File: Mathlib/Analysis/Complex/RealDeriv.lean

full_name: HasStrictDerivAt.real_of_complex
code: /-- 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 HasStrictDerivAt.real_of_complex (h : HasStrictDerivAt e e' z) :
    HasStrictDerivAt (fun x : ℝ => (e x).re) e'.re z 
goal: HasStrictDerivAt (fun x => (e ↑x).re) e'.re z

full_name: HasDerivAt.real_of_complex
code: /-- 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 HasDerivAt.real_of_complex (h : HasDerivAt e e' z) :
    HasDerivAt (fun x : ℝ => (e x).re) e'.re z 
goal: HasDerivAt (fun x => (e ↑x).re) e'.re z

full_name: ContDiffAt.real_of_complex
code: theorem ContDiffAt.real_of_complex {n : ℕ∞} (h : ContDiffAt ℂ n e z) :
    ContDiffAt ℝ n (fun x : ℝ => (e x).re) z 
goal: ContDiffAt ℝ n (fun x => (e ↑x).re) z

full_name: ContDiff.real_of_complex
code: theorem ContDiff.real_of_complex {n : ℕ∞} (h : ContDiff ℂ n e) :
    ContDiff ℝ n fun x : ℝ => (e x).re 
goal: ContDiff ℝ n fun x => (e ↑x).re

full_name: HasStrictDerivAt.complexToReal_fderiv'
code: theorem HasStrictDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E}
    (h : HasStrictDerivAt f f' x) :
    HasStrictFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x 
goal: HasStrictFDerivAt f (Complex.reCLM.smulRight f' + Complex.I • Complex.imCLM.smulRight f') x

full_name: HasDerivAt.complexToReal_fderiv'
code: theorem HasDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E} (h : HasDerivAt f f' x) :
    HasFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x 
goal: HasFDerivAt f (Complex.reCLM.smulRight f' + Complex.I • Complex.imCLM.smulRight f') x

full_name: HasDerivWithinAt.complexToReal_fderiv'
code: theorem HasDerivWithinAt.complexToReal_fderiv' {f : ℂ → E} {s : Set ℂ} {x : ℂ} {f' : E}
    (h : HasDerivWithinAt f f' s x) :
    HasFDerivWithinAt f (reCLM.smulRight f' + I • imCLM.smulRight f') s x 
goal: HasFDerivWithinAt f (Complex.reCLM.smulRight f' + Complex.I • Complex.imCLM.smulRight f') s x

full_name: HasStrictDerivAt.complexToReal_fderiv
code: theorem HasStrictDerivAt.complexToReal_fderiv {f : ℂ → ℂ} {f' x : ℂ} (h : HasStrictDerivAt f f' x) :
    HasStrictFDerivAt f (f' • (1 : ℂ →L[ℝ] ℂ)) x 
goal: HasStrictFDerivAt f (f' • 1) x

full_name: HasDerivAt.complexToReal_fderiv
code: theorem HasDerivAt.complexToReal_fderiv {f : ℂ → ℂ} {f' x : ℂ} (h : HasDerivAt f f' x) :
    HasFDerivAt f (f' • (1 : ℂ →L[ℝ] ℂ)) x 
goal: HasFDerivAt f (f' • 1) x

full_name: HasDerivWithinAt.complexToReal_fderiv
code: theorem HasDerivWithinAt.complexToReal_fderiv {f : ℂ → ℂ} {s : Set ℂ} {f' x : ℂ}
    (h : HasDerivWithinAt f f' s x) : HasFDerivWithinAt f (f' • (1 : ℂ →L[ℝ] ℂ)) s x 
goal: HasFDerivWithinAt f (f' • 1) s x

full_name: HasDerivAt.comp_ofReal
code: /-- If a complex function `e` is differentiable at a real point, then its restriction to `ℝ` is
differentiable there as a function `ℝ → ℂ`, with the same derivative. -/
theorem HasDerivAt.comp_ofReal (hf : HasDerivAt e e' ↑z) : HasDerivAt (fun y : ℝ => e ↑y) e' z 
goal: HasDerivAt (fun y => e ↑y) e' z

full_name: HasDerivAt.ofReal_comp
code: /-- If a function `f : ℝ → ℝ` is differentiable at a (real) point `x`, then it is also
differentiable as a function `ℝ → ℂ`. -/
theorem HasDerivAt.ofReal_comp {f : ℝ → ℝ} {u : ℝ} (hf : HasDerivAt f u z) :
    HasDerivAt (fun y : ℝ => ↑(f y) : ℝ → ℂ) u z 
goal: HasDerivAt (fun y => ↑(f y)) (↑u) z

full_name: DifferentiableAt.conformalAt
code: /-- 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. -/
theorem DifferentiableAt.conformalAt (h : DifferentiableAt ℂ f z) (hf' : deriv f z ≠ 0) :
    ConformalAt f z 
goal: ConformalAt f z

full_name: conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj
code: /-- A complex function is conformal if and only if the function is holomorphic or antiholomorphic
    with a nonvanishing differential. -/
theorem conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj {f : ℂ → ℂ} {z : ℂ} :
    ConformalAt f z ↔
      (DifferentiableAt ℂ f z ∨ DifferentiableAt ℂ (f ∘ conj) (conj z)) ∧ fderiv ℝ f z ≠ 0 
goal: ConformalAt f z ↔
  (DifferentiableAt ℂ f z ∨ DifferentiableAt ℂ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z)) ∧ fderiv ℝ f z ≠ 0