Problem 35 - Advanced Engineering Mathematics

Question

In Problems 35-38, give the points at which the given function will not be analytic. $$ f(z)=\frac{z}{z-3 i} $$

Step-by-Step Solution

Answer

The function is not analytic at \( z = 3i \).

Step 1: Identify the function form

The function given is \( f(z) = \frac{z}{z - 3i} \). This is a rational function where the numerator is \( z \) and the denominator is \( z - 3i \). We need to identify where this function may not be analytic.

Step 2: Determine where denominator is zero

A function is non-analytic where its denominator equals zero because division by zero is undefined. Solve for \( z \) in the equation \( z - 3i = 0 \).

Step 3: Solve for z

Add \( 3i \) to both sides of the equation \( z - 3i = 0 \) to get \( z = 3i \). This is the point at which the function is not analytic.

Step 4: Conclusion

The function \( f(z) = \frac{z}{z - 3i} \) is not analytic only at \( z = 3i \), because this is where the denominator becomes zero, and division by zero is not defined.

Key Concepts

Analytic Functions

Analytic functions are a fascinating type of function in complex analysis. These functions are essential because they are differentiable at every point in their domain. In simple terms, if you can take a derivative of a function across a region and it doesn't encounter any issues, it's considered analytic in that region.

• Being differentiated once means it can be differentiated infinitely.
• They have power series representations.
• Analytic functions are smooth and continuous.

These characteristics make analytic functions predictable and well-behaved, crucial for solving complex problems. They provide stability and insight in a vast array of applications, from engineering to physics.
When analyzing a function, knowing where it is not analytic, like the one specified in the exercise, can help identify potential problems or special features.

Rational Functions

Rational functions are ratios of polynomial expressions, much like fractions are ratios of numbers. In mathematics, they take the form \( f(z) = \frac{P(z)}{Q(z)} \),where both \( P(z) \) and \( Q(z) \) are polynomials.

• The numerator determines the general behavior of the function.
• The denominator dictates where the function is undefined.
• Identifying the zeros of the denominator is key.

In the given exercise, our rational function is \( f(z) = \frac{z}{z - 3i} \). Here, the polynomial in the numerator is simply \( z \), while the denominator is \( z-3i \).
The presence of a denominator makes these functions potentially undefined when it equals zero. Understanding where the denominator equals zero allows students to pinpoint where the function has issues, like becoming non-analytic due to singularities.

Singularity

A singularity is a point where a function stops behaving 'nicely'. For analytic functions, these are often places where the function fails to be analytic, usually where the function might explode to infinity or behave unpredictably.
In the realm of rational functions, singularities are typically found where the denominator is zero, as the function becomes undefined at these points.

• Poles are a typical type of singularity for rational functions.
• Pole of order 1 means a single degree mismatch between factors.
• Identifying and classifying singularities is crucial for complex analysis.

The exercise identifies a singularity at \( z = 3i \), where the function is not analytic because the denominator is zero, leading to a division by zero issue. Recognizing these points helps in understanding the deeper behavior of complex functions.

Division by Zero

Division by zero is a fundamental concept in mathematics that often signals a point of danger or special consideration. When dealing with rational functions, division by zero occurs if the value that makes the denominator zero is plugged into the function. This situation results in an undefined expression.

• It disrupts the continuity and differentiability of functions.
• Illegal operations in mathematics often hinge on this concept.
• The presence of division by zero points to a singularity.

In the exercise at hand, division by zero occurs at \( z = 3i \), which is why the function \( f(z) = \frac{z}{z - 3i} \) is not analytic there. Understanding this helps in navigating the landscape of complex analysis, as these points are where more nuanced mathematical techniques are required to parse the function's behavior.