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 of the textbook covers a wide range of topics in complex analysis, including arithmetic of complex numbers, powers and roots, and the application of polar coordinates via De Moivre's theorem. A significant focus is placed on establishing function differentiability via the Cauchy-Riemann equations and verifying analyticity. Additionally, it explores complex integration through contour integrals and the use of exponential, logarithmic, trigonometric, and hyperbolic functions in complex contexts. The exercises reinforce both computational skills and theoretical understanding essential for advanced studies in complex analysis.

Learning Objectives

1

Apply and manipulate complex numbers including computing powers, roots, and evaluating expressions in both rectangular and polar forms.

2

Demonstrate proficiency in using the Cauchy-Riemann equations to determine differentiability and analyticity of complex functions.

3

Evaluate contour integrals and apply integration techniques in the complex plane.

4

Utilize exponential, logarithmic, trigonometric, and hyperbolic functions in complex analysis to simplify and solve problems.

5

Interpret geometric sets in the complex plane and apply these concepts to solve real-world analytic and contour integration problems.

Key Concepts

CONCEPT

DEFINITION

Complex Number

A number of the form z = x + iy, where x and y are real numbers and i is the imaginary unit satisfying i² = -1.

Modulus and Argument

The modulus |z| is the distance of z from the origin given by √(x² + y²) while the argument Arg(z) is the angle formed with the positive real axis.

Powers and Roots

Operations taking a complex number to an integer power (using De Moivre's theorem) or finding n-th roots by expressing the complex number in polar form.

Cauchy-Riemann Equations

A set of two partial differential equations, ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x, that provide a necessary (and under smoothness conditions, sufficient) condition for a complex function f(z) = u(x,y) + iv(x,y) to be differentiable (analytic).

Contour Integral

An integral of a complex function taken along a path (contour) in the complex plane, analogous to line integrals in vector calculus.

Exponential and Logarithmic Functions

For complex numbers, the exponential function is defined via its power series; the logarithmic function is the inverse of the exponential function and is inherently multi-valued.

Trigonometric and Hyperbolic Functions

Functions defined on complex numbers by extending the usual sine and cosine formulae (or via exponentials) and similarly for hyperbolic functions like sinh and cosh.

Example Problems

Example 1

$$3+3 i$$

Example 2

$$-4 i$$

Example 3

$$i^{8}=\left(i^{2}\right)^{4}=(-1)^{4}=1$$

Example 4

$$i^{11}=i\left(i^{2}\right)^{5}=i(-1)^{5}=-i$$

Example 5

$$7-13 i$$

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Step-by-Step Explanations

QUESTION

How do you compute i raised to the 11th power (i^11)?

STEP-BY-STEP ANSWER:

Step 1: Recall that i^2 = -1.
Step 2: Express i^11 as i × (i^2)^5 because 11 = 1 + 2*5.
Step 3: Compute (i^2)^5 = (-1)^5 = -1.
Step 4: Multiply by the remaining i: i × (-1) = -i.
Final Answer: i^11 = -i.

Computing i^11

QUESTION

How do you evaluate the contour integral ∮_C z^2 dz on a given simple path?

STEP-BY-STEP ANSWER:

Step 1: Parameterize the contour C by writing z(t) in terms of a real parameter t.
Step 2: Express z^2 and dz in terms of t.
Step 3: Substitute these expressions into the integral ∮_C z^2 dz and identify the integration limits for t.
Step 4: Perform the integration term by term.
Final Answer: The computed value will depend on the specific parameterization and limits (e.g., if C is a line segment or circle).

Evaluating a Simple Contour Integral

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Common Mistakes

  • Incorrectly simplifying powers of i by failing to use the periodicity (i^4 = 1).
  • Ignoring the multi-valued nature of the complex logarithm and inverse trigonometric functions.
  • Not verifying that the conditions of the Cauchy-Riemann equations hold in an entire neighborhood, thereby incorrectly concluding analyticity.
  • Errors in parameterizing contours for integration, leading to miscalculations in complex integrals.
  • Confusing the treatment of real versus complex differentiation, especially when separating functions into their u(x,y) and v(x,y) components.