Please post the images of your work. The second image is used for reference for

Question

Please post the images of your work. The second image is used for reference for a few parts in this question.

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l be given. Use Exercise 1(a) to show that every sine coefficient, bn, n = 1,2,..., in the full Fourier series of feven must be zero. (Hint: Recall from calculus that if h(x) is odd, then Use Exercise 1(b) to show that the coefficients an for f even also satisfy (Hint: Recall from calculus that if h(x) is even, then Conclude that the full Fourier series of f even is equivalent to the Fourier cosine series of f. Use Exercise 1(a) to show that every cosine coefficient, an, n = ,1,2,..., in the full Fourier series of f odd must be zero. Use Exercise 1(c) to show that the coefficients bn for f odd also satisfy Conclude that the full Fourier series of f odd is equivalent to the Fourier sine series of f. Show that if f is even and g is odd. then f g is odd. Show that if f and g are even, then f g is even, c Show that if f and g are odd. then f g is even.

Explanation / Answer

[Large extbf{Problem 2a}]

Since (sin) is odd and (f_{ ext{even}}) is even, by Exercise 1(a), the product of those expressions is odd. Thus, by the hint, the integral is zero.

[Large extbf{Problem 2b}]

Since (cos) is even and (f_{ ext{even}}) is even, by Exercise 1(b), the product of those expressions is even. Thus, by the hint given, we have

[a_n = dfrac{1}{ell} int_{-ell}^{ell} f(x)cos(npi x/ell),dx = dfrac{2}{ell} int_0^{ell} f(x)cos(npi x/ell),dx]

[Large extbf{Problem 2c}]

Since (cos) is even and (f_{ ext{odd}}) is odd, by Exercise 1(a), the product of those expressions is odd. Thus, by the hint, the integral is zero.

[Large extbf{Problem 2d}]

Since (sin) is odd and (f_{ ext{odd}}) is odd, by Exercise 1(c), the product of those expressions is even. Thus, by the hint given, we have

[b_n = dfrac{1}{ell} int_{-ell}^{ell} f(x)sin(npi x/ell),dx = dfrac{2}{ell} int_0^{ell} f(x)sin(npi x/ell),dx]

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