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I was reading around stuff on differentiability in $\mathbb{C}$ and wondered whether it is same as differentiability in $\mathbb{R}^2$. I approached a professor and gave me an example and asked me to think over it.

$f:\mathbb{R}^2 \to \mathbb{R}^2$, $ f(x,y)=(x,-y)$

And $g:\mathbb{C} \to \mathbb{C}$, $ g(z)=\bar z$, the conjugate of z

If we could plot both the graphs, both would be exactly the same.

Clearly $g$ is not differentiable anywhere in $\mathbb{C}$

Also $f$ is a Linear Transformation (as we can find a Matrix of transformation for the given function) and hence differentiable.

Why is there such a difference.

I read Complex differentiability vs differentiability in $\mathbb{R}^2$ and couldn't understand much as I'm just a beginner in Linear Algebra.

Please, try to explain in the simplest way possible. Thanks.

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Swapnil Tripathi
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1 Answers1

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This is not so much a matter of limits, as a matter of linear algebra.

A general linear map $$A:\quad{\mathbb R}^2\to{\mathbb R}^2,\quad{\bf z}\to{\bf w}=A{\bf z}\ ,$$ being a map between real-two-dimensional spaces, appears in coordinates $(x,y)$, resp. $(u,v)$, as $$\left.\eqalign{u&=a_{11}x+a_{12}y\cr v&=a_{21}x+a_{22}y\cr}\right\}\ .$$ Here $\bigl[a_{ik}\bigr]=:\bigl[A\bigr]$ can be an arbitrary $(2\times2)$-matrix with real entries. In particular, the derivative $d{\bf f}({\bf p})$ of a map $$f:\quad{\mathbb R}^2\to{\mathbb R}^2$$ at a point ${\bf p}$ can be any such $A$, and therefore can have $4$ independently chosen entries in its matrix.

Contrasting this, a linear map $$T:\quad{\mathbb C}\to{\mathbb C},\quad z\to w=Tz\ ,$$ being a map between complex-one-dimensional spaces, appears in the complex coordinates $z\ (=x+iy)$, resp. $w\ (=u+iv)$, as $$w=\lambda\>z,\qquad\lambda=\mu+i\nu\in{\mathbb C}\ .\tag{1}$$ Separating real and imaginary parts in $(1)$ we have $$\left.\eqalign{u&=\mu x-\nu y\cr v&=\nu x+\mu y\cr}\right\}\ .$$ The "geometric" map $T$, which appeared as one-dimensional complex-linear map in $(1)$, is now two-dimensional real linear with matrix $$\bigl[T\bigr]=\left[\matrix{\mu&-\nu\cr \nu&\mu\cr}\right]\ .\tag{2}$$ The matrix appearing here is not an arbitrary real $(2\times2)$-matrix, but has a particular structure with only $2$ "real degrees of freedom". That the $\bigl[T\bigr]$ in $(2)$ has this form is the content of the famous CR differential equations for analytic functions of $z$.

Christian Blatter
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