3 ms·

I don't follow. The antiderivative fails because of division by zero. What limit? And what does symbolic manip do?

gus_massa 5y ago
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_{ε->0} [ integral{from t=0, to t=x} t^{-1+ε} dx ] that is just = lim_{ε->0} [ (t^ε - 1) / ε ] replacing t^ε = lim_{ε->0} [ (e^(ln(t)*ε) - 1) / ε ] by https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule = lim_{ε->0} [ ln(t)*e^(ln(t)*ε) / 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 5y ago
  Just for the sake of correctness: the differential has to be dt instead of dx here.
  - gus_massa 5y ago
    I agree, thanks. But I saw your comment just now and it's too late to edit. :(
- westurner 5y ago
  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.org/message/JQPNUQXELOYKP25XJ77SX3TVVG24QS72/ https://mail.python.org/archives/list/python-ideas@python.or...
  - hither_shores 5y ago
    > So the type of the return value changes at asymptotes/limits (already) and thus that's not a pure function in terms of math. Plenty of functions are discontinuous. Almost all of them, in fact. (This isn't true constructively, but GP is clearly doing classical analysis.) > 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? Depends on the smooth structures you've imposed on the domain and codomain, which requires much more powerful types to represent than are available in any mainstream language I'm aware of. (You might be able to contort Haskell into something sort of close, but it wouldn't be simple.)

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_{ε->0} [ integral{from t=0, to t=x} t^{-1+ε} dx ] that is just = lim_{ε->0} [ (t^ε - 1) / ε ] replacing t^ε = lim_{ε->0} [ (e^(ln(t)*ε) - 1) / ε ] by https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule = lim_{ε->0} [ ln(t)*e^(ln(t)*ε) / 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 5y ago
Just for the sake of correctness: the differential has to be dt instead of dx here.
- gus_massa 5y ago
  I agree, thanks. But I saw your comment just now and it's too late to edit. :(
westurner 5y ago
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.org/message/JQPNUQXELOYKP25XJ77SX3TVVG24QS72/ https://mail.python.org/archives/list/python-ideas@python.or...
- hither_shores 5y ago
  > So the type of the return value changes at asymptotes/limits (already) and thus that's not a pure function in terms of math. Plenty of functions are discontinuous. Almost all of them, in fact. (This isn't true constructively, but GP is clearly doing classical analysis.) > 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? Depends on the smooth structures you've imposed on the domain and codomain, which requires much more powerful types to represent than are available in any mainstream language I'm aware of. (You might be able to contort Haskell into something sort of close, but it wouldn't be simple.)