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 delves into advanced topics in complex analysis, emphasizing the powerful role of conformal mappings, linear fractional and Schwarz–Christoffel transformations in simplifying complex boundary value problems. It further bridges theory with applications through the Poisson integral formulas and complex potentials, showcasing how these methods translate into practical problem solving in fields such as fluid dynamics and electromagnetic theory. Understanding the mapping properties and ensuring careful treatment of singularities, branch points, and transformation constants are pivotal for mastering these techniques.

Learning Objectives

1

Describe and analyze the mapping properties of complex functions, including conformal mappings, linear fractional transformations, and Schwarz?Christoffel transformations.

2

Apply complex mapping techniques to solve Dirichlet problems and transform boundary conditions in various regions of the complex plane.

3

Interpret the geometric implications of mappings such as inversion, rotations, power functions, and translations on lines, circles, and wedges.

4

Utilize Poisson integral formulas and Gamma function properties in the context of boundary value problems and advanced applications.

5

Identify the conditions for conformality and analyze the behavior of singular points, branch cuts, and mapping distortions.

Key Concepts

CONCEPT

DEFINITION

Conformal Mapping

A function f(z) that preserves angles locally and the shapes of infinitesimal figures, though not necessarily their size.

Linear Fractional Transformation

A mapping of the form T(z) = (az + b)/(cz + d) that transforms circles and lines in the complex plane to circles or lines.

Schwarz–Christoffel Transformation

A formula that maps the upper half‐plane onto the interior of a polygon by integrating a function with factors corresponding to the polygon’s interior angles.

Poisson Integral Formula

A representation formula used to solve the Dirichlet problem for Laplace’s equation in the unit disk, or more generally in regions amenable to conformal mapping.

Gamma Function

An extension of the factorial function to complex numbers, defined by Γ(z) = ∫_0^∞ t^(z-1)e^(−t) dt for Re(z) > 0.

Complex Potential

An analytic function whose real part represents a potential function and the imaginary part gives a stream function in fluid flow applications.

Example Problems

Example 1

For $w=\frac{1}{z}, u=\frac{x}{x^{2}+y^{2}}$ and $v=\frac{-y}{x^{2}+y^{2}} .$ If $y=x, u=\frac{1}{2} \frac{1}{x}, v=-\frac{1}{2} \frac{1}{x},$ and so $v=-u .$ The image is the line $v=-u$ (with the origin (0,0) excluded.)

Example 2

If $y=1, u=\frac{x}{x^{2}+1}$ and $v=\frac{-1}{x^{2}+1} .$ It follows that $u^{2}+v^{2}=\frac{1}{x^{2}+1}=-v$ and so $u^{2}+\left(v+\frac{1}{2}\right)^{2}=\left(\frac{1}{2}\right)^{2} .$ This is a circle with radius $r=\frac{1}{2}$ and center at $\left(0,-\frac{1}{2}\right)=-\frac{1}{2} i .$ The circle can also be described by $\left|w+\frac{1}{2} i\right|=\frac{1}{2}.$

Example 3

For $w=z^{2}, u=x^{2}-y^{2}$ and $v=2 x y .$ If $x y=1, v=2$ and so the hyperbola $x y=1$ is mapped onto the line $v=2.$

Example 4

If $x^{2}-y^{2}=4, u=4$ and so the hyperbola $x^{2}-y^{2}=4$ is mapped onto the vertical line $u=4.$

Example 5

For $w=\operatorname{Ln} z, u=\log _{e}|z|$ and $v=\operatorname{Arg} z .$ The semi-circle $|z|=1, y>0$ may also be described by $r=1$ $0<\theta<\pi .$ Therefore $u=0$ and $0<v<\pi .$ The image is therefore the open line segment from $z=0$ to $z=\pi i.$
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Step-by-Step Explanations

QUESTION

Explain how the mapping w = 1/z transforms a given point z and describe the image of the line y = 0, taking z = x + iy.

STEP-BY-STEP ANSWER:

Step 1: Express z in Cartesian form as x + iy and compute |z|^2 = x^2 + y^2.
Step 2: Write w = 1/z = (x - iy)/(x^2 + y^2), hence the real and imaginary parts are u = x/(x^2 + y^2) and v = -y/(x^2 + y^2).
Step 3: Consider the specific case when y = 0. Then u = x/x^2 = 1/x and v = 0.
Step 4: Conclude that the line y = 0 in the z-plane is mapped onto the real axis (v = 0) in the w-plane, with the mapping excluding the point at the origin (since 1/z is undefined at z = 0).
Final Answer: The mapping w = 1/z transforms points on the line y = 0 into points on the real axis, with w = 1/x for x ≠ 0.

Mapping by Inversion: w = 1/z

QUESTION

Outline the process of mapping the upper half-plane onto a polygon using the Schwarz–Christoffel formula.

STEP-BY-STEP ANSWER:

Step 1: Identify the polygon by its vertices and determine the interior angles at each vertex.
Step 2: Compute the corresponding exponents in the mapping derivative f'(z) based on the formula f'(z) = A ∏ (z - z_k)^(α_k/π - 1), where α_k are the interior angles.
Step 3: Integrate f'(z) to obtain the mapping function f(z), up to additive constants.
Step 4: Use boundary conditions to determine the constants A and any integration constant, ensuring that the vertices of the polygon are mapped correctly from specific real points in the z-plane.
Final Answer: The Schwarz–Christoffel transformation maps the upper half‐plane onto a given polygon by constructing f(z) from its derivative and using appropriate boundary conditions to fix the constant parameters.

Application of the Schwarz–Christoffel Transformation

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Common Mistakes

  • Assuming that all mappings uniformly preserve shapes without accounting for distortions in size and orientation.
  • Confusing the roles of real and imaginary parts in determining the image sets of complex functions.
  • Forgetting to exclude singularities or poles (e.g., z = 0 in w = 1/z) when describing image regions.
  • Ignoring branch cuts and multi-valued nature of inverse functions like Ln z, leading to incorrect mapping of domains.
  • Misapplying the exponents in the Schwarz–Christoffel formula, resulting in incorrect transformation of boundary angles.