[HN Gopher] Logarithms yearning to be free
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Logarithms yearning to be free
Author : _Microft
Score : 68 points
Date : 2022-04-13 13:56 UTC (9 hours ago)
(HTM) web link (www.johndcook.com)
(TXT) w3m dump (www.johndcook.com)
| CyberRabbi wrote:
| A link between zeroth powers and the logarithm function is such a
| simple idea but potentially fruitful. Exponentials and logarithms
| often translate between additive groups and multiplicative
| groups. Maybe there are interesting isomorphisms where the
| logarithm can be generalized as a zeroth power.
| ewewr wrote:
| integral{from t=0, to t=x} t^{-1+0} dx
|
| but you calculate instead
|
| f(x) = lim_{e->0} [ integral{from t=0, to t=x} t^{-1+e} dx ]
|
| that is just
|
| = lim_{e->0} [ (t^e - 1) / e ]
|
| replacing t^e
|
| = lim_{e->0} [ (e^(ln(t) _e) - 1) / e ]
|
| by https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule
|
| = lim_{e->0} [ ln(t)_e^(ln(t) _e) / 1 ]
|
| that is easy to calculate
|
| = ln(t)_e^(ln(t) _0) / 1
|
| = ln(t)_1 / 1
|
| = ln(t)
|
| So f(t)=ln(t)
|
| reply
| codeflo wrote:
| If you want to make a statement like that, then logarithms might
| not be the zeroth power, but the epsilonth power, where e is an
| "infinitesimal".
|
| Bear with me here, I know infinitesimal math isn't a fully
| coherent thing. But there's a reason why Newton used it, it
| sometimes works surprisingly well to make intuitive analogies.
| Maybe there's a way to make it work here.
|
| For example: The function ln(x) grows strictly slower than x^a
| for any positive real a, but faster than x^0. Hence, it's x^e,
| where 0 < e < a for any positive real a.
| gradschool wrote:
| Where does that leave superpolylogarithmic subexponential
| functions (sung to the tune from Mary Poppins)?
|
| https://www.csee.umbc.edu/~sherman/Papers/superpoly.ps
| voxl wrote:
| Infinitesimal math is completely coherent, it just doesn't have
| one theory that is "thee algebra" like the real numbers do.
| _Microft wrote:
| You might want to appeal to Alexandra, patron saint of the
| paywalled scientist, for she might hear you.
| CodesInChaos wrote:
| Searching for the title of the paper, DDG shows scihub as the
| first result, while google only has a libgen hit on the second
| page (and it's the author's list of papers, not the paper
| itself).
| westurner wrote:
| > _The author opens with the example of finding the
| antiderivative of xn. When n [?] -1 the antiderivative is another
| power function, but when n = -1 it's a logarithm._
|
| What a neat limit. Probably best to leave the powerfn/logfn() as
| a dumb symbolic symbol until the end (until after later parameter
| substitution)?
| nh23423fefe wrote:
| I don't follow. The antiderivative fails because of division by
| zero. What limit? And what does symbolic manip do?
| gus_massa wrote:
| The idea is that you want to calculate
|
| integral{from t=0, to t=x} t^{-1+0} dx
|
| but you calculate instead
|
| f(x) = lim_{e->0} [ integral{from t=0, to t=x} t^{-1+e} dx ]
|
| that is just
|
| = lim_{e->0} [ (t^e - 1) / e ]
|
| replacing t^e
|
| = lim_{e->0} [ (e^(ln(t)*e) - 1) / e ]
|
| by https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule
|
| = lim_{e->0} [ ln(t)*e^(ln(t)*e) / 1 ]
|
| that is easy to calculate
|
| = ln(t)*e^(ln(t)*0) / 1
|
| = ln(t)*1 / 1
|
| = ln(t)
|
| So f(t)=ln(t)
| _Microft wrote:
| Just for the sake of correctness: the differential has to
| be dt instead of dx here.
| gus_massa wrote:
| I agree, thanks. But I saw your comment just now and it's
| too late to edit. :(
| westurner wrote:
| So the type of the return value changes at
| asymptotes/limits (already) and thus that's not a _pure
| function_ in terms of math. If a [math] function returns a
| more complex type signature instead of throwing a
| ZeroDivisionError (as Python core does) what is that then
| called? Is it differentiable or no, etc?
|
| We throw ZeroDivisionError instead of axiomatically
| defining a ranking for scalar*parameter*inf
| if x > 0: 2*x*inf > x*inf # because
| 2 > 1
|
| But basically every CAS just prematurely throws away all
| terms next to infinity (by replacing the information in
| that expression with just infinity)? And nothing yet
| implements e.g. Conway's _Surreal numbers_ infinities?
|
| Is negative infinity to the infinity greater or lesser than
| infinity? assert (-1*math.inf)**math.inf ==
| math.inf assert (-1*sympy.oo)**sympy.oo ==
| sympy.oo
|
| Here's a dumb Real/Function instead of prematurely
| discarding information that could be useful:
| from sympy import symbol from sympy.abc import x
| Infinity = symbol('Infinity', real=True) # * #
| from sympy.symbols import Wild
|
| All the axioms just change there.
| limit(-x**-1)
|
| An uphill battle for certain.
|
| "[Python-ideas] Re: 'Infinity' constant in Python"
| https://mail.python.org/archives/list/python-
| ideas@python.or...
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