In this problem we will prove one orthogonality relation used in Fourier analysis. a) Show that sin(mx) cos(nx) = ½ (sin((m +n)x) + sin((m − n)x)) using the angle addition and subtraction formulas. b) Use the result from part a) to evaluate the integral [ε sin (mx) cos(nx) dx c) Prove the following statement for integers m‡±n. (y = sin 3x cos 4x is graphed for reference) 2π sin(mx) cos(nx) dx = 0 որիան, M
In this problem we will prove one orthogonality relation used in Fourier analysis. a) Show that sin(mx) cos(nx) = ½ (sin((m +n)x) + sin((m − n)x)) using the angle addition and subtraction formulas. b) Use the result from part a) to evaluate the integral [ε sin (mx) cos(nx) dx c) Prove the following statement for integers m‡±n. (y = sin 3x cos 4x is graphed for reference) 2π sin(mx) cos(nx) dx = 0 որիան, M
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Orthogonality in Fourier Analysis**
In this problem, we will prove one orthogonality relation used in Fourier analysis.
### a) Show the Identity
Show that \( \sin(mx) \cos(nx) = \frac{1}{2} \left( \sin((m+n)x) + \sin((m-n)x) \right) \) using the angle addition and subtraction formulas.
### b) Evaluate the Integral
Use the result from part (a) to evaluate the integral
\[ \int \sin(mx) \cos(nx) \, dx \]
### c) Prove the Orthogonality Condition for Different Integers
Prove the following statement for integers \( m \neq \pm n \). *(Note: \( y = \sin(3x) \cos(4x) \) is graphed for reference)*
\[ \int_{0}^{2\pi} \sin(mx) \cos(nx) \, dx = 0 \]
#### Graph Explanation
The graph on the right illustrates the function \( y = \sin(3x) \cos(4x) \). The function oscillates around the x-axis, crossing it multiple times, which visually suggests that the area under one period from \( 0 \) to \( 2\pi \) sums to zero. This illustrates the orthogonality of \( \sin(3x) \) and \( \cos(4x) \) over the interval.
By following these steps, you should be able to establish these foundational concepts of Fourier analysis and their implications in orthogonality relations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F953e3ae3-1a36-4eca-a542-ddeb8c6ae72a%2Fc0924a4c-88eb-4718-820c-9965797ff3af%2Fs9rodpm_processed.png&w=3840&q=75)
Transcribed Image Text:**Orthogonality in Fourier Analysis**
In this problem, we will prove one orthogonality relation used in Fourier analysis.
### a) Show the Identity
Show that \( \sin(mx) \cos(nx) = \frac{1}{2} \left( \sin((m+n)x) + \sin((m-n)x) \right) \) using the angle addition and subtraction formulas.
### b) Evaluate the Integral
Use the result from part (a) to evaluate the integral
\[ \int \sin(mx) \cos(nx) \, dx \]
### c) Prove the Orthogonality Condition for Different Integers
Prove the following statement for integers \( m \neq \pm n \). *(Note: \( y = \sin(3x) \cos(4x) \) is graphed for reference)*
\[ \int_{0}^{2\pi} \sin(mx) \cos(nx) \, dx = 0 \]
#### Graph Explanation
The graph on the right illustrates the function \( y = \sin(3x) \cos(4x) \). The function oscillates around the x-axis, crossing it multiple times, which visually suggests that the area under one period from \( 0 \) to \( 2\pi \) sums to zero. This illustrates the orthogonality of \( \sin(3x) \) and \( \cos(4x) \) over the interval.
By following these steps, you should be able to establish these foundational concepts of Fourier analysis and their implications in orthogonality relations.
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