Problem 25 - Advanced Engineering Mathematics

Question

\(\operatorname{On} C,\left|\frac{e^{z}}{z^{2}+1}\right| \leq \frac{\left|e^{z}\right|}{|z|^{2}-1}=\frac{e^{5}}{24} \cdot\) Thus \(\left|\oint_{C} \frac{e^{z}}{z^{2}+1} d z\right| \leq \frac{e^{5}}{24} \cdot 10 \pi=\frac{5 \pi}{12} e^{5}\).

Step-by-Step Solution

Answer

The integral over the contour is bounded by \( \frac{5\pi}{12} e^5 \).

Step 1: Identify the Contour Path

The contour \( C \) is the circle \( |z| = 5 \) centered at the origin in the complex plane. We are evaluating the integral of a function \( \frac{e^{z}}{z^{2}+1} \) over this path.

Step 2: Calculate \(|z^2 + 1|\) and \(|z|^2 - 1|\)

For any \( z \) on \( C \), \(|z| = 5\). Thus, \( |z^2 + 1| \geq |z|^2 - 1 = 25 - 1 = 24 \), since \( \text{if } |z| = 5, \text{ then } |z^2 + 1| \geq \text{Re} (z^2) + 1 = |z|^2 - 1 \).

Step 3: Evaluate \(\left|\frac{e^{z}}{z^{2}+1}\right| \) on the Contour

Using the bound from step 2, we know \( \left|\frac{e^{z}}{z^{2}+1}\right| \leq \frac{|e^{z}|}{|z|^{2} - 1} = \frac{e^{5}}{24} \). This is because for \(|z| = 5\), the numerator \( |e^z| \) is maximized as \( e^5 \).

Step 4: Apply the Estimation Lemma

The estimation lemma (also known as ML inequality) gives the inequality \(|\oint_{C} f(z) \, dz| \leq \text{max}_{|z|=5} |f(z)| \cdot L(C)\). Here, \( f(z) = \frac{e^{z}}{z^{2}+1} \) and \( L(C) = 10\pi \) (since \( C \) is a circle with circumference \( 10\pi \)).

Step 5: Calculate the Maximum Integral Bound

According to ML inequality, \( \left|\oint_{C} \frac{e^{z}}{z^{2}+1} \, dz\right| \leq \frac{e^{5}}{24} \cdot 10\pi = \frac{5\pi}{12} e^{5} \). This gives us the upper bound of the integral over the contour.

Key Concepts

Contour Integration

Contour integration is an essential technique in complex analysis that evaluates integrals of complex functions over specific paths, known as contours, in a complex plane. This method is particularly useful when dealing with integrals that are challenging to solve using standard calculus techniques.
Imagine you have a curve, or contour, in the complex plane, and you want to know the value of a function along this path. A common scenario involves circular contours, like a circle centered at the origin or any other point. When integrating, one may encounter functions such as \(\frac{e^z}{z^2+1}\) along these contours.
Key aspects to consider when performing contour integration include the path's orientation (typically counter-clockwise) and the integrand's behavior along this path. Utilizing Cauchy's Integral Theorem, residues, or deformation techniques can simplify complex integrals, making contour integration a powerful tool in complex analysis.

Estimation Lemma

The Estimation Lemma, often called the "ML inequality," is a valuable result in complex analysis used to bound the value of a contour integral. This lemma facilitates the understanding of functions' behavior over a given path.
The lemma states that if \( f(z) \) is continuous on a contour \( C \) and parameterized by the arc length, then the integral \( \left|\int_C f(z) \, dz\right| \leq \max_{z \in C}|f(z)| \cdot L(C) \), where \( L(C) \) is the length of the contour.
In our case, with the contour \( |z| = 5 \), the function \( f(z) = \frac{e^z}{z^2+1} \) satisfies that condition with a maximum modulus \( \frac{e^5}{24} \). Thus, the lemma allows us to assert the integral's bound as \( \frac{5\pi}{12} e^5 \). This is particularly useful when precise evaluation of the integral is complex or algebraically intensive.

Maximum Modulus Principle

The Maximum Modulus Principle is a fundamental concept in complex analysis, primarily applied when analyzing holomorphic functions. It states that if a function \( f \) is holomorphic in a domain, the maximum value of \( |f(z)| \) must occur on the boundary of that domain, not in its interior unless it is constant.
This principle is crucial when assessing the maximum value of a function, like \( e^z \), on a specific contour such as \(|z|=5\). For contour integration, knowing the boundary behavior helps evaluate integrals more effectively and informs estimates like the one used in the estimation lemma.
For example, for \( |z|=5 \), \( |e^z| \leq e^5 \). The Maximum Modulus Principle indicates that the modulus is not exceeded within or on the boundary of the domain, solidifying our bounds for contour integrals.