Holomorphic functions #1

When we do analysis with real-valued functions of a single variable, differentiability at a point a \in \mathbb{R} says that there must exist a linear map (often called a “linearization” or “linear approximation”) L such that the limit \frac{f(a+\Delta x) - f(a) - L_a(\Delta x)}{\Delta x} = 0 as \Delta x \to 0. So we can basically see that this map, when supplied with \Delta x, gives us a (loosely speaking) “close” estimate of the difference between f(a+\Delta x) and f(a). In this single-variable estimation, this linear map is represented by a 1 \times 1 matrix, in other words a real number, and is denoted f'(a). Continuing with this idea, a function f : \mathbb{R}^n \to \mathbb{R}^m is differentiable if there is a linear map (this time from \mathbb{R}^n \to \mathbb{R}^m) which satisfies a similar limit. This linear map turns out to be the left-multiplication transformation associated to an m \times n matrix called the Jacobian and if m=1, that is, if our function is real-valued, it is a row matrix, which is the function’s so-called “gradient” \nabla f (in this way, a Jacobian can be thought of just as a matrix whose rows are the gradients of the component functions f^{(i)} for 1 \leq i \leq m).

In complex analysis, new functions are examined; namely functions f : D \to \mathbb{C}. For our purposes the set D is a nonempty, open, path-connected subset of \mathbb{C} (these are often referred to as domains). Intuitively then, such a function f 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 D 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 z_0 = x_0 + iy_0 \in D.

Definition. Suppose f : D \to \mathbb{C}, given by f(z) = u(x,y) + iv(x,y), is a complex-valued function of a complex variable z=x+iy. (u and v are the real and imaginary parts of f, respectively). Then we say f is differentiable at z_0 = x_0 + iy_0 \in D if the limit of \frac{f(z+\Delta z) - f(z)}{\Delta z} exists as \Delta z \to 0.

Definition. If f is differentiable on all of D it is said to be holomorphic on D. If it is differentiable in some neighbourhood of z_0 it is said to be holomorphic at z_0.

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 u and v 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.

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About mlbaker

just another guy trying to make the diagrams commute.
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2 Responses to Holomorphic functions #1

  1. crobert says:

    “The word “analytic” should be reserved for functions which have power series expansions on that set/neighbourhood.”

    …but every holomorphic function does have a power series (locally).

  2. dx7hymcxnpq says:

    Exactly, but I think it’s good to keep two separate names for the two properties, which (nontrivially) happen to imply one another. At least while you’re talking about elementary complex analysis – since after that you can probably assume everyone knows what’s going on. lol

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