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Searching for an asymptotic to exp[0.5]
I have considered many more things then I could ever post ...

I probably said it before but Im considering " fake analytic number theory ".

That might take some time to develop.

Here are some other ideas that inspire me and for which Im currently not sure how to continue.


Jay's function that approximates the binary partition function has become popular here.

I wonder about variants of type

DO these variants of " cubic type " solve anything ?



Initially fake function methods start with an OVERESTIMATE.

It is possible to use a method that is both GOOD and starts with an UNDERESTIMATE ?



Im wondering about fake fourier series and fake integrals.

A logical way , but possibly not the best , is to consider the fake function methods as fake n'th derivatives.

Then the fake integral is computed as the fake first derivative of the true second integral.

Or something like that ...



Lets try Jay's function J(x).

a_n x^n = J(x)

ln(a_n) + n ln(x) = ln(J(x))

ln(a_n) = min[ ln(J(x)) - n ln(x) ]

d/dx [ ln(J(x)) - n ln(x) ] = J(x/2)/J(x) - n/x

J(x/2)/J(x) = n/x

I really like the shape of this equation.
And ofcourse I wonder how good a_n will be compared to

ALthough Im not completely stuck here , Im also not completely sure how to proceed.
Use asymptotics , invent new special function , use contour integrals , numerical methods , ... ?


Under some trivial conditions I consider some ideas to improve finding a fake function without actually changing the method ...


Find fake exp(x).

Note : We already discussed the idea that we already have an entire function with positive derivatives , yet this is intresting.
And just an example , it applies to non-entire functions too.

Instead of
a_n x^n = exp(x)

We solve for

a_n x^n = 2 exp(x) / (1+x)

Then we multiply our result (taylor series with a_n , possibly scaled ) with (1+x)/2.

Notice multiplication by (1+x)/2 does not change the signs of the derivatives and is easy to compute !

The q-variant of this is also possible of course.

I call these methods

sx9(n) and qx9(n).

So for instance

Much more work needs to be done !


Comments and help is appreciated.



" Formally define useful and useless , but beware : take into account we are plain mortals and your an atheist who claims to be not obsessed by money nor by ego "
tommy1729 @sci.math

Messages In This Thread
RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 11/07/2014, 12:27 AM

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