Problem 8 - Complex Analysis for Mathematics and Engineering

Question

Suppose that \(f(z)=\sum_{n=0} c_{n} z^{n}\) is an entire function. (a) Find a series representation for \(\overline{f(\bar{z})}\), using powers of \(\mathrm{z}\). (b) Show that \(\overline{f(z)}\) is an entire function. (c) Does \(\overline{f(z)}=f(z) ?\) Why or why not?

Step-by-Step Solution

Answer

(a) \(\overline{f(\bar{z})} = \sum_{n=0} \overline{c_n} z^n\). (b) Yes, it's entire. (c) No, generally \(c_n\) are not real.

Step 1: Write the Expression for \(f(z)\)

The function \(f(z)\) is given by the power series sum, \(f(z) = \sum_{n=0}^{} c_n z^n\). This is the foundation for translating and understanding its complex conjugate and other operations.

Step 2: Express \(\overline{f(\bar{z})}\)

To find the series representation of \(\overline{f(\bar{z})}\), substitute \(\bar{z}\) into the series: \(f(\bar{z}) = \sum_{n=0} c_n (\bar{z})^n\). The complex conjugate of this function is: \(\overline{f(\bar{z})} = \sum_{n=0} \overline{c_n} z^n\).

Step 3: Verify the Function is Entire

An entire function is analytic everywhere in the complex plane. Since \(\overline{f(\bar{z})} = \sum_{n=0} \overline{c_n} z^n\) is represented as a power series with respect to \(z\) and power series are analytic wherever they converge, \(\overline{f(\bar{z})}\) is entire because it is a power series.

Step 4: Explore Equality \(\overline{f(z)} = f(z)\)

Checking \(\overline{f(z)} = f(z)\), we have \(\overline{f(z)} = \overline{\sum_{n=0} c_n z^n} = \sum_{n=0} \overline{c_n} \bar{z}^n\). The functions \(f(z)\) and \(\overline{f(z)}\) are equal if and only if each \(c_n = \overline{c_n}\) and the coefficients are real, which is not generally true for complex coefficients.

Key Concepts

Entire Function

An entire function is a complex-valued function that is complex differentiable at every point in the complex plane. This differentiability means that entire functions are also analytic everywhere on the complex plane. A well-known example of entire functions is polynomial functions, exponential functions, and entire power series like the power series given for the function \( f(z) = \sum_{n=0}^{\infty} c_n z^n \).

Entire functions possess special properties:

• They are continuous and differentiable throughout the entire complex plane, which implies both an infinite radius of convergence for their series representation and existence of Taylor series expansion everywhere.
• They follow Liouville's Theorem, which states that if an entire function is bounded, then it must be constant.
• Entire functions are crucial in complex analysis due to their role in understanding the behavior of analytic functions within different areas of the complex plane.

In the given problem, \( \overline{f(\bar{z})} = \sum_{n=0}^{\infty} \overline{c_n} z^n \) is also entire as it forms a power series which converges everywhere.

Power Series

A power series is an infinite series of the form \( \sum_{n=0}^{\infty} a_n z^n \), where \( a_n \) are coefficients and \( z \) is the variable. In complex analysis, power series play an essential role as they represent analytic functions within their radius of convergence.

Key aspects of power series include:

• Convergence: A power series converges within a certain distance from a center point, called the radius of convergence.
• Analyticity: Power series are representations of analytic functions, which means they can be differentiated term by term within their radius of convergence.
• Entire Function Example: When the radius of convergence is infinite, the power series defines an entire function, as it does in the given exercise for \( f(z) = \sum_{n=0}^{\infty} c_n z^n \).

In this exercise, the power series in question demonstrates how alterations of the variable, such as using \( \bar{z} \), affect the series but do not change its convergence in the sense of entire functions. This is also why \( \overline{f(\bar{z})} \) remains an entire function.

Complex Conjugate

The complex conjugate of a complex number \( z = x + yi \) is defined as \( \bar{z} = x - yi \). In essence, it flips the sign of the imaginary part while keeping the real part unchanged.

Properties and uses of complex conjugates include:

• They help simplify expressions involving complex numbers by neutralizing the imaginary part.
• In the context of functions, \( \overline{f(z)} = \overline{\sum_{n=0}^{\infty} c_n z^n} = \sum_{n=0}^{\infty} \overline{c_n} \bar{z}^n \) provides a symmetric version of the original function with conjugated coefficients.
• The complex conjugate is used to determine the equality and realness of functions in complex analysis.

Through this exercise, we find that \( \overline{f(z)} = f(z) \) holds if and only if the coefficients \( c_n \) are real numbers, ensuring that \( c_n = \overline{c_n} \). This reflects the unique ability of complex conjugation to establish properties of symmetry in complex functions.