2. In this exercise we show how the symmetries of a function imply certain properties of its Fourier coefficients. Let f be a 27-periodic Riemann integrable function defined on R. (a) Show that the Fourier series of the function f can be written as f(0) ~ ƒ(0) + £l (n) + ƒ(-n)] cos nº + i[f(n) – ƒ(-n)] sin nô. (b) Prove that if f is even, then f(n) = f(-n), and we get a cosine series. (c) Prove that if ƒ is odd, then f(n) = -ƒ(-n), and we get a sine series. (d) Suppose that f(0 +r) = f(0) for all 0 € R. Show that ƒ(n) = 0 for all odd n. %3D
2. In this exercise we show how the symmetries of a function imply certain properties of its Fourier coefficients. Let f be a 27-periodic Riemann integrable function defined on R. (a) Show that the Fourier series of the function f can be written as f(0) ~ ƒ(0) + £l (n) + ƒ(-n)] cos nº + i[f(n) – ƒ(-n)] sin nô. (b) Prove that if f is even, then f(n) = f(-n), and we get a cosine series. (c) Prove that if ƒ is odd, then f(n) = -ƒ(-n), and we get a sine series. (d) Suppose that f(0 +r) = f(0) for all 0 € R. Show that ƒ(n) = 0 for all odd n. %3D
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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c &d
![2. In this exercise we show how the symmetries of a function imply certain
properties of its Fourier coefficients. Let f be a 27-periodic Riemann integrable
function defined on R.
(a) Show that the Fourier series of the function f can be written as
f(0) ~ ƒ(0) + (n) + ƒ(-n)] cos no + i[f(n) – ƒ(-n)] sin nô.
(b) Prove that if ƒ is even, then f(n) = ƒ(-n), and we get a cosine series.
(c) Prove that if ƒ is odd, then f(n) = - ƒ(-n), and we get a sine series.
(d) Suppose that f(0+n) = f(0) for all 0 € R. Show that f(n) = 0 for all
odd n.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833481a2-df8c-4805-95a2-f24b64ba619f%2F9acf3100-cbd6-4474-8c99-9af28e3929bd%2F23we4yr_processed.png&w=3840&q=75)
Transcribed Image Text:2. In this exercise we show how the symmetries of a function imply certain
properties of its Fourier coefficients. Let f be a 27-periodic Riemann integrable
function defined on R.
(a) Show that the Fourier series of the function f can be written as
f(0) ~ ƒ(0) + (n) + ƒ(-n)] cos no + i[f(n) – ƒ(-n)] sin nô.
(b) Prove that if ƒ is even, then f(n) = ƒ(-n), and we get a cosine series.
(c) Prove that if ƒ is odd, then f(n) = - ƒ(-n), and we get a sine series.
(d) Suppose that f(0+n) = f(0) for all 0 € R. Show that f(n) = 0 for all
odd n.
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