Cauchy-Riemann Equations
The goal of this blog is for me to go through the derivation of Cauchy-Riemann Equations starting from derivability of a function on a complex variable. That said, this blog is aimed to improve my own understanding of the subject, and less aimed to convey the results to others, since I haven't mastered the concept.
Differentiability for R2→R2 and C→C
Definition. Let f:Ω⊆R2→R2 be a continuous function, where Ω is an open set. We say that f is differentiable at x0∈R2 if there exists a linear map df∣x0:R2→R2 such that
f(x0+h)=f(x0)+df∣x0(h)+o(h)
as ∥h∥→0 (for any norm ∥⋅∥).
To motivate the definition for complex differentials, we want to replace the linear map df∣x0 with a complex number df∣x0. This gives us the following definition.
Definition. Let f:Ω⊆C→C be a continuous function, where Ω is an open set. We say that f is differentiable at z0∈C (where we write z0=x0+iy0 so that x0,y0∈R) if there exists a complex number df∣z0∈C such that
f(z0+h)=f(z0)+df∣z0h+o(h)
as h→0.
Well, then, suppose f can be written as u+iv, so that f(x+iy)=u(x,y)+iv(x,y) for all x,y∈R such that x+iy∈Ω, and u,v:R2→R. Then what we see is that from
f(z0+h)=f(z0)+df∣z0h+o(h),
this means
h→0,h=0limhf(z0+h)−f(z0)=df∣z0
so when we restrict h to be real, we have
h→0,h=0,h∈Rlimhf(z0+h)−f(z0)=df∣z0.
That is,
h→0,h=0,h∈Rlimhu(x0+h,y0)+iv(x0+h,y0)−u(x0,y0)−iv(x0,y0)=df∣z0.
If we take the real part of both sides, we have
h→0,h=0,h∈Rlimhu(x0+h,y0)−u(x0,y0)=ℜ(df∣z0).
This is precisely ∂xu(x0,y0)=ℜ(df∣z0). Once we do the same for the imaginary part, we have
∂xv(x0,y0)=ℑ(df∣z0).
Now, instead of restricting h to be real, let us restrict h to be imaginary. We have
h→0,h=0,h∈iRlimhf(z0+h)−f(z0)=df∣z0.
or equivalently,
h→0,h=0,h∈Rlimihu(x0,y0+h)+iv(x0,y0+h)−u(x0,y0)−iv(x0,y0)=df∣z0.
Now, using the fact that i1=−i, we have
h→0,h=0,h∈Rlimh−iu(x0,y0+h)+v(x0,y0+h)+iu(x0,y0)−v(x0,y0)=df∣z0.
Taking real part of this equation, we have
∂yv(x0,y0)=ℜ(df∣z0),
and the imaginary part gives
−∂yu(x0,y0)=ℑ(df∣z0).
Comparing these two the previous results, we have shown that
∂xu(x0,y0)=ℜ(df∣z0)=∂yv(x0,y0),
and
∂xv(x0,y0)=ℑ(df∣z0)=−∂yu(x0,y0).
Translating these back into the notion of f, we have that these two equations
{∂xℜ(f(z0))=∂yℑ(f(z0))∂xℑ(f(z0))=−∂yℜ(f(z0))
hold true whenever f is complex-differentiable at z0∈Ω.
These equations are, indeed, the Cauchy-Riemann equations.