Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 8 - Integration Techniques

문제 8.3.71c

8장, 문제 8.3.71c

Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.
c.
π
∫ sin(mx) cos(nx) dx = 0, when |m + n| is even
0

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Step 1: Begin by recalling the trigonometric product-to-sum identities. Specifically, the identity for the product of sine and cosine:

sin(mx)cos(nx) = (1/2)[sin((m+n)x) + sin((m-n)x)]

. This will help simplify the integral.

Step 2: Substitute the product-to-sum identity into the integral. The integral becomes:

∫ sin(mx)cos(nx) dx = (1/2)∫ [sin((m+n)x) + sin((m-n)x)] dx

Step 3: Split the integral into two separate integrals:

(1/2)∫ sin((m+n)x) dx + (1/2)∫ sin((m-n)x) dx

. This allows us to evaluate each term independently.

Step 4: Evaluate each integral. Recall that the integral of

sin(kx)

over the interval

[0, π]

is zero when

k

is an integer and

k ≠ 0

. Since

m + n

and

m - n

are integers, both integrals evaluate to zero.

Step 5: Conclude that the original integral evaluates to zero:

∫ sin(mx)cos(nx) dx = 0

. This proves the orthogonality relation for the given conditions.

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Orthogonality of Functions

Orthogonality in the context of functions refers to the property that two functions are orthogonal if their inner product (integral of their product over a specified interval) equals zero. This concept is crucial in Fourier series, as it allows different sine and cosine functions to be treated independently, simplifying the analysis of periodic functions.

Fourier Series

A Fourier series is a way to represent a periodic function as a sum of sine and cosine functions. The coefficients of these sine and cosine terms are determined through integrals, and the orthogonality of these functions ensures that each coefficient can be calculated independently, leading to a unique representation of the original function.

Properties of Sine and Cosine Functions

Sine and cosine functions have specific properties, including periodicity and symmetry, which play a significant role in their orthogonality. For instance, the integral of the product of sine and cosine functions over a complete period is zero, particularly when the frequencies (m and n) are different, which is essential for proving the orthogonality relations in Fourier series.

Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.c.π∫ sin(mx) cos(nx) dx = 0, when |m + n| is even0