GSoC Week 8 and 9

I couldn’t write a blog post last week so including progress of week 8 and 9 both here.

Week 8

I continued working on the PR #11360. We added functionality to store a different type of initial condition for regular singular points other than the usual [y(0), y'(0), ...]. The exact format is described here in master, though it is changed to a more elegant form in #11422. This type of initial condition provides more information at regular singular points and is helpful in converting to expressions. Examples on how to use it:

In [22]: expr_to_holonomic(sin(x)/x**2, singular_ics={-1: [1, 0, -1]}).to_expr()
Out[22]:
sin(x)
──────
   2
  x

I also added method to compute this type of initial condition for algebraic functions of the form P^r, for some Polynomial P and a Rational Number R.

In [25]: expr_to_holonomic(sqrt(x**2+x))
Out[25]: HolonomicFunction((-x - 1/2) + (x**2 + x)Dx, x), {1/2: [1]}

In [26]: _25.to_expr()
Out[26]:
     _______
√x⋅╲╱ x + 1

After that I made some changes in `to_meijerg()` to return the polynomial itself if the `meijerg` function represents a polynomial instead of raising `NotImplementedError`.

In [40]: expr_to_holonomic(4*x**3 + 2*x**2, lenics=3).to_meijerg().expand()
Out[40]:
   3      2
4⋅x  + 2⋅x

I also added code to return the general solution in `_frobenius()` if none of the roots of indicial equation differ by an integer.

Week 9

I wasn’t able to do much this week because my college started. I travelled back and had some college related stuff to do.

I opened #11422 and first added a basic method to determine the domain for polynomial coefficients in the differential equation.

In [77]: expr_to_holonomic(sqrt(y*x+z), x=x, lenics=2).to_expr()
Out[77]:
  _________
╲╱ x⋅y + z 

In [78]: expr_to_holonomic(1.1329138213*x)
Out[78]: HolonomicFunction((-1.1329138213) + (1.1329138213*x)Dx, x), f(0) = 0

Then I added support for the new type of initial condition on regular singular points in .integrate().

In [83]: expr_to_holonomic(sin(x)/x**3, singular_ics={-2: [1, 0, -1]}).integrate(x).to_expr()
Out[83]:
 ⎛ 2                          ⎞
-⎝x ⋅Si(x) + x⋅cos(x) + sin(x)⎠
────────────────────────────────
                 2
              2⋅x               

Also added support for the same in addition.

In [6]: expr_to_holonomic(sqrt(x)) + expr_to_holonomic(sqrt(2*x))
Out[6]: HolonomicFunction((-1/2) + (x)Dx, x), {1/2: [1 + sqrt(2)]}

In [7]: _6.to_expr()
Out[7]: √x⋅(1 + √2)

I plan to continue my work on this PR and add more support for this initial condition.

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