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Very curious question
#1
What If I told you I can find infinite functions that equal their own derivative?

Take some fractional differentiation method which differentiates f across s, t times. Now assume that:

for some s in some set , which can be easily constructed using some theorems I have.

Then:



If you differentiate by the continuity of this improper integral

What does this mean? How did I get this? Where is the mistake?
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#2
Let's make another function that equals its own derivative. I'm very curious as to why this is happening!




Differentiate and watch for your self!

Does this mean the function cannot converge? I know the integral converges, not sure about the summation though.

Using the other method I can easily create a function that converges for some domain... What's going on?
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#3
(08/19/2013, 05:39 PM)JmsNxn Wrote: Let's make another function that equals its own derivative. I'm very curious as to why this is happening!




Differentiate and watch for your self!

Does this mean the function cannot converge? I know the integral converges, not sure about the summation though.

Using the other method I can easily create a function that converges for some domain... What's going on?

The summation does not look like it converges. Try graphing the integrand for s = 1 and look what happens as n increases.

Also, using a numerical integration from to (roughly centers around the "peak", at least for relatively small n), one can approximate the integral and see the divergence:

n = 1, s = 1: 0.38446
n = 2, s = 1: 0.042752
n = 3, s = 1: -0.082158
n = 4, s = 1: 0.26084
n = 5, s = 1: -0.83652
n = 6, s = 1: 2.2210
n = 7, s = 1: 2.4999
n = 8, s = 1: -149.51

So the sum of these values approaches no limit. While the values do shrink for negative , the sum also includes the problematic positive values.

Note that this numerical test is not a proof of divergence, but it strongly indicates that is what is happening.
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#4
Aww thank you mike. I've been coming across a lot of these functions and I've yet to see one that converges so I think I'm not doing anything too wrong.

Btw, you should look at my continuum sum thread, I know you were looking into the method earlier, I found a way using fractional calculus, but I'm a little mirky on some of the formal fine tunings, help would be greatly appreciated Smile
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