Problem 12 - 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. $$ f(z)=z-\sin z ; z=0 $$

Step-by-Step Solution

Answer

The zero of \( f(z) \) at \( z=0 \) is of order 3.

Step 1: Determine Maclaurin Series for f(z)

The Maclaurin series for a function is given by expanding the function around zero. The function given is \( f(z) = z - \sin z \). We know the Maclaurin series for \( \sin z \) is \( \sin z = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \ldots \). Therefore, the Maclaurin series for \( f(z) \) is \( f(z) = z - (z - \frac{z^3}{3!} + \frac{z^5}{5!} - \ldots) \).

Step 2: Simplify the Series Expression

Simplifying the series, we find that the \( z \) terms cancel out: \( f(z) = z - z + \frac{z^3}{3!} - \frac{z^5}{5!} + \ldots \) which simplifies to \( f(z) = \frac{z^3}{3!} - \frac{z^5}{5!} + \ldots \).

Step 3: Identify the Order of the Zero

The order of a zero is determined by the first non-zero term in the series expansion of the function. From the series \( f(z) = \frac{z^3}{3!} - \frac{z^5}{5!} + \ldots \), the first non-zero term is \( \frac{z^3}{3!} \). This indicates that \( z = 0 \) is a zero of order 3, as the first non-zero derivative at \( z=0 \) is of degree 3.

Key Concepts

Maclaurin Series

The Maclaurin series is a special case of the Taylor series, used for approximating functions around the point zero. It's derived from the Taylor series by setting the center of the expansion at 0, thereby simplifying the process. To construct a Maclaurin series, you repeatedly take derivatives of the function and evaluate them at zero.

For instance, the Maclaurin series of a function \( f(z) \) is represented as:

• \( f(0) + \frac{f'(0)}{1!}z + \frac{f''(0)}{2!}z^2 + \frac{f'''(0)}{3!}z^3 + \ldots \)

This approach gives you an infinite series that approximates \( f(z) \) around \( z = 0 \). This is handy for functions where direct computation might be complex.

In providing an approximation for functions like \( \sin z \), the series becomes rather simple:

• \( \sin(z) = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \ldots \)

Thus, when constructing the Maclaurin series for \( f(z) = z - \sin z \), you're essentially subtracting the series of \( \sin z \) from \( z \), effectively canceling out the linear terms.

Taylor Series

The Taylor series is a powerful tool in complex analysis, giving us a way to represent functions as an infinite sum of terms computed from the values of their derivatives at a single point. While this series can be centered at any point \( a \), when \( a = 0 \) it specifically becomes the Maclaurin series.

The general formula for a Taylor series expansion of a function \( f \) around the point \( a \) is:

• \( f(a) + \frac{f'(a)}{1!}(z-a) + \frac{f''(a)}{2!}(z-a)^2 + \ldots \)

This allows us to model changes around the reference point \( z = a \) using the polynomial form. Taylor series are fundamental because they can simplify the calculation of complex functions, making them easier to analyze and integrate.

This series is particularly helpful in determining behavior near certain points. For example, the exercise demonstrates how using a Taylor or Maclaurin series reveals the order of a zero by highlighting the first non-zero term in its expansion.

Order of Zero

In mathematics, the term 'order of zero' refers to the smallest power of \( z \) that appears in a function’s expansion, which is not zero. This provides insight into how many times a function 'touches' or 'crosses' the zero axes at a particular point.

To determine the order of zero for a function like \( f(z) = z - \sin z \), you must analyze its series expansion around the point \( z = 0 \). As seen in the previous steps, if the series expansion begins with \( \frac{z^3}{3!} \), it means the function has a zero of order 3 at that point.

The significance of the order of zero can be seen in:

• Identifying critical points where graph behavior changes.
• Solving equations where such zeros could dramatically alter outcomes.
• Determining multiplicity in solutions, which can have implications in stability analysis in engineering and physics.

Hence, knowing the order of zero plays a critical role in understanding the dynamic behavior of complex functions.