Question

The zeroes and singularities of \(\frac{z^{2}+1}{1-z^{2}}\) ure

Short answer

Zeroes: \( i, -i \). Singularities: \( 1, -1 \).

Step-by-step solution

Define Zeroes and Singularities

In complex analysis, zeroes of a function are points where the function equals zero. A singularity is where the function does not have a well-defined value, often where the denominator equals zero.

Find the Zeroes

The zeroes occur where the numerator is zero. For the function \( \frac{z^2 + 1}{1 - z^2} \), we set the numerator \( z^2 + 1 = 0 \). Solving this, \( z^2 = -1 \), results in \( z = i \) and \( z = -i \). So, the zeroes are \( i \) and \( -i \).

Identify Singularities

Singularities occur where the denominator is zero. For \( \frac{z^2 + 1}{1 - z^2} \), this happens when \( 1 - z^2 = 0 \). Solve \( z^2 = 1 \), producing \( z = 1 \) and \( z = -1 \). Consequently, these values are the singularities of the function.

Verification

Consider points \( i \) and \( -i \) for zeroes and substitute into the original function to confirm the numerator is zero and the denominator is non-zero. Similarly, substitute \( 1 \) and \( -1 \) into the denominator and ensure it results in zero, confirming singularities.

Key concepts

Zeroes of Complex Functions

Complex functions often have zeroes, which are points where the function's value ends up being zero. An easier way to understand this is to think of zeroes like solving a familiar algebraic equation where you find the x-values that make the equation equal to zero. In our case, the complex function is given as \( \frac{z^2 + 1}{1 - z^2} \). To find these zeroes, we set the numerator \( z^2 + 1 \) to zero and solve for \( z \). This equation simplifies to \( z^2 = -1 \), whose solutions are \( z = i \) and \( z = -i \). These are your zeroes, often called roots in regular algebra. These zeroes are the points on the complex plane where the complex function touches the real axis.

Singularities in Complex Functions

Singularities are like stumbling blocks for complex functions. These are points where the function does not have a well-defined value, often because you end up dividing by zero. With our function \( \frac{z^2 + 1}{1 - z^2} \), singularities happen where the denominator \( 1 - z^2 \) equals zero. When you solve \( 1 - z^2 = 0 \), you find \( z^2 = 1 \), leading to \( z = 1 \) and \( z = -1 \). These are your singular points. The function really misbehaves at these points, meaning it can shoot off to infinity or be undefined in various ways. Recognizing singularities is important for understanding the behavior and limitations of complex functions.

Complex Function Analysis

Analyzing complex functions is like peering into a deeper world of numbers, where imaginary and real parts coexist. When working with functions like \( \frac{z^2 + 1}{1 - z^2} \), you'll explore where it zeros out and where it misbehaves, which provides a fuller understanding of its landscape.

• Zeroes: Check where the numerator equals zero for roots.
• Singularities: Focus on the denominator equaling zero to locate singularities.

These analyses allow you to sketch behavior and dynamics in complex systems, predicting where problems might arise due to singularities or what values the function might achieve at particular zeroes. This scrutiny is key in not just mathematics but also physics and engineering, where predicting and controlling systems depends heavily on such functions.