[Merged by Bors] - feat(ring_theory/polynomial): show gaussian form of hermite n
#18896
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…gaussian_mul_gaussian, remove hermite_gauss def, change eval to aeval
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| { simp_rw [aeval_def, eval₂_eq_eval_map], | ||
| apply polynomial.differentiable }, | ||
| { simp_rw [aeval_def, eval₂_eq_eval_map], | ||
| apply differentiable.const_mul, | ||
| apply polynomial.differentiable }, | ||
| { simp } } |
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These side goals go away if you work with has_deriv_at instead of deriv:
induction n with n ih generalizing x,
{ rw [function.iterate_zero_apply, pow_zero, one_mul, hermite_zero, C_1, map_one, one_mul] },
{ replace ih : (deriv^[n] _) = _ := _root_.funext ih,
have has_deriv_at_gaussian : ∀ x,
has_deriv_at (λ y, real.exp (-(y^2 / 2))) ((-x) * real.exp (-(x^2 / 2))) x,
{ intro x,
simpa [mul_div_cancel_left x (two_ne_zero), neg_mul, mul_comm]
using (((has_deriv_at_id' x).pow 2).div_const 2).neg.exp },
rw [function.iterate_succ_apply', ih],
have := ((polynomial.has_deriv_at_aeval _ _).const_mul ((-1 : ℝ) ^ n)).mul (has_deriv_at_gaussian x),
refine this.deriv.trans _,
sorry },
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Actually also deriv_const_mul_field doesn't generate a side goal, I didn't notice it before (versus deriv_const_mul).
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The new version is pretty close to this, let's just go with it.
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Is this PR ready to merge? |
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My hope was to get #18945 in first since I think that golfs a few proofs slightly here. |
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This duplicates every lemma about differentiation of `polynomial.eval` for `polynomial.aeval` too. Some of these turned out to be needed in #18896.
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Show the equivalence of the polynomial form of the Hermite polynomials to the explicit form involving derivatives of the gaussian function. Co-authored-by: Jake Levinson <levinson.jake@gmail.com> Co-authored-by: Jake Levinson <levinson.jake@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Johan Commelin <johan@commelin.net> Co-authored-by: jakelev <levinson.jake@gmail.com>
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Thanks 🎉 bors merge |
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Show the equivalence of the polynomial form of the Hermite polynomials to the explicit form involving derivatives of the gaussian function. Co-authored-by: Jake Levinson <levinson.jake@gmail.com> Co-authored-by: lukemantle <lukeomantle@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Johan Commelin <johan@commelin.net>
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hermite nhermite n
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Show the equivalence of the polynomial form of the Hermite polynomials to the explicit form involving derivatives of the gaussian function.
Co-authored-by: Jake Levinson levinson.jake@gmail.com
We are interested in feedback about two parts of this PR:
- Should the definition(removed)gaussianbe removed and replaced with its definition written out explicitly? Or ifgaussianis kept, should it be placed somewhere else in mathlib?- Should the auxiliary definition(removed)hermite_gaussalso be removed the same way? If not, should it be hidden usingprotected defor something along these lines?