When we do analysis with real-valued functions of a single variable, differentiability at a point says that there must exist a linear map (often called a “linearization” or “linear approximation”) such that the limit as . So we can basically see that this map, when supplied with , gives us a (loosely speaking) “close” estimate of the difference between and . In this single-variable estimation, this linear map is represented by a matrix, in other words a real number, and is denoted . Continuing with this idea, a function is differentiable if there is a linear map (this time from ) which satisfies a similar limit. This linear map turns out to be the left-multiplication transformation associated to an matrix called the Jacobian and if , that is, if our function is real-valued, it is a row matrix, which is the function’s so-called “gradient” (in this way, a Jacobian can be thought of just as a matrix whose rows are the gradients of the component functions for ).
In complex analysis, new functions are examined; namely functions . For our purposes the set is a nonempty, open, path-connected subset of (these are often referred to as domains). Intuitively then, such a function can be thought of as a transformation of a subset of the complex plane. Since two quantities (usually the modulus and the argument of the resulting complex number) are associated to each point in by this function, the graph is 4-dimensional (as opposed to the 2-dimensional graphs of single-variable analysis), therefore it is quite difficult to represent it properly without resorting to 2D color maps, and so on. Often, an image of a grid in the complex plane is shown, and then another diagram shows the same grid after it is transformed by the function. We first say what it means for such a function to be differentiable at a point .
Definition. Suppose , given by , is a complex-valued function of a complex variable . ( and are the real and imaginary parts of , respectively). Then we say is differentiable at if the limit of exists as .
Definition. If is differentiable on all of it is said to be holomorphic on . If it is differentiable in some neighbourhood of it is said to be holomorphic at .
So in any case we see that holomorphicity is a stronger condition than differentiability, as it requires differentiability not only at a given point, but at least on some neighbourhood around that point. Some authors use the term analytic or regular rather than holomorphic, but I disagree with this. The word “analytic” should be reserved for functions which have power series expansions on that set/neighbourhood.
Now that we know what it takes for a function to be complex-differentiable, we now ask whether there is some connection with the more familiar world of real-valued functions. There is: a certain condition on the functions and turns out to be linked to complex differentiability. These are called the Cauchy-Riemann equations, and I will speak about this in a follow-up post. I am currently reviewing a few notions of differentiability from multivariable calculus that are needed.