Problem 14 - A First Course in Complex Analysis with Applications

Question

The indicated number is a zero of the given function. Use a Maclaurin or Taylor series to determine the order of the zero. $$ f(z)=1-\pi i+z+e^{z} ; z=\pi i $$

Step-by-Step Solution

Answer

The order of the zero at \( z = \pi i \) is 2.

Step 1: Understanding the Function and the Zero

The given function is \( f(z) = 1 - \pi i + z + e^{z} \). The indicated zero is \( z = \pi i \). This means that \( f(\pi i) = 0 \). Our goal is to expand the function using a series expansion and determine the order of the zero at \( z = \pi i \).

Step 2: Find the Taylor Series Expansion

The Taylor series expansion of a function \( f(z) \) about a point \( a \) is given by \( f(z) = f(a) + f'(a)(z-a) + \frac{f''(a)}{2!}(z-a)^2 + \cdots \). Here, we need the expansion about \( z = \pi i \). First, evaluate the function and its derivatives at \( z = \pi i \).

Step 3: Evaluate Function and Derivatives at \( z = \pi i \)

Calculate \( f(\pi i) = 1 - \pi i + \pi i + e^{\pi i} = 1 + (-1) = 0 \). Now find the first few derivatives.- First derivative: \( f'(z) = 1 + e^{z} \), and \( f'(\pi i) = 1 + e^{\pi i} = 1 - 1 = 0 \).- Second derivative: \( f''(z) = e^{z} \), and \( f''(\pi i) = e^{\pi i} = -1 \).

Step 4: Analyzing the Orders of Zero

The Taylor series around \( z = \pi i \) becomes \( f(z) = 0 + 0(z - \pi i) + \frac{f''(\pi i)}{2!}(z - \pi i)^2 + \cdots \). Since the zeroth and first derivative terms are zero but the second derivative term is non-zero, the first non-vanishing term is of order 2. This indicates that \( z = \pi i \) is a zero of order 2.

Key Concepts

Zero of a Function

A zero of a function is a point where the function evaluates to zero. Imagine it as a spot on the graph of the function where it just touches or crosses the horizontal axis (the axis representing zero output). For complex functions, which have complex numbers as inputs and outputs, finding zeros can be more abstract.
In simpler terms, if you plug a specific value into the function and the result is zero, you've found a zero of the function. In the given problem, the zero is at the point \( z = \pi i \), which means substituting this value into the function \( f(z) = 1 - \pi i + z + e^{z} \) results in a zero output.
Understanding the zeros of a function is crucial because they often represent fundamental properties or behavior of the function. Zeros can help identify factors, symmetry, and other analytical features important in both real and complex function studies.

Complex Analysis

Complex Analysis is a fascinating field of mathematics dealing with complex numbers and their functions. A complex number includes a real part and an imaginary part and is expressed as \( a + bi \) where \( i \) is the imaginary unit \( \sqrt{-1} \). This field extends the idea of functions of real numbers to the complex plane, offering richer and more intricate equations and relationships.
One key tool in complex analysis is the Taylor series, which lets you approximate functions using polynomials. This becomes essential when studying functions like \( f(z) = 1 - \pi i + z + e^{z} \) near certain points on the complex plane. Additionally, complex analysis helps in exploring singularities, zeros, and other properties of complex functions. In our exercise, it allows us to express the function and its behavior near the zero \( z = \pi i \) conveniently, so we can analyze and understand its features more deeply.
Working with complex numbers and functions often involves evaluating derivatives and considering convergence, which can deepen the understanding of mathematical phenomena across various disciplines.

Order of Zero

The order of a zero is a concept that tells us about how a function behaves near its zeros. Specifically, it tells us how the function approaches zero. In simple terms, the order of a zero is the smallest positive integer \( n \) for which the \( n \)-th derivative at that zero is non-zero, while all preceding derivatives are zero.
In our example, for the function \( f(z) = 1 - \pi i + z + e^{z} \) at the zero \( z = \pi i \):

• The function value itself \( f(\pi i) \) is 0, which shows it is indeed a zero.
• The first derivative also equals 0, \( f'(\pi i) = 0 \).
• The second derivative is the first non-zero derivative, \( f''(\pi i) = -1 \).

Thus, the order of the zero is 2. This means the function and its derivative go to zero faster as they approach \( z = \pi i \), but not the second derivative. Understanding the order of a zero can help us better analyze the function's behavior in its vicinity. It also can assist in solving equations and understanding how the solutions relate to the behavior of the function graph.