Iterated Deriv resolvent
Physlib.Resolvent.iteratedDeriv_resolvent
Plain-language statement
Closed form for the iterated derivatives of the resolvent: the n-th derivative is (-1)ⁿ · n! · (resolvent z)ⁿ⁺¹.
Exact Lean statement
lemma iteratedDeriv_resolvent (hz : z.im ≠ 0) (n : ℕ) :
iteratedDeriv n (resolvent (R := ℝ) z)
= fun t ↦ (-1) ^ n * (n ! : ℂ) * resolvent z t ^ (n + 1)Formal artifact
Lean source
lemma iteratedDeriv_resolvent (hz : z.im ≠ 0) (n : ℕ) : iteratedDeriv n (resolvent (R := ℝ) z) = fun t ↦ (-1) ^ n * (n ! : ℂ) * resolvent z t ^ (n + 1) := by induction n with | zero => simp | succ n ih => funext t have hd := ((spectrum.hasDerivAt_resolvent_const_left (mem_resolventSet_of_im_ne_zero hz t)).pow (n + 1)).const_mul ((-1) ^ n * (n ! : ℂ)) simp only [Pi.pow_apply] at hd rw [iteratedDeriv_succ, ih, hd.deriv] push_cast [Nat.factorial_succ] ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Mathematics/Resolvent.lean:86-98
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