Doctest: Add more logic flags to more functions

[rws]

Some functions are handled with some flags in ​https://github.com/pynac/pynac/blob/master/ginac/function.cpp#L1581

This should be extended (in Pynac) and doctests added (here). This ticket should list which questions can be answered definitely to the positive. To start:

• f(real).is_real() for f in sin/cos/tan/cot/sec/csc

[vdelecroix]

The gamma function

sage: gamma(pi).is_real()
False

You might also want to deal with is_positive which would be easily handled for analytic functions whose power series have non-negative coefficients.

Are there flags in ginac for properties like: "power series with non-negative coefficients", "power series with integer coefficients", etc?

[vdelecroix]

Hyperbolic trigonometric functions cosh/sinh/tanh and inverses arccosh/arcsinh/arctanh

sage: cosh(-3).is_real()
False
sage: cosh(-3).is_positive()
False

[vdelecroix]

But instead of hardcoding all these informations in functions, it would be better to actually deduce it from some other properties (like Taylor series as I suggested before).

[vdelecroix]

Will this approach be recursive, namely cos(exp(-3) + log(2)).is_real()?

[rws]

Replying to vdelecroix:

it would be better to actually deduce it from some other properties (like Taylor series as I suggested before).

Not possible. Taylor series are determined in general by differentiation. There is no builtin formula which we can inspect.

Will this approach be recursive, namely cos(exp(-3) + log(2)).is_real()?

Yes.

[dimpase]

as discussed on sage-develop, the best way IMHO would be to be able to return "Unknown" instead of false "False"...

[rws]

Replying to dimpase:

as discussed on sage-develop, the best way IMHO would be to be able to return "Unknown" instead of false "False"...

See #22162.

[rws]

Improvements in ​https://github.com/pynac/pynac/commit/acec86167e7a200a383199d0e23e1eea07ae634f

[vdelecroix]

Nice! Could you add doctest to check

1) compatibility with assumptions

sage: x = SR.var('x')
sage: assume(x, 'real')
sage: cos(x).is_real()

2) complex stuff is actually False

cos(I).is_real()
sin(2 - I).is_real()
etc

3) What is the current status of?

sage: x = SR.var('x')
sage: assume(x, 'real')
sage: (cos(x) + 1).is_positive()

---

New commits:

​4a7499b

22155: doctest the improved domain logic

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​6e1c056

22155: address reviewer's comments

[rws]

Replying to vdelecroix:

3) What is the current status of?

sage: x = SR.var('x')
sage: assume(x, 'real')
sage: (cos(x) + 1).is_positive()

Not possible using current machinery. Note the information kept is boolean (or maybe tri-state in the future) for speed reasons, as it is routinely used with manipulation of expressions. Any slowdown like numeric computations that are potentially calling Python/Cython code from C++ should be avoided.

[vdelecroix]

What does mean "dependencies pynac-0.7.4"?

[rws]

There is no ticket atm for the 0.7.4 upgrade. When there is I'll replace that with the ticket number.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​fc0c64d

Merge branch 'develop' into t/22155/add_more_logic_flags_to_more_functions

[rws]

Two failing doctests.

[rws]

Actually the doctests were wrong: realpositive is not real but positvepositive is. Also (cos(exp(real) + log(pos))^pos^) is not real if the log argument is less than one. I replace the second pos with 8. Sage teaches me math!

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​39f0336	Merge branch 'develop' into t/22155/add_more_logic_flags_to_more_functions
​350580a	22155: fixes