Problem 20 - Complex Analysis

Question

Let \(f=u+\mathrm{i} v: D \rightarrow \mathbb{C}\) be a totally differentiable function (in the sense of real analysis), defined on an open region \(D \subset \mathbb{C} .\) Define the operators $$ \begin{aligned} &\frac{\partial f}{\partial z}:=\frac{1}{2}\left(\frac{\partial f}{\partial x}-\mathrm{i} \frac{\partial f}{\partial y}\right) \\ &\frac{\partial f}{\partial \bar{z}}:=\frac{1}{2}\left(\frac{\partial f}{\partial x}+\mathrm{i} \frac{\partial f}{\partial y}\right) \end{aligned} $$ Show: \(f\) is analytic if and only if \(\frac{\partial f}{\partial \bar{z}}=0\), and when this is the case one has \(f^{\prime}=\frac{\partial f}{\partial z}\) Remark. These differential operators \(\partial:=\frac{\partial}{\partial z}\) and \(\bar{\partial}:=\frac{\partial}{\partial z}\) were originally introduced by H. POINCARÉ (1899). A systematic calculus was developed by W. WirtingER (1927) - the so-called Wirtinger calculus. However, it is not very important in the one-variable function theory; its full significance lies in manyvariable function theory, for which it was originally developed by WIrTINGER.

Step-by-Step Solution

Answer

The function \(f\) is analytic if and only if \(\frac{\partial f}{\partial \bar{z}} = 0\), and when it is analytic, \(f' = \frac{\partial f}{\partial z}\).

Step 1: Understanding the Operators

Begin by detailing the definitions of the operators: \(\frac{\partial f}{\partial z}\) and \(\frac{\partial f}{\partial \bar{z}}\). These operators are expressed in terms of partial derivatives with respect to the real components \(x\) and \(y\). The function \(f\) is given as \(f = u + iv\), where \(u\) and \(v\) are real-valued functions.

Step 2: Interpreting Analytic Functions

Recall that a function is analytic if the Cauchy-Riemann equations \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\) are satisfied throughout \(D\). This is equivalent to the function having a complex derivative.

Step 3: Applying Analytic Condition to \(\frac{\partial f}{\partial \bar{z}}\)

Utilize the definitions of the operators: \(\frac{\partial f}{\partial \bar{z}} = \frac{1}{2}\left(\frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y}\right) = \frac{1}{2}\left(\frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} + i\frac{\partial u}{\partial y} - \frac{\partial v}{\partial y}\right)\). Set this expression to zero to verify analyticity.

Step 4: Confirming Analytic Condition

If \(\frac{\partial f}{\partial \bar{z}} = 0\), then \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\), which are exactly the Cauchy-Riemann equations, confirming that \(f\) is analytic.

Step 5: Relating to Complex Derivative

Given that \(f\) is analytic, show that \(f' = \frac{\partial f}{\partial z}\). Use the definition \(\frac{\partial f}{\partial z} = \frac{1}{2}\left(\frac{\partial f}{\partial x} - i \frac{\partial f}{\partial y}\right) = \frac{1}{2}\left(\frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} - i\frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}\right)\). With the Cauchy-Riemann equations in place, this simplifies to the derivative of \(f\).

Key Concepts

Analytic Functions

In complex analysis, an **analytic function** is foundational. A function is considered analytic on a domain if it fulfills two key criteria: it has a complex derivative at every point in its domain, and it conforms to the Cauchy-Riemann equations. These equations provide a bridge between the real and imaginary parts of a complex function. A complex function \(f = u + iv\) (where \(u\) and \(v\) are functions of \(x\) and \(y\)) is analytic if it is differentiable, meaning the limit defining the derivative is not just finite, but exists in every direction in the complex plane. To distill this further, an analytic function smoothly varies without abrupt changes or discontinuities in its behavior. Analyticity is crucial because it allows complex functions to be uniquely represented as power series, stretching the power of simple polynomial expressions across infinite terms.

Cauchy-Riemann Equations

The **Cauchy-Riemann equations** serve as a criterion for a function to be analytic. The function \(f = u + iv\) defined on a domain \(D\) deeply ties its analyticity to these conditions: \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\). In essence, these equations ensure the continuity and smooth transition between the real and imaginary components of the complex function. The equations originate from the need to express complex differentiation in terms of real partial derivatives. For a function to satisfy these equations everywhere within an open domain denotes that it can locally be approached using calculus, similar to how real functions work, allowing for complex integration and series expanses. Through the Cauchy-Riemann equations, we see that achieving zero for \(\frac{\partial f}{\partial \bar{z}}\) implies the complete adherence to derivative conditions, ratifying analyticity.

Wirtinger Calculus

The **Wirtinger calculus** is a stylish and skillful tool in complex analysis. It simplifies how derivatives with respect to complex variables are handled by using the operators \(\frac{\partial}{\partial z}\) and \(\frac{\partial}{\partial \bar{z}}\). Originally developed by Paul Poincaré and extended by Wilhelm Wirtinger, this calculus helps rewrite the classic equations into more manageable forms, especially in multi-variable complex analysis. In the context of single-variable problems, as seen here, it's used to calculate and establish conditions for zeros in \(\frac{\partial f}{\partial \bar{z}}\), thus identifying analytic behavior. This method offers a streamlined approach, focusing on complex rather than just real differentiation and providing insights into multi-dimensional complex spaces. Although not as crucial for basic one-variable functions, it lays groundwork for understanding the intricate dance of functions in high dimensions.