Problem 29 - Advanced Engineering Mathematics

Question

Let \(f(z)=z^{n} g(z)\), where \(n\) is a positive integer, \(g(z)\) is entire, and \(g(z) \neq 0\) for all \(z\). Let \(C\) be a circle with center at the origin. Evaluate \(\oint_{C} \frac{f^{\prime}(z)}{f(z)} d z\).

Step-by-Step Solution

Answer

The integral evaluates to \(2\pi ni\).

Step 1: Identify the Derivative

We begin by finding the derivative of the function \(f(z) = z^n g(z)\). The product rule gives us \( f'(z) = (z^n)'g(z) + z^n g'(z) = nz^{n-1}g(z) + z^n g'(z)\).

Step 2: Simplify the Function

The integral we need to evaluate is \( \oint_{C} \frac{f'(z)}{f(z)} \, dz\). Therefore, \(\frac{f'(z)}{f(z)} = \frac{nz^{n-1}g(z) + z^n g'(z)}{z^n g(z)} = \frac{nz^{n-1}}{z^n} + \frac{z^n g'(z)}{z^n g(z)}\). This simplifies to \( \frac{n}{z} + \frac{g'(z)}{g(z)} \).

Step 3: Use the Cauchy Integral Theorem

Now evaluate \( \oint_{C} \left( \frac{n}{z} + \frac{g'(z)}{g(z)} \right) \, dz\). The integral \(\oint_{C} \frac{n}{z} \, dz\) evaluates to \(2\pi ni\) by the residue theorem, since \(\frac{n}{z}\) has a pole of order 1 at the origin with residue \(n\).

Step 4: Evaluate the Remaining Portion

For the second integral, \(\oint_{C} \frac{g'(z)}{g(z)} \, dz\), since \(g(z)\) is entire and nowhere zero, the integral over a closed contour \(C\) is zero by the property that the integral of the logarithmic derivative of an analytic function around a closed curve is \(2\pi i\) times the total increase in the argument of \(g(z)\), which is zero, because \(g(z)\) doesn't wind around the origin.

Step 5: Sum the Integrals

So the total integral is \(\oint_{C} \left( \frac{n}{z} + \frac{g'(z)}{g(z)} \right) \, dz = 2\pi ni + 0 = 2\pi ni\).

Key Concepts

Cauchy Integral Theorem

The Cauchy Integral Theorem is a fundamental result in complex analysis, stating that if a function is analytic and defined everywhere inside and on some closed contour, then the integral of the function over this contour is zero.
This powerful theorem can significantly simplify the evaluation of complex integrals.

In the given problem, while the function \( \frac{f'(z)}{f(z)} \) is not necessarily analytic due to the presence of the \( \frac{n}{z} \) term, using the Cauchy Integral Theorem helps us understand why certain components of the integral can be set to zero. Specifically, the part of the integral involving \( \frac{g'(z)}{g(z)} \) is zero, as explained in the original solution.

• Marine Exploration: Helps in investigating functions that behave nicely within a given region.
• Application Convenience: Can reduce complex problems to simple evaluations.

Understanding and properly applying this theorem is crucial for solving many problems in complex analysis efficiently.

residue theorem

The residue theorem is an extension of the Cauchy Integral Theorem. It tells us how to evaluate integrals of functions around closed contours, especially when those functions have singularities (poles) inside the contour.

It states that the integral of a function over a closed contour is \(2\pi i\) times the sum of residues of the function's poles within the contour.

For example, in the current exercise, the function \( \frac{n}{z} \) has a pole at the origin. The residue of \( \frac{n}{z} \) is \( n \), which provides us with an easy way to compute the integral as \( 2 \pi n i \).
This theorem is especially useful in cases where direct integration seems challenging or tedious.

• Key for dealing with singularities within contours.
• Focuses computation onto points where the function exhibits interesting behavior.

The residue theorem directs attention to poles and dramatically reduces complexity in calculating integrals.

product rule

The product rule is a basic differentiation rule used when taking the derivative of a product of two functions. For functions \( u(z) \) and \( v(z) \), the product rule states:

\[\frac{d}{dz}[u(z)v(z)] = u'(z)v(z) + u(z)v'(z).\]

In the original solution, we apply this rule to \( f(z) = z^n g(z) \) to find its derivative.
The derivative becomes \( f'(z) = n z^{n-1} g(z) + z^n g'(z) \).

This step is crucial as it leads to further simplification of the integrand and allows us to apply other theorems like the residue theorem.

• Essential for breaking down the differentiation of complex functions.
• Provides structured approach to handle products of two or more functions.

The product rule is a fundamental tool that appears across various fields of calculus and gives us insights into changes in dependent quantities.

logarithmic derivative

In this context, the logarithmic derivative \( \frac{g'(z)}{g(z)} \) appears in the simplified form of the integrand. The logarithmic derivative of a function measures how rapidly or slowly the function is changing with respect to its variable.

This form is particularly useful in complex analysis, as it provides insight into the behavior of functions that are nowhere zero, like \( g(z) \) in our original example.

The logarithmic derivative simplifies to \( \frac{g'(z)}{g(z)} \), which integrates to zero around any closed contour if \( g(z) \) is entire and nowhere zero. It's a powerful concept because it connects changes in the function to its structure and zeros:

• Gives direct insight into phase and amplitude changes of analytic functions.
• Helps detect winding numbers over closed contours.

Understanding this derivative simplifies many aspects of evaluating complex integrals by clarifying how the function behaves overall.