Problem 38 - Advanced Engineering Mathematics

Question

Give the points at which the given function will not be analytic. $$ f(z)=\frac{z-4+3 i}{z^{2}-6 z+25} $$

Step-by-Step Solution

Answer

The function is not analytic at \( z = 3 + 4i \) and \( z = 3 - 4i \).

Step 1: Identify Points of Non-Analyticity

For the function \( f(z) = \frac{z-4+3i}{z^2-6z+25} \), the points where the function is not analytic are the poles, which occur where the denominator equals zero. Thus, we need to solve the equation: \[ z^2 - 6z + 25 = 0 \] to find these points.

Step 2: Solve the Quadratic Equation

Use the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -6 \), and \( c = 25 \). This gives: \[ z = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(25)}}{2(1)} = \frac{6 \pm \sqrt{36 - 100}}{2} = \frac{6 \pm \sqrt{-64}}{2}. \] Simplifying further, we find \( \sqrt{-64} = 8i \), thus: \[ z = \frac{6 \pm 8i}{2}. \]

Step 3: Find Points of Non-Analyticity

The solutions from the quadratic formula are \( z = 3 + 4i \) and \( z = 3 - 4i \). These are the points where the function \( f(z) \) is not analytic, as the denominator is zero at these points.

Key Concepts

Analytic Functions

In complex analysis, an analytic function is a function that is locally representable by a convergent power series.
This means in some region around every point within the function's domain, the function can be expressed as a series of powers of the variable. Analytic functions are also called holomorphic functions. They exhibit smoothness and are differentiable at every point in their domain.
Here are some key attributes of analytic functions:

• They are infinitely differentiable and conform to the Cauchy-Riemann equations.
• If a function is analytic over its entire domain, it is termed entire. For example, exponential functions like \( e^z \), sine \( \sin(z) \), and cosine \( \cos(z) \) are entire functions.
• Analytic functions demonstrate the property of infinite differentiability, making them predictable and smooth throughout their region of analyticity.

Being non-analytic at certain points, referred to as singularities, does not preclude a function from being analytic elsewhere. Understanding such points is crucial in various applications of complex analysis.

Poles of a Function

In complex analysis, poles are a type of singularity where the function takes on an infinite value.
Specifically, if a function \( f(z) \) has a pole at point \( z_0 \), then the function can be expressed as \( f(z) = \frac{g(z)}{(z - z_0)^n} \), where \( g(z) \) is analytic and non-zero at \( z_0 \), and \( n \) is a positive integer.
Key things about poles include:

• They represent points where the function is not analytic.
• The order of a pole helps define how the function behaves near that point. A pole of order 1 is known as a simple pole.
• Pole locations are essential for evaluating complex integrals, particularly in using the residue theorem.

In the provided exercise, the poles are found by identifying where the function's denominator equals zero, showing the points of non-analyticity. Understanding the behavior near poles is pivotal for complex function analysis and helps in computations involving limits and integrals.

Quadratic Equations

A quadratic equation takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable or unknown. Solving quadratic equations can be done using a variety of methods, including:

• Factoring: Expressing the quadratic as a product of two binomials can sometimes allow for straightforward solutions.
• Completing the square: This method involves forming a perfect square trinomial, making it simpler to solve.
• Quadratic Formula: For \( ax^2 + bx + c = 0 \), the values of \( x \) are given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is universal and works in all cases.

The quadratic formula is particularly useful when the equation does not factor neatly or when it results in complex numbers, as seen in the original exercise.
Complex solutions arise when the discriminant \( b^2 - 4ac \) is negative, necessitating the use of imaginary numbers. Understanding these methods ensures robust problem-solving skills, crucial for tackling quadratic equations across various fields in mathematics.