Problem 25 - Advanced Engineering Mathematics

Question

Find an upper bound for the absolute value of the given integral along the indicated contour. \(\oint_{C} \frac{e^{z}}{z^{2}+1} d z\), where \(C\) is the circle \(|z|=5\)

Step-by-Step Solution

Answer

The approximate upper bound for the integral is 194.32.

Step 1: Define the Maximum on the Contour

The first step is to estimate the maximum value of the integrand \(\frac{e^{z}}{z^2 + 1}\) on the contour, \(|z| = 5\). We need to evaluate the maximum possible value of \(|e^z|\) and \(|z^2 + 1|\). Along the contour, \(|z| = 5\), we know:- The maximum value of \(|e^z|\) is \(e^5\), because the maximum modulus theorem tells us the maximum of \(|e^z|\) on a circle is at its center, and here, it's constant.- The minimum value of \(|z^2 + 1|\) can be determined, but what matters more is that \(|z^2 + 1| \geq |z|^2 - 1 = 24\) by the reverse triangle inequality and substituting \(|z| = 5\), thus the minimum value becomes \(24\).

Step 2: Apply the ML Inequality

Using the maximum modulus principle and ML inequality, the contour integral \[ \left| \oint_C \frac{e^{z}}{z^{2} + 1} \, dz \right| \leq M \cdot L \]where \(M\) is the maximum value of the integrand along the contour and \(L\) is the length of the contour. - Here, \(M = \frac{e^5}{24}\), as this is the maximum value of the integrand \(\frac{e^z}{z^2+1}\).- The length of contour \(L\) is the circumference of the circle, which is \(2\pi\times5 = 10\pi\).Thus, the upper bound:\[ \left| \oint_C \frac{e^{z}}{z^{2} + 1} \, dz \right| \leq \frac{e^5}{24} \times 10\pi \].

Step 3: Compute and Simplify

Finally, calculate the approximate upper bound by substituting \(e^5\) as approximately \( 148.413\) and \(\pi \approx 3.1416\). Thus:\[ \frac{e^5}{24} \times 10\pi \approx \frac{148.413}{24} \times 31.416 \]When performing the calculations:- \(\frac{148.413}{24} \approx 6.184\)- \(6.184 \times 31.416 \approx 194.32\)Hence, the upper bound for the absolute value of the integral is approximately \(194.32\).

Key Concepts

Contour Integration

Contour integration is a fundamental concept in complex analysis, crucial for evaluating integrals of complex functions over specific paths or 'contours'. In this scenario, the contour is the circle defined by \(|z|=5\). Contour integrals are used when functions are complex-valued, and you have to integrate around a closed path.

• Contours can be circles, ellipses, or any closed curve in the complex plane.
• They are applicable in evaluating integrals that cannot easily be solved by real analysis methods.
• In this example, the contour is circular, which simplifies the use of various complex analysis theorems.

Contour integration also leverages complex function properties like analyticity. Analyzing functions over closed contours allows for insightful results using powerful theorems. Understanding the contour's particular properties, such as symmetry, helps simplify calculations and apply relevant theorems effectively.

Maximum Modulus Theorem

The Maximum Modulus Theorem is a key theorem in complex analysis, which states that if a function is holomorphic (complex differentiable) within and on some closed contour, then its maximum modulus on that contour occurs on the boundary. It is used in this problem to analyze the integrand \ \(\frac{e^{z}}{z^{2}+1}\) along the circular contour \(|z| = 5\).

• Evaluating \(|e^z|\): For \(|z| = 5\), we expect the modulus \(|e^z| = e^{Re(z)} = e^5|\), as \(Re(z)\) is constant along the contour.
• The theorem ensures the maximum value of \(|e^z|\) is obtained on the boundary.

The theorem is sometimes paired with the reverse triangle inequality, providing estimates for derived functions like \(z^2 + 1\). This theorem simplifies the search for maximum values over complex-valued functions.

ML Inequality

The ML Inequality is a crucial result in complex analysis used to provide bounds for the modulus of contour integrals. It combines information about the maximum value of the integrand and the total length of the contour. The inequality states that for a function \(f(z)\) integrated along a contour \(C\), the inequality is:
\[ \left| \oint_C f(z) \, dz \right| \leq M \cdot L \]

• \(M\) represents the maximum modulus of the function on the contour, here \(\frac{e^5}{24}\).
• \(L\) represents the length of the contour in this scenario, the circumference \(10\pi\).

The ML Inequality is particularly useful when exact evaluation of integrals is complex or intractable. It provides a way to estimate the upper bound of the absolute value of an integral, ensuring we stay accurate without delving into intricate computations. This principle was effectively used in our problem to calculate the upper bound of the integral of interest.

Circular Contour

A circular contour is a path around a circle in the complex plane, specifically in this scenario, where \(|z| = 5\). It's a fundamental type of contour that facilitates the application of several essential concepts and theorems in complex analysis.

• Helps in focusing on the boundary behavior of holomorphic functions, utilizing the Maximum Modulus Theorem.
• Makes computations straightforward for evaluations involving constants like \(e^5\).
• Acts as a controlled environment to apply the ML Inequality for bounds.

The circular contour is key to understanding and applying these analysis techniques, providing a clear path for evaluation and simplification of integrals. Its symmetric nature often simplifies calculations and is pivotal in exercises as it lays down a manageable framework to work with.