Proof of The Cauchy–Riemann Equations

Routine arithmetic and bookkeeping facts are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity). Throughout, $\Phi$ and $V$ are as in The Complex Plane as ℝ²: Standing Notation, whose background is used without further citation: $\Phi$ is a real-linear isometric bijection, and $V$ is open in $(\mathbb{R}^{2},d_{E})$, so the notions of Differentiability at a Point for Functions Between Euclidean Spaces (i) apply to $F$ on $V$. For $w\in U$ with $w\neq z_{0}$ we write

$$Q(w)=\frac{f(w)-f(z_{0})}{w-z_{0}}$$

for the difference quotient of $f$ at $z_{0}$. In real coordinates, $\Phi(F(q))=f(\Phi(q))$ for every $q\in V$, since $F(q)=\Phi^{-1}\bigl(f(\Phi(q))\bigr)$ by The Complex Plane as ℝ²: Standing Notation (iv).

Step 1: multiplication by $a+ib$ as a matrix. For real numbers $a$ and $b$ let $A_{a,b}\in\mathcal{M}_{2\times2}(\mathbb{R})$ be the matrix with rows $(a,-b)$ and $(b,a)$. Then

$$\Phi(A_{a,b}h)=(a+ib)\,\Phi(h)\qquad(h\in\mathbb{R}^{2}).$$

Indeed, for $h=(s,t)$ the matrix-vector product is $A_{a,b}h=(as-bt,\ bs+at)$, while multiplication of complex numbers in canonical form (The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii)) gives

$$(a+ib)(s+it)=(as-bt)+i\,(bs+at).$$

Step 2: the two error terms agree. Let $h\in\mathbb{R}^{2}$ with $p_{0}+h\in V$, and put $w=\Phi(p_{0}+h)$; by linearity of $\Phi$, $w=z_{0}+\Phi(h)$, that is, $\Phi(h)=w-z_{0}$. Applying $\Phi$ to $F(p_{0}+h)-F(p_{0})-A_{a,b}h$ and using its linearity together with the identity $\Phi(F(q))=f(\Phi(q))$ and Step 1 gives $f(w)-f(z_{0})-(a+ib)(w-z_{0})$. Moreover $\lVert x\rVert=|\Phi(x)|$ for every $x\in\mathbb{R}^{2}$, by the isometry property (The Complex Plane as $\mathbb{R}^2$: the Map $(x,y)\mapsto x+iy$ Is a Real-Linear Isometric Bijection (iii)) with $p=x$ and $q=0$ and since $\Phi(0)=0$ by linearity (The Complex Plane as $\mathbb{R}^2$: the Map $(x,y)\mapsto x+iy$ Is a Real-Linear Isometric Bijection (ii)); therefore

$$\lVert F(p_{0}+h)-F(p_{0})-A_{a,b}h\rVert=\bigl|f(w)-f(z_{0})-(a+ib)(w-z_{0})\bigr|,\qquad\lVert h\rVert=|w-z_{0}|.$$

So the error of the linear approximation of $F$ at $p_{0}$ by $A_{a,b}$ is exactly the error of the complex-linear approximation of $f$ at $z_{0}$ by $a+ib$, measured at corresponding points.

Step 3: complex differentiability gives the derivative matrix. This proves (i) from left to right. Suppose $f$ is complex differentiable at $z_{0}$ with $f'(z_{0})=a+ib$. Since $U$ is open and $z_{0}\in U$, first fix $r>0$ with $B_{d_{\mathbb{C}}}(z_{0},r)\subseteq U$ (Open Subset of a Metric Space (i)); $r$ does not depend on what follows. Now let $\varepsilon>0$ be given. Take $\delta_{0}>0$ as in The Complex Derivative at a Point (i) for this $\varepsilon$, so that $|Q(w)-(a+ib)|<\varepsilon$ whenever $w\in U$ and $0<|w-z_{0}|<\delta_{0}$, and put $\delta=\min(r,\delta_{0})$.

Let $h\in\mathbb{R}^{2}$ with $\lVert h\rVert<\delta$, and put $w=z_{0}+\Phi(h)=\Phi(p_{0}+h)$. Then $|w-z_{0}|=\lVert h\rVert<r$, so $w\in U$, and hence $p_{0}+h\in V$. If $h=0$, the error in Step 2 is $0$. If $h\neq0$, then $0<|w-z_{0}|<\delta_{0}$, and writing $f(w)-f(z_{0})-(a+ib)(w-z_{0})=(w-z_{0})\bigl(Q(w)-(a+ib)\bigr)$, Step 2 gives

$$\lVert F(p_{0}+h)-F(p_{0})-A_{a,b}h\rVert=|w-z_{0}|\,\bigl|Q(w)-(a+ib)\bigr|\le\varepsilon\,\lVert h\rVert.$$

As $\varepsilon>0$ was arbitrary, $F$ is differentiable at $p_{0}$ with derivative matrix $A_{a,b}$ (Differentiability at a Point for Functions Between Euclidean Spaces (i)).

Step 4: the derivative matrix gives complex differentiability. This proves (i) from right to left. Suppose $F$ is differentiable at $p_{0}$ with derivative matrix $A_{a,b}$, and let $\varepsilon>0$ be given. Take $\delta>0$ as in Differentiability at a Point for Functions Between Euclidean Spaces (i) for $\varepsilon/2$ in place of $\varepsilon$. Let $w\in U$ with $0<|w-z_{0}|<\delta$, and put $h=\Phi^{-1}(w-z_{0})$. Then $\lVert h\rVert=|w-z_{0}|<\delta$ and $\Phi(p_{0}+h)=z_{0}+(w-z_{0})=w\in U$, so $p_{0}+h\in V$, and Step 2 gives

$$\bigl|f(w)-f(z_{0})-(a+ib)(w-z_{0})\bigr|\le\frac{\varepsilon}{2}\,|w-z_{0}|.$$

Dividing by $|w-z_{0}|>0$ yields $|Q(w)-(a+ib)|\le\varepsilon/2<\varepsilon$. Hence $a+ib$ is a complex derivative of $f$ at $z_{0}$ (The Complex Derivative at a Point (i)), so $f$ is complex differentiable at $z_{0}$ and $f'(z_{0})=a+ib$ (The Complex Derivative at a Point (ii)).

Step 5: the Cauchy–Riemann equations and the derivative. This proves (ii) and (iii). Recall that $u=F_{1}$, $v=F_{2}$, and that $\partial/\partial x$ and $\partial/\partial y$ are the partial derivatives $\partial_{1}$ and $\partial_{2}$ (The Complex Plane as ℝ²: Standing Notation (v)).

Suppose first that $f$ is complex differentiable at $z_{0}$, and write $f'(z_{0})=a+ib$ with $a,b$ real. By (i), $F$ is differentiable at $p_{0}$ with derivative matrix $A_{a,b}$, and by The Derivative Is the Jacobian Matrix, and Is Unique (ii) the partial derivatives of $u$ and $v$ at $p_{0}$ exist and are the entries of $A_{a,b}$:

$$\frac{\partial u}{\partial x}(p_{0})=a,\qquad\frac{\partial u}{\partial y}(p_{0})=-b,\qquad\frac{\partial v}{\partial x}(p_{0})=b,\qquad\frac{\partial v}{\partial y}(p_{0})=a.$$

So the Cauchy–Riemann equations hold, and $f'(z_{0})=a+ib$ is the formula of (iii).

Conversely, suppose $F$ is differentiable at $p_{0}$, say with derivative matrix $A$ (Differentiability at a Point for Functions Between Euclidean Spaces (iii)), and that the Cauchy–Riemann equations hold. By The Derivative Is the Jacobian Matrix, and Is Unique (ii), the entries of $A$ are the partial derivatives of $u$ and $v$ at $p_{0}$; with $a=\frac{\partial u}{\partial x}(p_{0})$ and $b=\frac{\partial v}{\partial x}(p_{0})$ the Cauchy–Riemann equations say exactly that

$$A=\begin{pmatrix}\dfrac{\partial u}{\partial x}(p_{0})&\dfrac{\partial u}{\partial y}(p_{0})\\ \dfrac{\partial v}{\partial x}(p_{0})&\dfrac{\partial v}{\partial y}(p_{0})\end{pmatrix}=\begin{pmatrix}a&-b\\ b&a\end{pmatrix}=A_{a,b}.$$

Now (i) shows that $f$ is complex differentiable at $z_{0}$ with $f'(z_{0})=a+ib$, which is the formula of (iii).