Question

If \(f(z)\) and \(\overline{f(z)}\) are analytic functions, then \(f(z)\) is conutant.

Short answer

\( f(z) \) is constant.

Step-by-step solution

Understanding the Problem

We need to determine if the function \( f(z) \) is constant given that both \( f(z) \) and its complex conjugate \( \overline{f(z)} \) are analytic functions.

Recall Cauchy-Riemann Equations

For a function to be analytic in a region, it must satisfy the Cauchy-Riemann equations. These equations are \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} \), where \( f(z) = u(x, y) + i v(x, y) \).

Analyticity of \( \overline{f(z)} \)

If \( \overline{f(z)} \) is analytic, it also satisfies Cauchy-Riemann equations. Considering \( \overline{f(z)} = u(x, y) - i v(x, y) \), this leads to \( \frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = \frac{\partial v}{\partial x} \).

Compare Equations

By comparing equations from the two sets of conditions: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y} \), as well as \( \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} \) and \( \frac{\partial u}{\partial y} = \frac{\partial v}{\partial x} \), we see these imply \( \frac{\partial u}{\partial x} = 0 \) and \( \frac{\partial v}{\partial y} = 0 \).

Conclusion on Constancy

The conditions \( \frac{\partial u}{\partial x} = 0 \) and \( \frac{\partial v}{\partial y} = 0 \) suggest that \( u \) and \( v \) are constant functions. Thus, \( f(z) = u + iv \) is a constant function.

Key concepts

Cauchy-Riemann equations

The Cauchy-Riemann equations are fundamental to the study of analytic functions in complex variables. These equations are necessary to determine whether a complex function is differentiable at a point. Specifically, a function \( f(z) = u(x, y) + iv(x, y) \) is considered analytic if it satisfies these equations:

• \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \)
• \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

Here, \( u \) and \( v \) are real-valued functions of the real variables \( x \) and \( y \). This pair of partial differential equations ensures that the real and imaginary parts are intertwined appropriately, which is necessary for the complex function to be smooth and differentiable in a region. If a function satisfies the Cauchy-Riemann equations in a region, it's an indication of its holomorphic nature in that area.

complex conjugate

In the realm of complex functions, the complex conjugate is a crucial concept that deals with pairing the imaginary component of a complex number with its negative. For a given complex number \( z = x + yi \), its complex conjugate is \( \overline{z} = x - yi \).
Applying this to functions, for \(f(z) = u(x, y) + iv(x, y) \), the complex conjugate \( \overline{f(z)} \) becomes \( u(x, y) - iv(x, y) \).
The behavior of the complex conjugate is particularly interesting in the context of analyticity. If both \( f(z) \) and \( \overline{f(z)} \) are analytic, the Cauchy-Riemann equations lead to a surprising result: the function must be constant, as we later observe.
It is this property of the complex conjugate that heavily influences the characteristics of the original complex function \( f(z) \).

constant function

When we talk about constant functions, we refer to functions that take the same value regardless of the input \( z \). Mathematically, this means \( f(z) = c \) for any \( z \), where \( c \) is a constant.
In the context of complex variables, if a complex function like \( f(z) \) is not influenced by any changes in its input values, it's considered constant.
Returning to our original problem, when both \( f(z) \) and its complex conjugate are analytic, the Cauchy-Riemann equations come into play to make an important deduction. Both sets of equations (for \( f(z) \) and \( \overline{f(z)} \)) enforce conditions such that any changes in the input do not affect the function's value, thereby confirming that \( f(z) \) is indeed a constant function.
This results from the necessity to fulfill two different, yet similar, conditions for derivability as prescribed by the Cauchy-Riemann equations.

function of a complex variable

A function of a complex variable is one of the foundational blocks of complex analysis, defined with inputs from the complex number system. These functions, such as \( f(z) \), where \( z = x + yi \), map complex numbers to complex outputs.
These functions often expressed in terms of their real (\( u(x, y) \)) and imaginary (\( v(x, y) \)) parts, allow us to explore deeper mathematical properties and applications. They're essential for fields ranging from electrical engineering to theoretical physics.
One of the key aspects of these functions is their differentiability, which is where the Cauchy-Riemann equations become crucial. Differentiability with respect to complex variables requires a more nuanced understanding than in the real-analytic case.
In cases where a function and its complex conjugate are both analytic, further exploration of their properties can lead to significant conclusions about their behavior and forms. Understanding such functions gives insight into purely theoretical concerns as well as complex physical phenomena.