Book cover for Advanced Engineering Mathematics

Advanced Engineering Mathematics

Dennis G. Zill, Michael R. Cullen

ISBN #9780763740955

3rd Edition

4,310 Questions

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17,647 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers various methods for integrating complex functions over contours, including direct parameterization and separation into real and imaginary components, application of the Cauchy–Goursat theorem to argue the vanishing of integrals for analytic functions over closed paths, and the use of Cauchy’s integral formulas to compute explicit values of integrals. Additionally, the exercises extend into the study of sequences and series, highlighting convergence issues and residue techniques. Mastery of these techniques is essential for advanced studies in complex analysis and for applications in physics and engineering.

Learning Objectives

1

Describe and compute contour integrals in the complex plane, including parameterization and separation into real and imaginary parts.

2

Explain and apply the Cauchy–Goursat theorem to evaluate integrals over closed contours and identify regions of analyticity.

3

Utilize Cauchy’s integral formulas to compute integral values and higher derivatives of analytic functions.

4

Analyze convergence of complex sequences and series, and understand the role of residues in evaluating integrals around singularities.

Key Concepts

CONCEPT

DEFINITION

Contour Integral

An integral of a complex function taken over a specified path (or contour) in the complex plane.

Parameterization

The process of representing a contour using a parameter (often t) so that z becomes a function of t, facilitating integration.

Analytic Function

A complex function that is differentiable in a neighborhood of every point in its domain; such functions obey powerful integral theorems.

Cauchy–Goursat Theorem

A theorem stating that if a function is analytic within and on a closed contour, then the integral over that contour is zero.

Cauchy’s Integral Formula

A fundamental result that expresses the value of an analytic function at any point inside a contour in terms of an integral over the contour.

Residue

The coefficient of (1/(z-z0)) in the Laurent series expansion of a function about a singular point z0, used in evaluating integrals via the residue theorem.

Example Problems

Example 1

$\int_{C}(z+3) d z=(2+4 i)\left[\int_{1}^{3}(2 t+3) d t+i \int_{1}^{3}(4 t-1) d t\right]=(2+4 i)[14+14 i]=-28+84 i$

Example 2

$$\int_{C}(2 \bar{z}-z) d z=\int_{0}^{2}\left[-t-3\left(t^{2}+2\right) i\right](-1+2 t i) d t=\int_{0}^{2}\left(6 t^{3}+13 t\right) d t+i \int_{0}^{2}\left(t^{2}+2\right) d t=50+\frac{20}{3} i$$

Example 3

$$\int_{C} z^{2} d z=(3+2 i)^{3} \int_{-2}^{2} t^{2} d t=\frac{16}{3}(3+2 i)^{3}=-48+\frac{736}{3} i$$

Example 4

$$\int_{C}\left(3 z^{2}-2 z\right) d z=\int_{0}^{1}\left(-15 t^{4}+4 t^{3}+3 t^{2}-2 t\right) d t+i \int_{0}^{1}\left(-6 t^{5}+12 t^{3}-6 t^{2}\right) d t=-2+0 i=-2$$

Example 5

Using $z=e^{i t},-\pi / 2 \leq t \leq \pi / 2,$ and $d z=i e^{i t} d t, \quad \int_{C} \frac{1+z}{z} d z=-\int_{-\pi / 2}^{\pi / 2}\left(1+e^{i t}\right) d t=(2+\pi) i$.
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Step-by-Step Explanations

QUESTION

How do we evaluate the integral ∮C (z + 3) dz over a given contour?

STEP-BY-STEP ANSWER:

Step 1: Identify the parameterization of the contour. In our example, the contour might be split into segments where z = x + iy with appropriate limits.
Step 2: Express the integrand (z + 3) in terms of the parameter. Separate z and the constant into their real and imaginary parts if needed.
Step 3: Compute dz using the derivative of the parameterization with respect to the chosen parameter.
Step 4: Split the integral into real and imaginary parts if the integrand becomes complex-valued, and integrate term by term.
Step 5: Combine the results to yield the final answer. (In the textbook example, the final result obtained was –28 + 84i.)
Final Answer: –28 + 84i.

Evaluating a Contour Integral

QUESTION

How can we compute ∮C 4/(z - 3i) dz using Cauchy’s integral formula?

STEP-BY-STEP ANSWER:

Step 1: Confirm that the function f(z) = 4 is analytic on and inside the contour, and that the singularity z = 3i lies inside the contour.
Step 2: Recognize that Cauchy’s integral formula states that ∮C f(z)/(z - z0) dz = 2πi · f(z0).
Step 3: Substitute f(z0) = 4, so the integral becomes 2πi · 4.
Step 4: Simplify to obtain the final answer.
Final Answer: 8Ï€i.

Applying Cauchy’s Integral Formula

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Common Mistakes

  • Misparameterizing the contour, leading to incorrect expressions for z and dz.
  • Failing to verify that the integrand is analytic in the region, which is essential for invoking the Cauchy–Goursat theorem and Cauchy’s integral formulas.
  • Making sign errors, particularly when dealing with complex conjugates and separating real and imaginary parts.
  • Overlooking the proper handling of singularities (poles) and consequently mis-evaluating residues.
  • Ignoring the role of the integration path in determining convergence and the correct application of deformation principles.