Can one define 0 Log[0] = 0?

Question

With reference to this post, can one define 0 Log[0] = 0 in Mathemtica?

That being said, I would use Function[r, Piecewise[{{Log[r^r], 0 < r < 1}, {r Log[r], 1 < r}}, 0], Listable]. — J. M.'s missing motivation, Commented Mar 8, 2019 at 11:08
The problem I'm facing is that in my code I have terms like x Log[x]; when x becomes very small, Mathematica shows indeterminate when it should show 0. — H. Kenan, Commented Mar 8, 2019 at 11:13
IF the coefficient of Log[x] always vanishes with an order higher than some x^r, r > 0, as x -> 0, then the following should work: log[x_?NumericQ /; x != 0] := Log[x]; expr /. Log -> log /. Thread[{x, y, z} -> 0.] /. log -> Log. Some tweaking may be needed depending on the application. — Michael E2, Commented Mar 8, 2019 at 13:39
A similar approach could be used with @J.M.'s function but defining xlogx[x] and replacing expr /. Log -> (xlogx[#]/# &) // Simplify — Michael E2, Commented Mar 8, 2019 at 13:43

John Doty · Accepted Answer · 2019-03-08 13:37:24Z

2

You can use Inactive here.

expr = (1 - x) Log[1 - x];
iexpr = expr /. Log -> Inactive[Log];

Now, insert a number, activate after numeric evaluation of everything but Log.

iexpr /. x -> -2. // Activate
(* 3.29584 *)

Because you've controlled the order of evaluation, the troublesome case becomes zero.

iexpr /. x -> 1. // Activate
(* 0. *)

Activate not needed in this special case, but harmless.

This works for exact numbers and machine numbers. For controlled precision, you'll need to do more.

answered Mar 8, 2019 at 13:37

John Doty

14k1 gold badge23 silver badges43 bronze badges

2

$\begingroup$ This is similar to the approach in my comment, and probably should have the same priviso, "IF the coefficient of Log[x] always vanishes with an order higher than some x^r, r > 0, as x -> 0,..." $\endgroup$
– Michael E2
Commented Mar 8, 2019 at 16:09

Add a comment |

Roman · Accepted Answer · 2019-03-08 12:55:21Z

0

You can define

s[0|0.] = 0;
s[x_] = -x*Log[x];
SetAttributes[s, Listable]

and then use the function s instead of x Log[x]. If you also have values of $x<0$ then other definitions should be added.

answered Mar 8, 2019 at 12:55

Roman

49.8k2 gold badges57 silver badges131 bronze badges

$\begingroup$ Thanks, @Roman. How can one put further restrictions like s[x]=0 for say x<10^(-15). $\endgroup$
– H. Kenan
Commented Mar 8, 2019 at 16:31
$\begingroup$ @TobiasFritzn take a look at the Picewise function. $\endgroup$
– gothicVI
Commented Mar 9, 2019 at 0:09
$\begingroup$ I don't think you need to make a separate definition for, say, x<10^(-15), as the function x*log(x) is well-behaved around x=0. There's only really a need for a special definition at the very point x=0. $\endgroup$
– Roman
Commented Mar 9, 2019 at 1:23

Add a comment |

2 Answers 2