Question

Show that (a) the sum and the product of two entire functions are entire, and (b) the ratio of two entire functions is analytic everywhere except at the zeros of the denominator.

Short answer

The sum, product, and ratio (provided the denominator is non-zero) of two entire functions are also entire functions. The ratio of two entire functions is analytic everywhere except at the zeros of the denominator where the function is not defined.

Step-by-step solution

Proving the Sum of Entire Functions is Entire

Let's consider two entire functions, \(f(z)\) and \(g(z)\). The sum of these functions is \(h(z) = f(z) + g(z)\). Now by the rules of differentiation, \(h'(z) = f'(z) + g'(z)\), indicates that the derivative of the sum exists. Therefore, the sum of two entire functions is also an entire function.

Proving the Product of Entire Functions is Entire

Now, let's address the second part of the task: proving that the product of two entire functions is entire. Say we have the same functions \(f(z)\) and \(g(z)\), and we form a function \(h(z)\) from their product: \(h(z) = f(z)g(z)\). Using the properties of derivatives, we get \(h'(z) = f'(z)g(z) + f(z)g'(z)\). This demonstrates that the derivative of the product also exists, and thus the product of two entire functions is also entire.

Proving the Ratio of Entire Functions is Analytic

Let us define \(h(z) = f(z) / g(z)\), where \(f(z)\) and \(g(z)\) are entire functions and \(g(z) ≠ 0\). Now, the derivative of \(h(z)\) is \(h'(z) = (f'(z)g(z) - f(z)g'(z)) / (g(z))^2\) and since all terms in the equation for \(h'(z)\) exist, \(h(z)\) is differentiable, except where \(g(z) = 0\). Therefore, the ratio \(f(z) / g(z)\) is analytic everywhere except at zeros of the denominator, where \(g(z) = 0\).

Key concepts

Analytic Functions

Analytic functions are a pivotal concept in complex analysis, a field of mathematics focused on functions of complex numbers. These functions are characterized by their ability to be expressed as a power series about any point within their domain. This implies that analytic functions are infinitely differentiable, a property that makes them incredibly smooth and predictable.
To be analytic, a function must be complex differentiable, which is a stronger condition than real differentiability because the derivative must be the same regardless of the path chosen in the complex plane. This nuanced requirement leads to fascinating behaviors and properties that set analytic functions apart from their real counterparts.

Differentiability

In the context of complex analysis, differentiability refers to a function's ability to have a derivative at every point within its domain. Understanding differentiability in complex analysis requires a shift from the real numbers to the intricate world of complex functions.
A complex function is said to be differentiable at a point if the limit \( \lim_{{\Delta z \to 0}} \frac{{f(z + \Delta z) - f(z)}}{{\Delta z}} \) exists and is the same regardless of how \( \Delta z \) approaches zero in the complex plane. This condition is more stringent than in real analysis because it must hold for any direction in the complex plane. As a result, once a complex function is differentiable in an open set, it's automatically infinitely differentiable and analytic, revealing the beautiful interconnectedness of these properties.

Complex Analysis

Complex analysis is an expansive branch of mathematics that explores functions of complex variables. It merges algebraic and geometric methods to analyze these functions, which are described by complex numbers of the form \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit.
This field is instrumental in various applications, from engineering to physics and beyond, due to its ability to provide a deeper understanding of functions that are not only differentiable but also inherently predictable. Key concepts in complex analysis include analytic functions, contour integrals, and residues. These concepts play an essential role in solving complex equations and understanding phenomena such as wave propagation and electrical currents.

Zeros of Denominator

When dealing with the ratio of entire functions, a crucial aspect to consider is the zeros of the denominator. If the denominator of a function ratio is zero at a certain point, it makes the function undefined at that point. However, outside these specific points, the function can still maintain analyticity.
This idea is particularly significant in complex analysis when examining the limits and behavior of functions. If \( h(z) = \frac{f(z)}{g(z)} \), where both \( f(z) \) and \( g(z) \) are entire, the ratio is analytic everywhere except at points where \( g(z) = 0 \). These points are termed 'poles,' and understanding their influences is key when studying complex functions. When exploring such functions, it's common to use techniques like Laurent series expansion to understand behavior near these poles, providing deeper insights into their contributions to the overall function.