provide missing function expansions of power series

[rwst]

Some functions do not support rings/power-series*:

sage: R.<x> = PowerSeriesRing(ZZ)
sage: sqrt(1-4*x^2)
1 - 2*x^2 - 2*x^4 - 4*x^6 - 10*x^8 - 28*x^10 - 84*x^12 - 264*x^14 - 858*x^16 - 2860*x^18 + O(x^20)
sage: sin(1+4*x^2)
...
TypeError: cannot coerce arguments: no canonical coercion from Power Series Ring in x over Integer Ring to Symbolic Ring

What is missing:

• acos, acosh, asin, asinh, atan, atanh, cos, cosh, cotanh, dilog, gamma, intformal, lngamma, psi, sin, sinh, tan, tanh

Component: calculus

Keywords: function, series expansion

Issue created by migration from https://trac.sagemath.org/ticket/16197

[kcrisman]

comment:1

Similarly to my other comment on this, I don't know if we want n(pi/4) in here or not...

Also, the regular SR one works fine:

sage: ex.series(X,16)
(sin(1)) + (4*cos(1))*X^2 + (-8*sin(1))*X^4 + (-32/3*cos(1))*X^6 + (32/3*sin(1))*X^8 + (128/15*cos(1))*X^10 + (-256/45*sin(1))*X^12 + (-1024/315*cos(1))*X^14 + Order(X^16)

so I'm not sure why you referenced it, though I agree that the error you get for power series rings is not ideal. Is there a way to either use SR for this (perhaps not "correct" for power series rings though?) or to get Pari to return a symbolic constant term?

sage: atan(4*X^2+1)
arctan(4*X^2 + 1)
sage: _.series(X,16)
(1/4*pi) + 2*X^2 + (-4)*X^4 + 16/3*X^6 + (-128/5)*X^10 + 256/3*X^12 + (-1024/7)*X^14 + Order(X^16)

What if the coefficients weren't rationals? Or integers? Note that in your example there is a 16/3 coefficient, which presumably isn't in the power series ring over integers. These may be dumb questions, but I'm just trying to explore what is really the desired behavior - certainly wrapping more Pari stuff is not a bad idea!

[rwst]

comment:2

Pari's symblic is not up to this. I'd consider it a bug that there is no default precision to SR series results (like with PowerSeries), so yes, if your last example would work without the 16 this ticket would rather be about my example sin(1+4*x^2)---it should simply work like series() and give back one instead of SR.

The rest of the ticket concerns possibly missing functions and I will move this to another ticket.

[kcrisman]

comment:3

Pari's symblic is not up to this. I'd consider it a bug that there is no default precision to SR series results (like with PowerSeries), so yes, if your last example would work without the 16 this ticket would rather be about my example sin(1+4*x^2)---it should simply work like series() and give back one instead of SR.

Hmm, that's an interesting suggestion. One could imagine it's a bug the other way around, but I have no vested interest in this - I think a default for either one could be useful, in principle, and (importantly) wouldn't be backward-incompatible. But what would the default be? It's hard to imagine one non-arbitrary... hmm. What do Mathematica and/or Maple and/or Magma do with this? If there is a standard one could use that.

[rwst]

comment:4

I wrote earlier:

The rest of the ticket concerns possibly missing functions and I will move this to another ticket.

Nothing missing there except the a.g.m., fortunately.

Replying to @kcrisman:

Hmm, that's an interesting suggestion. One could imagine it's a bug the other way around, but I have no vested interest in this - I think a default for either one could be useful, in principle, and (importantly) wouldn't be backward-incompatible. But what would the default be? It's hard to imagine one non-arbitrary... hmm.

This is now #16201

What do Mathematica and/or Maple and/or Magma do with this? If there is a standard one could use that.

If I ask for "cosine power series" in Wolfram Alpha I get 1-x<sup>2/2+x</sup>4/24-x<sup>6/720+O(x</sup>7)

[rwst]

comment:5

Edit: my copy of another comment was unrelated so I deleted it

[rwst]

comment:7

Closing as wontfix since all issues mentioned have their own tickets.

[kcrisman]

comment:8

I only see one ticket mentioned. Just for completeness, can you mention them (if there are others)?

[rwst]

comment:9

Default symbolic expression series precision is #16201, and symbolic agm function is #16202.