Introduction
Even though this is a mathematics blog, I’ve been writing only about Texas Hold’em lately, so I felt like I should try writing about mathematics, which is the original purpose of the blog (although in the sense that this is just a hobby page for the author, Texas Hold’em might fit the purpose). In my book, I have stated that original complex functions are functions of two variables, and that complex functions of one variable (holomorphic functions) that are dealt with in general complex analysis and complex function theory are special cases. Let me briefly describe the differentiation of these complex functions (holomorphic functions).
Differentiation of Real Functions
If we accept the validity of limit-taking operations such as the completeness of the space, the derivative of a real function \(\small f(x)\) can be defined as follows:
\[ \small \frac{df(x)}{dx} = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}. \]
For example, if \(\small f(x)=x^2\), then it can be calculated as:
\[ \small \frac{df(x)}{dx} = \lim_{h\rightarrow 0} \frac{2xh+h^2}{h} = 2x + \lim_{h\rightarrow 0} h = 2x. \]
I wrote \(\small h\rightarrow 0\), but this operation can be approached from \(\small h < 0\) or from \(\small h > 0\). The differential coefficient when approaching from a negative value is called the left-hand derivative, and the differential coefficient when approaching from a positive value is called the right-hand derivative. If a function \(\small f(x)\) is differentiable with respect to the coordinate \(\small x\), then the left-hand derivative and the right-hand derivative will be the same. An example of where this does not match is when considering the derivative at \(\small x=0\) for \(\small f(x) = |x|\). When \(\small h_+>0\), it can be calculated as:
\[ \small \frac{df(0)}{dx} = \lim_{h_+\rightarrow 0}\frac{|h_+|-0}{h_+} = 1, \]
while when \(\small h_-<0\), the result is:
\[ \small \frac{df(0)}{dx} = \lim_{h_-\rightarrow 0}\frac{|h_-|-0}{h_-} = -1, \]
and the differential coefficients are different. In the usual sense, this would mean that \(\small f(x) = |x|\) is not differentiable at \(\small x=0\).
Differentiation of Complex Functions
In the case of real functions, we only need to consider two possibilities: approaching from the positive direction or approaching from the negative direction. However, when considering complex functions \(\small f(z),z=x+yi\), we must consider the space in two dimensions (complex plane), so when performing limit operations, there are infinite variations in which direction to approach from. When considering the differentiation of a complex function, we set \(\small \Delta z = \Delta x + \Delta y i\) and consider the limit:
\[ \small h = |\Delta z| = \sqrt{\Delta x^2+\Delta y^2} \rightarrow 0. \]
The combination of \(\small \Delta x, \Delta y\) exists on a sphere of radius \(\small h\). In the differentiation of a complex function of one variable, if the points are in this neighborhood, regardless of the choice of \(\small \Delta x, \Delta y\), and the differential coefficients calculated from any direction are equal, then the function \(\small f(z)\) is said to be differentiable if and only if. If we consider real functions, it is easy to understand that this is a highly restrictive condition. This corresponds to the condition that for one of the two variables, all directional derivatives must be equal in the vicinity of the point where the derivative is calculated (as will be described later, when a complex function is actually expressed as \(\small f(z,\bar{z})\), the condition is \(\small \partial f/\partial \bar{z}=0\)).
Let us find the condition under which the differential coefficient does not change even if we take the limit from any direction. First, define a complex function as:
\[ \small f(z) = u(x,y) + v(x, y)i, \quad z = x + yi. \]
The definition of differentiation is:
\[ \small \frac{df(z)}{dz} = \lim_{\Delta z \rightarrow0} \frac{f(z+\Delta z)-f(z)}{\Delta z}, \]
and the range when performing limit operations is defined as:
\[ \small \Delta z = h (\epsilon +\sqrt{1-\epsilon^2}\:i) = h (\cos \theta+i\sin\theta) = h e^{i\theta}. \]
If we find the conditions under which the derivative can be calculated independently of the values of \(\small \epsilon\) or \(\small \theta\), we will be able to determine the conditions under which differentiation is possible.
For the numerator of the differential calculation, put \(\small \Delta x = h\cos\theta, \Delta y = h\sin\theta\) and calculate to get
\[ \small \begin{align*} f(z+\Delta z)-f(z) &= u(x+\Delta x, y+\Delta y)+v(x+\Delta x, y+\Delta y)i \\ &-(u(x,y) + v(x, y)i). \end{align*} \]
Since \(\small u(x,y),v(x,y)\) are real functions,
\[ \small \begin{align*} &u(x+\Delta x, y+\Delta y) \approx u(x,y)+\frac{\partial u(x,y)}{\partial x}\Delta x+\frac{\partial u(x,y)}{\partial y}\Delta y \\ &v(x+\Delta x, y+\Delta y) \approx v(x,y)+\frac{\partial v(x,y)}{\partial x}\Delta x+\frac{\partial v(x,y)}{\partial y}\Delta y \end{align*} \]
can be obtained by Taylor expansion (assuming that orders higher than second order are negligibly small). By substituting,
\[ \small \begin{align*} f(z+\Delta z)-f(z) &=\frac{\partial u(x,y)}{\partial x}\Delta x+\frac{\partial u(x,y)}{\partial y}\Delta y \\ &+ \left(\frac{\partial v(x,y)}{\partial x}\Delta x+\frac{\partial v(x,y)}{\partial y}\Delta y\right)i \end{align*} \]
can be obtained.
To simplify by \(\small \Delta z\), we rearrange the equation to get
\[ \small \begin{align*} f(z+\Delta z)-f(z) &=\frac{\partial u(x,y)}{\partial x}\Delta x+\frac{\partial v(x,y)}{\partial y}\Delta yi \\ &+ \left(\frac{\partial v(x,y)}{\partial x}\Delta x-\frac{\partial u(x,y)}{\partial y}\Delta yi\right)i. \end{align*} \]
Then, if
\[ \small \frac{\partial u(x,y)}{\partial x} =\frac{\partial v(x,y)}{\partial y},\;\;\frac{\partial v(x,y)}{\partial x}=-\frac{\partial u(x,y)}{\partial y} \]
holds, each term can be combined into \(\small \Delta x + \Delta y i\), which can be expressed as:
\[ \small f(z+\Delta z)-f(z) =\left(\frac{\partial u(x,y)}{\partial x}+\frac{\partial v(x,y)}{\partial x}i\right)(\Delta x + \Delta y i). \]
The parentheses after the right-hand side are \(\small \Delta z\), so we get
\[ \small \frac{df(z)}{dz} = \lim_{\Delta z \rightarrow0} \frac{f(z+\Delta z)-f(z)}{\Delta z}=\frac{\partial u(x,y)}{\partial x}+\frac{\partial v(x,y)}{\partial x}i, \]
which means we can calculate the derivative without depending on the values of \(\small \epsilon\) or \(\small \theta\).
The condition:
\[ \small \frac{\partial u(x,y)}{\partial x} =\frac{\partial v(x,y)}{\partial y},\;\;\frac{\partial v(x,y)}{\partial x}=-\frac{\partial u(x,y)}{\partial y} \]
introduced to ensure that the differential coefficient does not depend on \(\small \theta\) is called the Cauchy-Riemann relation, and satisfying this relation is a necessary and sufficient condition for a complex function to be differentiated. The Cauchy-Riemann relation is often used as a condition for the derivatives to match when approached from the real axis and the imaginary axis, but if this relation is satisfied, the derivatives will also match when approached from any direction in the complex plane.
Conversely, in the case of a non-holomorphic function, note that the value of the derivative will differ depending on the direction from which it is approached. For example, in the case of \(\small f(z)=\bar{z}\), if we write
\[ \small \begin{align*} &\Delta z = h \\ &\Delta z = hi \\ &\Delta z = h\frac{1+i}{\sqrt{2}} \end{align*} \]
for the case where we approach from the real axis, the imaginary axis, or the case where we approach at an angle, then
\[ \small \begin{align*} &\frac{df(z)}{dz} = \lim_{\Delta z \rightarrow0} \frac{f(z+\Delta z)-f(z)}{\Delta z}=\frac{\overline{z+h}-z}{h} = \frac{h}{h} = 1 \\ &\frac{df(z)}{dz} = \lim_{\Delta z \rightarrow0} \frac{f(z+\Delta z)-f(z)}{\Delta z}=\frac{\overline{z+ih}-z}{ih} = \frac{-ih}{ih} = -1 \\ &\frac{df(z)}{dz} = \lim_{\Delta z \rightarrow0} \frac{f(z+\Delta z)-f(z)}{\Delta z}=\frac{\overline{\Delta z}}{\Delta z} = \frac{1-i}{1+i} = -i \end{align*} \]
holds and each case will take a different value. The differentiation of a non-holomorphic function is not impossible in the sense that there are no values, but rather that it is impossible to differentiate because there are an infinite number of values and it cannot be uniquely determined.
Summary
Textbooks on complex function theory describe mysterious properties related to holomorphic functions, but the fundamental cause of the difference between real functions and holomorphic functions is thought to stem from the above. When considering the differentiation of a complex function, if we consider the differentiation as a function of two variables on the real axis and imaginary axis, just like with real functions, we simply transform the variable:
\[ \small f(z,\bar{z}) = f(z(x,y), \bar{z}(x,y)) = F(x,y), \]
so we can calculate the partial derivative as if \(\small z\) and \(\small \bar{z}\) were independent variables. This method of considering the differentiation of complex functions is called Wirtinger differentiation, named after its proponent. A holomorphic function is a function that satisfies
\[ \small \frac{\partial f(z,\bar{z})}{\partial \bar{z}} \equiv 0, \quad \forall z,\bar{z}, \]
and the Cauchy-Riemann relation can also be derived from this formula. It seems that equation:
\[ \small \frac{\partial f(z,\bar{z})}{\partial \bar{z}} = g(z,\bar{z}) \]
is sometimes used when considering the properties of holomorphic and non-holomorphic functions, and this type of equation is said to be a \(\small \bar{\partial}\)-equation (d-bar equation).
As mentioned in the article on the normal distribution of complex numbers:
when considering the probability distribution of complex numbers or random variables, the probability density function is generally not a holomorphic function. For example, the probability density function of the normal distribution of complex numbers is calculated in
\[ \small p(z, \bar{z}) = \frac{1}{2\pi\sigma}\exp \left(-\frac{(z-\mu_z)(\bar{z}-\mu_{\bar{z}})}{2\sigma^2} \right). \]
This function is clearly not a holomorphic function, so in order to analyze such functions, the theory of differentiation and differential equations for non-holomorphic functions would be required. The same is true when dealing with the Schrödinger equation, and the author has written an e-book that presents a hypothesis about what would happen if the Schrödinger equation were considered using Wirtinger differentiation.
Reference
[1] Ahlfors, Lars V. (1979), Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable 3rd edition, McGraw-Hill, Inc.
[2] Hirano, Kaname (2024), Complex Functions and Schrödinger Equation, Amazon Kindle Store.
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