Problem 7 - Advanced Engineering Mathematics

Question

In Problems 3-8, determine the zeros and their orders for the given function. $$ f(z)=e^{2 z}-e^{z} $$

Step-by-Step Solution

Answer

The function has a zero at \( z = 0 \) with order 1.

Step 1: Identify the Expression for Zeros

To find where the function has zeros, set the function equal to zero:\[ e^{2z} - e^z = 0 \]

Step 2: Factor the Expression

Factor the equation by recognizing that this is a difference:\[ e^z(e^z - 1) = 0 \]

Step 3: Solve for Zeros

Solve the factored equation by setting each factor equal to zero:1. \( e^z = 0 \)2. \( e^z - 1 = 0 \)The first equation \( e^z = 0 \) has no solution since the exponential function never equals zero.For the second equation, \( e^z - 1 = 0 \), solve to get \( e^z = 1 \), which gives \( z = 0 \).

Step 4: Determine the Order of the Zero

To find the order of the zero at \( z = 0 \), consider the factored form:\[ e^z(e^z - 1) \]Note that \( e^z - 1 \) has a simple zero at \( z = 0 \), which indicates the zero is of order 1 at \( z = 0 \).

Key Concepts

Complex analysis

Complex analysis is a fascinating branch of mathematics that studies functions of complex numbers. These functions are not just a simple extension of real functions; they exhibit properties and behaviors unique to the complex plane. The zeros of a complex function are points where the function evaluates to zero. Identifying these zeros helps in understanding the behavior of the function in complex analysis.

Let's take the function from the exercise, \( f(z) = e^{2z} - e^z \). To find its zeros, we set \( f(z) = 0 \), leading to the equation \( e^{2z} = e^z \). This process of finding zeros is crucial since it allows mathematicians to determine where a function does not have an inverse, among other important properties.

Remember, the challenge in complex analysis is not just in solving these equations, but in appreciating the broader implications of these zeros within the context of complex numbers and their geometrical interpretations on the complex plane.

Factorization

Factorization in complex functions involves breaking down an expression into simpler components, called factors, that when multiplied together give the original expression.

In our exercise, the expression \( e^{2z} - e^z \) was factored as \( e^z(e^z - 1) \). Factorization is a vital technique here because it simplifies solving equations by transforming them into products which are easier to analyze. This step reduces the original complex function into mathematical parts that reveal potential solutions when individually set to zero.

When factoring, it’s important to look for common patterns or terms that can be grouped together, as seen in this exercise where the exponential terms allowed us to factor the expression neatly. Being adept at recognizing these patterns can save time and provide deeper insights into the function's properties.

Order of zero

The order of a zero of a complex function refers to the number of times a particular zero is repeated for that function. It tells us how the function behaves as it approaches the zero.

In our example, once the function was factored to \( e^z(e^z - 1) \), the solution revealed a zero at \( z = 0 \). To determine its order, we analyze the factor \( e^z - 1 \), which indicates a simple zero with an order of one at this point.

• A simple zero is one where the corresponding term is not squared or raised to a higher power.
• If a factor were something like \((e^z - 1)^n\), then \( n \) would indicate the order of the zero.

Understanding the order is essential because it determines the function's nature near the zero, affecting graph shape and symmetry in the complex plane.