limit of function with dirac delta

asked 2017-03-09 12:15:01 +0100

maaaaaaartin
35 ●2 ●7

Consider the following code:

sage.var(x)
f = (1-x) * sage.unit_step(x)
df = f.diff(x)    ->    -unit_step(x) + (1-x)*dirac_delta(x)

Trying to compute df.limit(x=0, dir="right"), which should return -1, but instead returns limit(-unit_step(x) + (1-x)*dirac_delta(x), x, 0, plus).

Even df.limit(x=0.5) return limit(-unit_step(x) + (1-x)*dirac_delta(x), x, 0.5) instead of -1.

Any idea on what might be going on? Thanks!

Comments

My guess is that Maxima doesn't know how to take limits of these functions.

kcrisman ( 2017-03-09 14:50:09 +0100 )edit

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answered 2017-03-09 22:09:05 +0100

paulmasson
1580 ●1 ●24 ●45 http://analyticphysics...

updated 2017-03-09 23:28:59 +0100

To expand on the comment by @kcrisman, limits are sent to Maxima by default. Maxima can evaluate the limit of the step function: entering

unit_step(x).limit(x=0,dir='right')

returns 1 as expected. Maxima cannot evaluate the limit of the Dirac delta: entering

dirac_delta(x).limit(x=0,dir='right')

returns an unevaluated expression. Since part of your limit cannot be evaluated by Maxima, it all comes back unevaluated.

The other option for limit evaluation is to set algorithm='sympy': entering

dirac_delta(x).limit(x=0,algorithm='sympy')

returns zero as expected. Unfortunately, entering

dirac_delta(x).limit(x=0,dir='right',algorithm='sympy')

gives the message

sympy does not support one-sided limits

Even more problematic, entering

unit_step(x).limit(x=0,algorithm='sympy')

gives the message

SymPy function 'unit_step' doesn't exist

so SymPy won't get the complete job done either.

Not exactly the answer you want, but hopefully it helps you understand what's happening.

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Comments

the existence problem can probably be fixed by setting a custom "conversion" dictionary in src/sage/functions/generalized.py

mforets ( 2017-03-10 07:54:15 +0100 )edit

the problem is that there is really no such UnitStep function in SymPy. Here it says that it's possible to define it at $0$ passing a 2nd argument. see also in github. But in my local sage installation, it doesn't work; don't know if it's because of a difference in the SymPy versions..

mforets ( 2017-03-10 13:04:50 +0100 )edit

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