Differentiability and Cauchy-Riemann.
(Lecture 7 of Mathematical Methods II.)
We have seen last time how to differentiate the complex function $w$ of the complex variable $z$. We have also seen on the second lecture how such a function also gives rise to two real-valued functions $u$ and $v$ of the real variables $x$ and $y$ through:
$$w(z)=u(x,y)+iv(x,y)\,.$$
In this lecture, we will bring these two approaches together, the derivative of $w$ seen as a function with a real and imaginary part of two variables $x$ and $y$ making up $z=x+iy$.
Let us assume that $w'(z_0)$ exists (that is, $w$ is differentiable at $z_0$ and, so far, at $z_0$ only). That is, by last time's definition:
$$\lim_{z\rightarrow z_0}\frac{w(z)-w(z_0)}{z-z_0}$$
exists (and is equal to $w'(z_0)\in\mathbf{C}$). Here we will use the fact that if the limit of a complex function exists, the limit of its real part and imaginary part also exist and are equal to the real and imaginary parts of the limit, respectively (this can be proved as an exercise). This allows us to write the above expression as:
$$w'(z_0)=\lim_{z\rightarrow z_0}\frac{u(x,y)-u(x_0,y_0)}{z-z_0}+i\lim_{z\rightarrow z_0}\frac{v(x,y)-v(x_0,y_0)}{z-z_0}\,.$$
Now by definition of the derivative, $z$ can approach $z_0$ from any path, and still the result must be the same number, namely, $w'(z_0)$. We now take particular cases:
Say $z\rightarrow z_0$ horizontally, i.e., $z=x+iy_0$ and $x\rightarrow x_0$. By mere substitution in the above formula:
$$w'(z_0)=\lim_{x\rightarrow x_0}\frac{u(x,y_0)-u(x_0,y_0)}{x-x_0}+i\lim_{x\rightarrow x_0}\frac{v(x,y_0)-v(x_0,y_0)}{x-x_0}\,.$$
We recognize the objects that appeared in this particular case from the calculus of several variables as the partial derivative $\partial$. Therefore:
\begin{equation} \tag{1} w'(z_0)=\frac{\partial u}{\partial x}(x_0,y_0)+i\frac{\partial v}{\partial x}(x_0,y_0)\,. \end{equation}
This is already an interesting result. If we know that $w$ is differentiable at $z_0$, we can compute it through Eq.~(1). There is more. If we compute the derivative approaching $z_0$ vertically, that is, $z=x_0+iy$ and $y\rightarrow y_0$, doing the same, we arrive at the counterpart:
\begin{equation} \tag{2} w'(z_0)=-i\frac{\partial u}{\partial y}(x_0,y_0)+\frac{\partial v}{\partial y}(x_0,y_0)\,. \end{equation}
In the computation of $z-z_0$ along the vertical direction, we gathered a factor $i$ that reappears in the above expression.
If we now compare Eqs.~(1) and~(2), that are equal (from their lhs), we see that if $w$ is differentiable at $z_0=x_0+iy_0$, then:
\begin{align} &\frac{\partial u}{\partial x}(x_0,y_0)=\frac{\partial v}{\partial y}(x_0,y_0)\,,\\ &\frac{\partial u}{\partial y}(x_0,y_0)=-\frac{\partial v}{\partial x}(x_0,y_0)\,. \end{align}
These equations are known as the Cauchy-Riemann equations.
There is an equally important counterpart for the reverse relationship, namely, if the partial derivatives of $u$ and $v$ exist at $(x_0,y_0)$ and are differentiable in the sense of a function of multiple variables, then $w=u+iv$ is differentiable at~$x_0+iy_0$.
From $\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}=\frac{\partial w}{\partial x}$ and the same for $\partial/\partial_y$ we see that the partial derivatives of $w$ exist. If $u$ and $v$ are differentiable there, we can linearise $w$ as function of multiple variables:
$$w(x,y)=w(x_0,y_0)+\frac{\partial w}{\partial x}(x-x_0)+\frac{\partial w}{\partial y}(y-y_0)+\epsilon(x-x_0,y-y_0)$$
with $\epsilon(x,y)$ a function such that $\lim_{x,y\rightarrow 0}\frac{\epsilon(x,y)}{\sqrt{x^2+y^2}}=0$.
- (Here careful with notations. We should really be writing, e.g., $\frac{\partial w}{\partial x}(x_0,y_0)\times(x-x_0)$ instead of simply $\frac{\partial w}{\partial x}(x-x_0)$ but we will use the shorter version in the intermediate steps, remembering of the full version for the final result.)
The above means that $w$ is differentiable as a real function of two variables. From the Cauchy-Riemann equations, however, we find that $\partial w/\partial y=i\partial w/\partial x$, so that the linearization can be written:
$$w(x,y)=w(x_0,y_0)+\frac{\partial w}{\partial x}\big((x-x_0)+i(y-y_0)\big)+\epsilon\,.$$
Now the term in parentheses is $z-z_0$, so that:
$$\frac{w(z)-w(z_0)}{z-z_0}=\frac{\partial w}{\partial x}+\epsilon/(z-z_0)\,.$$
Here we see that regardless of the trajectory with which $z$ approaches $z_0$, the quotient on the lhs is unique (namely, $\frac{\partial w}{\partial x}(z_0)$). Taking the limit, which thus exists, we find that:
$$\lim_{z\rightarrow z_0}\frac{w(z)-w(z_0)}{z-z_0}=\frac{\partial w}{\partial x}(z_0)=\color{blue}{-i\frac{\partial w}{\partial y}(z_0)}$$
which proves that $w$ is differentiable as a complex function (independently of the path of approach). We also find the value of the derivative (we added in blue the expression if we had used partial derivation along $y$ instead; the last equality can also be seen as the Cauchy-Riemann equations, written more economically).
- You may find in some textbooks that continuity of the partial derivatives is required. This is in fact too strong a condition. While it is true that the sole existence of partial derivatives satisfying the Cauchy-Riemann equations is not enough to ensure complex differentiability, it is enough that $u$ and $v$ be real differentiable, which is a stronger condition than the existence of the partial derivatives, but is a weaker one than these partial derivatives themselves also be continuous.~[1][2]
If the Cauchy-Riemann conditions holds on an entire open, the function is, by definition, holomorphic there (and therefore, also analytical).
We conclude with an insight into a more involved notion of complex derivation. Introducing $\partial w/\partial z^*$, with $z^*$ the conjugate of $z$. Using the chain rule:
\begin{align}\frac{\partial w}{\partial z^*}&=\frac{\partial w}{\partial x}\frac{\partial x}{\partial z^*}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial z^*}\\ &=\left(\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}\right)\frac{1}{2}+\left(\frac{\partial u}{\partial y}+i\frac{\partial v}{\partial y}\right)\frac{i}{2}\\ &=\frac{1}{2}\left(\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\right)+\frac{i}{2}\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right) \end{align}
where we have used $x=(z+z^*)/2$ and $y=-i(z-z^*)/2$ to compute (formally) the partial derivative of $z^*$. If the Cauchy-Riemann conditions hold, $\partial w/\partial z^*=0$. This suggests that complex-differentiability means that the function depends on $z$ only, not $z^*$. Holomorphicity is a notion of $z$ seen not as a two-dimensional variable but as a monolithic, one-dimensional object of complex dimension.
