Derivatives and analyticity.
(Lecture 6 of Mathematical Methods II.)
Much like for a real function, the derivative at $z_0\in\mathbf{C}$ of a complex function $f$ is the limit, if it exists:
$$f'(z_0)=\lim_{z\rightarrow z_0}\frac{f(z)-f(z_0)}{z-z_0}\,.$$
While the definition is exactly the same as for a real function of a real variable, the algebra is now that of complex numbers. In particular with respect to the limit, the fraction must be (by definition) independent of the path taken by $z$ to go towards $z_0$.
While the derivative is difficult to visualize in term of a slope to the function, it is precisely what it achieves also in the complex plane: a linear expansion locally to a point. The fact that the derivative must be the same along any direction through which the limit is approached will result in strong properties for the function.
For this reason, we will qualify of holomorphic at $z_0$ any function which has a derivative for all points $z$ in a neighborhood of $z_0$. The reason for this name will be explained later when the said properties will have been characterized. More generally, we will otherwise say that a function is differentiable when it merely has a derivative (say, on a single point but not compulsorily on all those in an open containing this point).
This does not mean that differentiability alone says nothing about the function. For instance, just like in the real case, differentiability implies continuity, since $f$ differentiable at $z_0$ means that $\lim_{z\rightarrow z_0}\big(f(z)-f(z_0)\big)/(z-z_0)$ exists, which in turns implies that
$$\lim_{z\rightarrow z_0}f(z)=\lim_{z\rightarrow z_0}\Big\{\Big[\big(f(z)-f(z_0)\big)\big /(z-z_0)\Big](z-z_0)+f(z_0)\Big\}=f(z_0)$$
(by the properties of the limit $\lim(f+g)=\lim f+\lim g$ and $\lim fg=\lim f\lim g$ when these limits exist). But that $\lim_{z\rightarrow z_0} f(z)=f(z_0)$ is precisely the definition of $f$ being continuous at $z_0$.
By letting $z=z_0+\Delta z$ in the previous definition, the derivative can also be written in a form that allows for explicit calculations:
$$f'(z_0)=\lim_{\Delta z\rightarrow 0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}\,.$$
For example, for $f(z)=z^2$, we find that $f'(z_0)=\lim_{\Delta z\rightarrow 0}\big((z_0+\Delta z)^2-z_0^2\big)/\delta z=2z_0+\Delta z$ which, for all $\Delta z$ in the complex plane, goes to $2z_0$ as $\Delta z\rightarrow 0$, so that:
$$(z^2)'=2z$$
for all $z\in\mathbf{C}$. The general case for the monomial $z^n$ with $n\in\mathbf{N}$ reads $\big((z_0+\Delta z)^n-z_0^n\big)/\Delta z$ which brings us to:
$$(z^n)'=nz^{n-1}$$
from the binomial expansion $(x+y)^n = \sum_{k=0}^n {n \choose k}x^{k}y^{n-k}$.
As a non-polynomial example, if $f(z)=\exp(z)$, since $[\exp(z_0+\Delta z)-\exp(z_0)]/\Delta z=\exp(z_0)[\exp(\Delta z)-1]/\Delta z$, we find that:
$$\big(\exp(z)\big)'=\exp(z)$$
since $\exp(\Delta z)=1+\Delta z+\sum_{n=2}^\infty(\Delta z)^n/n!$. Just like for the real variable, the exponential is the function whose rate of growth is equal to itself.
As a rule, so far, complex derivative are the same as real ones with the promotion of the variable to the complex plane.
The function $f(z)=z^*$ (complex conjugation) is an example of a function that has no derivative. Indeed, $\big((z_0+\Delta z)^*-z_0^*\big)/\Delta z=\Delta z^*/\Delta z$ which has no limit as $\Delta z\rightarrow 0$ since, precisely, it depends on the path (as can be seen by taking the particular cases $\Delta z=\Delta x$ and $i\Delta y$).
We have used the exponential as a non-polynomial example, although we relied on the expansion $\exp(x)=\sum_{n=0}^\infty x^n/n!$ that we could have used directly to compute the derivative. In this way, for instance, we can see that
$$(\sin z)'=\left(\sum_{n=1}^\infty (-1)^{n+1}\frac{z^{2n-1}}{(2n-1)!}\right)'=\cos z\,.$$
Again, a result identical to that of the real variable. This is, not surprisingly, thanks to the same properties for the real variables of the types of functions we have been considering, namely, so-called analytic functions, which are those that are locally given by a convergent power series, i.e., those that can be expanded around $z_0$ in the form:
$$f(z)=\sum_{n=0}^\infty c_n(z-z_0)^n$$
for some coefficients $c_n\in\mathbf{C}$ which, one can check, are given by $f^{(n)}(z_0)/{n!}$. Note that $n\ge0$; later we will encounter important series where the summation extends over negative $n$. Functions that are holomorphic except at single points where the function diverges (called poles) are called Meromorphic.
One of the most important theorems of complex analysis is that holomorphic functions are analytic, and vice-versa. It is a very strong result. In particular, since power series can be iteratively differentiated, so are holomorphic functions infinitely differentiable: if the first derivative exists, so do all higher order derivatives.
We will keep encountering incredibly strong properties of this type in complex analysis. We will conclude this first encounter with their differential calculus with another classical character inherited from the real variable, namely, the derivative of the logarithm. Calling $w=\log z$, and remembering that by definition of the logarithm, this means that $e^w=z$, we compute:
\begin{align} (\log z_0)'&=\lim_{z\rightarrow z_0}\frac{w-w_0}{z-z_0}\\ &=\lim_{z\rightarrow z_0}\frac{1}{\frac{z-z_0}{w-w_0}}\\ &=\lim_{w\rightarrow w_0}\frac{1}{\frac{z-z_0}{w-w_0}}\\ &=\lim_{w\rightarrow w_0}\frac{1}{\frac{e^w-e^{w_0}}{w-w_0}}\\ &=\frac{1}{(e^{w_0})'}=\frac{1}{e^{w_0}}=\frac{1}{z_0}\,. \end{align}
We have used the continuity of the logarithm to pass from $z\rightarrow z_0$ to $w\rightarrow w_0$ (since the former implies the latter). Remarkably, the derivative of the logarithm (elsewhere than at zero where it is undefined), is the inverse function.
