Stanislav Maslovski wrote:
>
>>
>> 1/x = PP 1/x + %i*%pi*Dirac_delta(x)
>
> By PP 1/x you mean
the Cauchy principal value.
> Secondly, this definition for sure can be useful in physics when you
> consider integrals of some physical quantities like polarizabilities or
> dielectric functions. But Maxima is a mathematical package. It should
> not depend, for instance, on the type of the time dependence that one
> uses in physical problems (exp(+i omega t) or exp(-i omega t), this
> will change the sign of your second term).
This is true.
>
> I think that the real reason for the imaginary part in the answer must
> be something really stupid
I think it may use the Cauchy residue theorem and choose arbitrarily one way
to close the contour. In physics this choice is usually dictated by the
choice of sign you mentioned. For example if you integrate on omega and you
have exp(+i omega t) then you close on upper complex omega half-plane when
t>0 (this will produce exp(- infinity) ) and on lower half plane for t<0.
So it may be that we are hitting here a difficulty occurring in different
parts of Maxima where it is unable to make good use of the sign of certain
quantities.
Or it may be that it does the indefinite integral using logs and then takes
the value of logs at negative values of the argument, which, if one makes
analytic continuation easily produces %i*%pi terms. The difficulty being of
course that if you do the continuation from above or below you find a
different sign, but this is a reflection of the same causality problem.
For my own taste, i don't have any problem with formal results obtained from
a formal computation program. I am more unhappy when i see
diff(t*sqrt(t),t)= sqrt(t) when under a taylor expansion as i experienced
today.
--
Michel Talon