mathlib documentation

analysis.complex.real_deriv

Real differentiability of complex-differentiable functions #

has_deriv_at.real_of_complex expresses that, if a function on is differentiable (over ), then its restriction to is differentiable over , with derivative the real part of the complex derivative.

differentiable_at.conformal_at states that a real-differentiable function with a nonvanishing differential from the complex plane into an arbitrary complex-normed space is conformal at a point if it's holomorphic at that point. This is a version of Cauchy-Riemann equations.

conformal_at_iff_differentiable_at_or_differentiable_at_comp_conj proves that a real-differential function with a nonvanishing differential between the complex plane is conformal at a point if and only if it's holomorphic or antiholomorphic at that point.

TODO #

Warning #

We do NOT require conformal functions to be orientation-preserving in this file.

Differentiability of the restriction to of complex functions #

theorem has_strict_deriv_at.real_of_complex {e : } {e' : } {z : } (h : has_strict_deriv_at e e' z) :
has_strict_deriv_at (λ (x : ), (e x).re) e'.re z

If a complex function is differentiable at a real point, then the induced real function is also differentiable at this point, with a derivative equal to the real part of the complex derivative.

theorem has_deriv_at.real_of_complex {e : } {e' : } {z : } (h : has_deriv_at e e' z) :
has_deriv_at (λ (x : ), (e x).re) e'.re z

If a complex function is differentiable at a real point, then the induced real function is also differentiable at this point, with a derivative equal to the real part of the complex derivative.

theorem times_cont_diff_at.real_of_complex {e : } {z : } {n : with_top } (h : times_cont_diff_at n e z) :
times_cont_diff_at n (λ (x : ), (e x).re) z
theorem times_cont_diff.real_of_complex {e : } {n : with_top } (h : times_cont_diff n e) :
times_cont_diff n (λ (x : ), (e x).re)
theorem has_deriv_at.complex_to_real_fderiv {f : } {f' x : } (h : has_deriv_at f f' x) :
has_fderiv_at f (f' 1) x
theorem has_deriv_within_at.complex_to_real_fderiv {f : } {s : set } {f' x : } (h : has_deriv_within_at f f' s x) :

Conformality of real-differentiable complex maps #

theorem differentiable_at.conformal_at {E : Type u_1} [normed_group E] [normed_space E] [normed_space E] [is_scalar_tower E] {z : } {f : → E} (hf' : fderiv f z 0) (h : differentiable_at f z) :

A real differentiable function of the complex plane into some complex normed space E is conformal at a point z if it is holomorphic at that point with a nonvanishing differential. This is a version of the Cauchy-Riemann equations.

A complex function is conformal if and only if the function is holomorphic or antiholomorphic with a nonvanishing differential.