allow constant != 1 in log(power series)

[rws]

log(power series) fails when the constant coefficient is not 1:

sage: R.<x> = PowerSeriesRing(RR)
sage: log(2+x)
---------------------------------------------------------------------------
ArithmeticError                           Traceback (most recent call last)
<ipython-input-4-49ca3295e0d6> in <module>()
----> 1 log(Integer(2)+x)

/usr/local/src/sage-git/local/lib/python2.7/site-packages/sage/functions/log.pyc in __call__(self, *args, **kwds)
    311         if base is None:
    312             if len(args) == 1:
--> 313                 return GinacFunction.__call__(self, *args, **kwds)
    314             # second argument is base
    315             base = args[1]

/usr/local/src/sage-git/local/lib/python2.7/site-packages/sage/symbolic/function.so in sage.symbolic.function.BuiltinFunction.__call__ (build/cythonized/sage/symbolic/function.cpp:9120)()

/usr/local/src/sage-git/local/lib/python2.7/site-packages/sage/rings/power_series_ring_element.so in sage.rings.power_series_ring_element.PowerSeries.log (build/cythonized/sage/rings/power_series_ring_element.c:15894)()

ArithmeticError: constant term of power series is not 1

Pari has no problems:

? log(2-x)
%1 = 0.69314718055994530941723212145817656807 - 1/2*x - 1/8*x^2 - 1/24*x^3 - 1/64*x^4 - 1/160*x^5 - 1/384*x^6 - 1/896*x^7 - 1/2048*x^8 - 1/4608*x^9 - 1/10240*x^10 - 1/22528*x^11 - 1/49152*x^12 - 1/106496*x^13 - 1/229376*x^14 - 1/491520*x^15 + O(x^16)

Effectively the log of the constant is taken but this simply changes the base ring from whatever to SR if the constant is not equal to one.

[kcrisman]

I don't think Ginac is actually involved. Take a look at the last piece of the traceback, which points to ​here. In particular, in sage/rings/power_series_ring_element.pyx,

        if prec is None:
            prec = self._parent.default_prec()

        if not self[0].is_one():
            raise ArithmeticError("constant term of power series is not 1")

        zero = self.parent().zero()
        t = zero.solve_linear_de(prec,b=self.derivative()/self,f0=0)
        return t

So this is actually hard-coded in for some reason, referring eventually to _solve_linear_de which was factored out of this stuff a very long time ago.

It could be worth asking around why/whether this is really necessary. I don't know that I want the above instead of

sage: log(2-x)
log(-x + 2)
sage: _.taylor(x,0,16)
-1/1048576*x^16 - 1/491520*x^15 - 1/229376*x^14 - 1/106496*x^13 - 1/49152*x^12 - 1/22528*x^11 - 1/10240*x^10 - 1/4608*x^9 - 1/2048*x^8 - 1/896*x^7 - 1/384*x^6 - 1/160*x^5 - 1/64*x^4 - 1/24*x^3 - 1/8*x^2 - 1/2*x + log(2)

note the log(2) instead of an approximation.

[rws]

So it looks like any constant from exp(n), n in NN0 will leave this in PowerSeries(ZZ) but anything else will make it necessary to rebase in PowerSeries(SR).

[rws]

This was ill-conceived from the beginning, because only SR is a superset of the coefficients in the Pari result (floating point and rational). That's why Pari is questionable, even if it's convenient.

As I find it not useful to duplicate symbolics functionality in the power series ring, the best way to get such a power series is via SR and the newly available #16203:

            sage: R.<x> = PowerSeriesRing(SR)
            sage: ex=(log(2-y)).series(y,4); R(ex)
            log(2) - 1/2*x - 1/8*x^2 - 1/24*x^3 + O(x^4)
            sage: ex=(gamma(1-y)).series(y,3); R(ex)
            1 + euler_gamma*x + (1/2*euler_gamma^2 + 1/12*pi^2)*x^2 + O(x^3)

[jdemeyer]

Replying to rws:

So it looks like any constant from exp(n), n in NN0 will leave this in PowerSeries(ZZ) but anything else will make it necessary to rebase in PowerSeries(SR).

If the base ring is RR, then log() is perfectly defined for non-one constant terms, so that should certainly work.

[rws]

New commits:

​ee40e93

16198: allow constant != 1 in log(power series)