Documentation

Mathlib.Analysis.Complex.RealDeriv

Real differentiability of complex-differentiable functions #

HasDerivAt.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.

DifferentiableAt.conformalAt 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.

conformalAt_iff_differentiableAt_or_differentiableAt_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 HasStrictDerivAt.real_of_complex {e : } {e' : } {z : } (h : HasStrictDerivAt e e' z) :
HasStrictDerivAt (fun (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 HasDerivAt.real_of_complex {e : } {e' : } {z : } (h : HasDerivAt e e' z) :
HasDerivAt (fun (x : ) => (e x).re) e'.re z

If a complex function e is differentiable at a real point, then the function ℝ → ℝ given by the real part of e is also differentiable at this point, with a derivative equal to the real part of the complex derivative.

theorem ContDiffAt.real_of_complex {e : } {z : } {n : ℕ∞} (h : ContDiffAt n e z) :
ContDiffAt n (fun (x : ) => (e x).re) z
theorem ContDiff.real_of_complex {e : } {n : ℕ∞} (h : ContDiff n e) :
ContDiff n fun (x : ) => (e x).re
theorem HasStrictDerivAt.complexToReal_fderiv' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {x : } {f' : E} (h : HasStrictDerivAt f f' x) :
theorem HasDerivAt.complexToReal_fderiv' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {x : } {f' : E} (h : HasDerivAt f f' x) :
HasFDerivAt f (Complex.reCLM.smulRight f' + Complex.I Complex.imCLM.smulRight f') x
theorem HasDerivWithinAt.complexToReal_fderiv' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {f : E} {s : Set } {x : } {f' : E} (h : HasDerivWithinAt f f' s x) :
HasFDerivWithinAt f (Complex.reCLM.smulRight f' + Complex.I Complex.imCLM.smulRight f') s x
theorem HasStrictDerivAt.complexToReal_fderiv {f : } {f' : } {x : } (h : HasStrictDerivAt f f' x) :
theorem HasDerivAt.complexToReal_fderiv {f : } {f' : } {x : } (h : HasDerivAt f f' x) :
HasFDerivAt f (f' 1) x
theorem HasDerivWithinAt.complexToReal_fderiv {f : } {s : Set } {f' : } {x : } (h : HasDerivWithinAt f f' s x) :
HasFDerivWithinAt f (f' 1) s x
theorem HasDerivAt.comp_ofReal {e : } {e' : } {z : } (hf : HasDerivAt e e' z) :
HasDerivAt (fun (y : ) => e y) e' z

If a complex function e is differentiable at a real point, then its restriction to is differentiable there as a function ℝ → ℂ, with the same derivative.

theorem HasDerivAt.ofReal_comp {z : } {f : } {u : } (hf : HasDerivAt f u z) :
HasDerivAt (fun (y : ) => (f y)) (↑u) z

If a function f : ℝ → ℝ is differentiable at a (real) point x, then it is also differentiable as a function ℝ → ℂ.

Conformality of real-differentiable complex maps #

theorem DifferentiableAt.conformalAt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace E] {z : } {f : E} (h : DifferentiableAt f z) (hf' : deriv f z 0) :

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.