asked Ascribing values to Gamma of negative integers
Oct
17
comment Dirac Delta definition in non-standard analysis?
Delta is even function so its derivative should be odd. That is zero at x=0.
Oct
17
comment Dirac Delta definition in non-standard analysis?
@Hurkyl if derivative of delta is negative at x<0 and positive at x>0, this means that delta is non-positive around and in zero, which would not give positive integral. It is evident you made a mistake.
Oct
17
comment Dirac Delta definition in non-standard analysis?
@Hurkyl it seems you confused the right and left parts of the function in this definition of derivative. Its derivative should be positive at negative part and negative at positive area, otherwise its integral would be negative. Also note that this definition has a major disadvantage: delta is even function so its derivative shoud be odd. And also it should be 0 in x=0 because delta has maximum there.
Oct
17
comment Dirac Delta definition in non-standard analysis?
it is differentiable in a sense. At least, the notion of derivative of dirac delta (and further derivatives) is widely used. See my answer for the for that I found is employed in non-standard analysis. Its derivative thus would be $-\frac{2 w^3 z e^{-(w z)^2}}{\sqrt{\pi }}$
Oct
17
revised Dirac Delta definition in non-standard analysis?
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Oct
17
comment Dirac Delta definition in non-standard analysis?
this has only a few properties of Dirac Delta and does not have others (i.e. differentiability).
Oct
17
revised Dirac Delta definition in non-standard analysis?
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Oct
17
comment Dirac Delta definition in non-standard analysis?
How would u differentiate such delta function?
Oct
17
comment Dirac Delta definition in non-standard analysis?
@Semiclassical this is what is non-standard analysis. And also $\delta(0)=\frac{\omega}{\sqrt{\pi}}$
Oct
17
revised Dirac Delta definition in non-standard analysis?
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