Can you explain to me starting at the sentence, "The denominator of the right hand side is larger than the strictly positive number........". I know the DCT deals with being able to pull the limit inside the integral, which is what we want to do here, but I'm not exactly sure how we know the right hand side is uniformly bounded.
Can you explain to me starting at the sentence, "The denominator of the right hand side is larger than the strictly positive number........". I know the DCT deals with being able to pull the limit inside the integral, which is what we want to do here, but I'm not exactly sure how we know the right hand side is uniformly bounded.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you explain to me starting at the sentence, "The denominator of the right hand side is larger than the strictly positive number........".
I know the DCT deals with being able to pull the limit inside the integral, which is what we want to do here, but I'm not exactly sure how we know the right hand side is uniformly bounded.
![Certainly! Here is a transcription of the provided image text suitable for an educational website:
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### Analysis of the Limit as \( R \to 1 \)
Given the equation:
\[ u(re^{i\theta}) = \frac{1}{2\pi} \int_0^{2\pi} u(Re^{i\phi}) \frac{R^2 - r^2}{R^2 - 2Rr \cos(\phi - \theta) + r^2} \, d\phi \]
for all \( R < 1 \).
Now, we observe what happens as \( R \to 1 \). Evidently, the limit of the left side of the equation is \( u(re^{i\theta}) \) since none of the quantities on the left-hand side of the equation depend on \( R \). The denominator of the right-hand side is larger than the strictly positive number \( (R - r)^2 \), so the dominated convergence theorem applies since the right-hand side is uniformly bounded. Thus, we can pull the limit inside the integral and replace \( R \) by 1.
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This section deals with the analysis of a function \( u \) expressed in terms of integrals and trigonometric functions. The focus is on comprehending the behavior of the function as the variable \( R \) approaches 1, utilizing theories such as the dominated convergence theorem to justify the manipulations of limits and integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e1e2e4-888b-4182-8c02-fd46dda7f6b1%2F88499649-d3f9-4c0a-8284-4898111ebb4a%2Fptdm3om_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here is a transcription of the provided image text suitable for an educational website:
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### Analysis of the Limit as \( R \to 1 \)
Given the equation:
\[ u(re^{i\theta}) = \frac{1}{2\pi} \int_0^{2\pi} u(Re^{i\phi}) \frac{R^2 - r^2}{R^2 - 2Rr \cos(\phi - \theta) + r^2} \, d\phi \]
for all \( R < 1 \).
Now, we observe what happens as \( R \to 1 \). Evidently, the limit of the left side of the equation is \( u(re^{i\theta}) \) since none of the quantities on the left-hand side of the equation depend on \( R \). The denominator of the right-hand side is larger than the strictly positive number \( (R - r)^2 \), so the dominated convergence theorem applies since the right-hand side is uniformly bounded. Thus, we can pull the limit inside the integral and replace \( R \) by 1.
---
This section deals with the analysis of a function \( u \) expressed in terms of integrals and trigonometric functions. The focus is on comprehending the behavior of the function as the variable \( R \) approaches 1, utilizing theories such as the dominated convergence theorem to justify the manipulations of limits and integrals.
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