n In the interval (−л, π), ³₂ (x) : = √ : exp(-n²x²). (a) Expand &n (x) as a Fourier cosine series. (b) Show that your Fourier series agrees with a Fourier expansion of 8 (x) in the limit as n → ∞. (c) Confirm the delta function nature of your Fourier series by showing that for any f(x) that is finite in the interval [—ë, ë] and continuous at x = 0, T ** f (x) [Fourier expansion of 8% (x)] dx = ƒ(0). -T
n In the interval (−л, π), ³₂ (x) : = √ : exp(-n²x²). (a) Expand &n (x) as a Fourier cosine series. (b) Show that your Fourier series agrees with a Fourier expansion of 8 (x) in the limit as n → ∞. (c) Confirm the delta function nature of your Fourier series by showing that for any f(x) that is finite in the interval [—ë, ë] and continuous at x = 0, T ** f (x) [Fourier expansion of 8% (x)] dx = ƒ(0). -T
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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![n
1.15.18 In the interval (-7, π), Sn (x) =
√r
: exp(-n²x²).
(a)
Expand 8, (x) as a Fourier cosine series.
(b)
Show that your Fourier series agrees with a Fourier expansion of 8 (x) in
the limit as n → ∞.
(c)
Confirm the delta function nature of your Fourier series by showing that
for any f(x) that is finite in the interval [-ï, ´] and continuous at x = 0,
T
f(x) [Fourier expansion of 8(x)] dx = ƒ(0).
-T](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F23659403-042f-4d3a-bc57-2dc1fb48c869%2F7937a0c4-bc8d-413d-b8bb-fbcad517ef39%2Fiqjvrnf_processed.png&w=3840&q=75)
Transcribed Image Text:n
1.15.18 In the interval (-7, π), Sn (x) =
√r
: exp(-n²x²).
(a)
Expand 8, (x) as a Fourier cosine series.
(b)
Show that your Fourier series agrees with a Fourier expansion of 8 (x) in
the limit as n → ∞.
(c)
Confirm the delta function nature of your Fourier series by showing that
for any f(x) that is finite in the interval [-ï, ´] and continuous at x = 0,
T
f(x) [Fourier expansion of 8(x)] dx = ƒ(0).
-T
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