Abstract Nonsense

Complex Differentiability and Holomorphic Functions (Pt. I)

Point of Post: In this post we define the notion of a function $f:\Omega\to\mathbb{C}$ to be holmorphic on some domain $\Omega\subseteq\mathbb{C}$.

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Motivation

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We are going to start discussing complex analysis in preparation for later discussion on Riemann surfaces. We start this discussion, naturally, with the notion of differentiability for functions mapping $\mathbb{C}\supseteq\Omega\to\mathbb{C}$. There is a standard amount of amazement associated to functions which are differentiable in the complex sense since, as we shall see (and, as I’m sure you well-know), they are MUCH nicer then any kind of real differentiable function $U\to\mathbb{R}^2$. In particular, we shall see that any once differentiable function shall be infinitely differentiable and, moreover, locally be expressable as a power series. Think about how different this is from standard real differentiable functions, say, even just $\mathbb{R}\to\mathbb{R}$ where we can find functions that have any number $N$ of deriviatives we desire, yet it’s $N^{\text{th}}$ derivative not even be continuous, let alone differentiable. We can even find functions that are infinitely differentiable yet whose Taylor series at a point doesn’t converge to the function!

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If you are shocked by this (which, as I already said, I doubt you are) then you will be even more shocked to learn that this is just the beginning of a long line of facts about complex differentiable functions which will blow your mind. Theorems like Liouville’s theorem, the Maximum Modulus Principle, the Argument Principle, Cauchy’s integral formula, etc. show you awesomely powerful the condition of being complex differentiable really is.

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This all begs the obvious quetsion: why? Why are functions which are complex differentiable SO incredibly more powerful then their real differentiable counterparts? This is a very good question, and one that I will (without a doubt, badly) attempt to explain. The rough idea behind the difference between differentiable functions $\mathbb{R}^2\to\mathbb{R}^2$ and complex differentiable functions $\mathbb{C}\to\mathbb{C}$ is commutation with rotations. In particular, similar to how we defined the total derivative we shall define complex differentiable functions to be those that are locally approximatable by linear functions. Where’s the difference? The devil is in the details, for we shall want our approximating linear functions in the complex case to be complex linear. The real difference then being between the total derivative $D_f(x,y)$ and the “complex total derivative” $\widetilde{D}_f(x+iy)$ is that while the relation

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$\widetilde{D}_f(x+iy)(iz)=i\widetilde{D}_f(x+iy)(z)$

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one doesn’t necessarily require that $D_f(x,y)$ commute with $i$.

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What exact does this commutation mean? It’s a common geometrical fact that if one thinks about multiplication by $i$ as an $\mathbb{R}$-linear transformation $T:\mathbb{R}^2\to\mathbb{R}^2$ then it’s just the transformation given in the standard basis by $\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$ which (as should be well-known from linear algebra) is just counterclockwise rotation by ninety degrees. Thus, we see that relation held by the complex total derivative is that

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$D_f(x+iy)\circ T=T\circ D_f(x+iy)$

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Now, we know exactly what the matrix of $D_f(x_0,y_0)$ looks like, it’s just

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$\displaystyle \text{Jac}_f(x_0,y_0)=\begin{pmatrix}\displaystyle \frac{\partial u}{\partial x}(x_0,y_0) & \displaystyle \frac{\partial u}{\partial y}(x_0,y_0)\\ \displaystyle \frac{\partial v}{\partial x}(x_0,y_0) & \displaystyle\frac{\partial v}{\partial y}(x_0,y_0)\end{pmatrix}$

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if $f(x,y)=(u(x,y),v(x,y))$. Thus, I leave it in the readerships capable hands to check the relation

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$\text{Jac}_f(x_0,y_0)\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}=\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}\text{Jac}_f(x_0,y_0)$

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Implies the set of equations

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$\begin{cases}\displaystyle \frac{\partial u}{\partial x}(x_0,y_0)=\frac{\partial v}{\partial y}(x_0,y_0) \\ \text{ }\\ \displaystyle \frac{\partial u}{\partial y}(x_0,y_0)=-\frac{\partial v}{\partial x}(x_0,y_0) \end{cases}$

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known as the Cauchy-Riemann equations. Thus, we see immediately that being complex differentiable imposes some serious restrictions on the function that are certainly not present in the real differentiable case.

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So, modulo some small details, the functions that are complex differentiable are precisely those which are real differentiable and which commute with $i$, or because of our analysis it’s precisely the real differentiable functions which satisfy the Cauchy-Riemann equations. Since the Cauchy-Riemann equations are kind of messy, it would be nice if we could represent them in a simpler fashion. Note now that $\displaystyle x=\frac{z+\bar{z}}{2}$ and $\displaystyle y=\frac{z+\bar{z}}{2i}$. Intuitively we can think about the correspondence $\displaystyle \frac{z+\bar{z}}{2}\leftrightarrow x$ and $\displaystyle \frac{z-\bar{z}}{2i}\leftrightarrow y$ as being a linear change of coordinates. I claim that this change of coordinates is exactly the fix we need to make the Cauchy-Riemann equations simpler. Indeed, I want to consider things like $\displaystyle \frac{\partial f}{\partial z}$ and $\displaystyle \frac{\partial f}{\partial \bar{z}}$ which is a totally sensible thing to do, and in fact is easily evaluatable. To find $\displaystyle \frac{\partial f}{\partial z}$ we merely make the change of coordinates to get that $\displaystyle f(x,y)=f\left(\frac{z+\bar{z}}{2},\frac{z-\bar{z}}{2i}\right)$ and then differentiate according to normal partial rules. I have yet to say why this coordinate change makes the Cauchy-Riemann equations less messy. The reason is that if one follows $\displaystyle \frac{\partial f}{\partial \bar{z}}$ back along the coordinate change one finds that it is equal to $\displaystyle \frac{1}{2}\left(\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\right)$. Thus, writing $f=u+iv$ we see that

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$\displaystyle \frac{\partial f}{\partial\bar{z}}=\frac{1}{2}\left(\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\right)=\frac{1}{2}\left(\left(\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\right)+i\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right)$

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Aha! Thus, we see that $f$ satisfying the Cauchy-Riemann equations is equivalent to the statement that $\displaystyle \frac{\partial f}{\partial\bar{z}}=0$! What this means intuitively is what if we write $\displaystyle f(x,y)$ as $\displaystyle f\left(\frac{z+\bar{z}}{2},\frac{z-\bar{z}}{2i}\right)$ and “expand” all the $\bar{z}$‘s should cancel–there is no dependence on $\bar{z}$ (test that this is true for functions such as $f(x,y)=(x^2-y^2)+2ixy$ [you will end up getting just $z^2$–no $\bar{z}$]). Thus, while we shall almost always think about problems in the usual $x-y$-coordinate system it is often convenient, to adopt notation from the $z-\bar{z}$-coordinate system, like defining complex differentiable functions to be those that satisfy $\displaystyle \frac{\partial f}{\partial\bar{z}}=0$.

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I hope this gives an intuition as to a) why complex differentiable functions are expected to be stronger than their real differentiable counterparts (they respect translations, i.e. commute with $i$), b) why the Cauchy-Riemann equations are the analytical interpretation of these statements, and c) why the $\displaystyle \frac{\partial f}{\partial z}$ and $\displaystyle \frac{\partial f}{\partial \bar{z}}$ notations are both intuitive and useful.

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References:

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[1] Greene, Robert Everist, and Steven George Krantz. Function Theory of One Complex Variable. Providence: American Mathematical Society, 2006. Print.

[2] Conway, John B. Functions of One Complex Variable I. New York: Springer-Verlag, 1978. Print.

[3] Rudin, W. Real and Complex Analysis. New York,NY: McGraw-Hill, 1988. Print.

[4]  Ahlfors, Lars V. Complex Analysis; an Introduction to the Theory of Analytic Functions of One Complex Variable. New York: McGraw-Hill, 1966. Print.

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