Let us assume that given function f can be represented on the interval [-T,7] as a finite linear combination of constant, sines, and cosines: f(r) = ao + (ak cos kz + bg sin kr). k=1 Using part (c) find the expressions for ao, ak, br in terms of f(x). Hint: Multiply both sides of the equality with an appropriate function and integrate from -a to n. A lot of terms should disappear.
Let us assume that given function f can be represented on the interval [-T,7] as a finite linear combination of constant, sines, and cosines: f(r) = ao + (ak cos kz + bg sin kr). k=1 Using part (c) find the expressions for ao, ak, br in terms of f(x). Hint: Multiply both sides of the equality with an appropriate function and integrate from -a to n. A lot of terms should disappear.
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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PART D ONLY PLEASE
![Basic version of Fourier polynomials.
(a) Write down the trig identities for
cos(x + y) = ..., cos(x – y) = ..., sin(r+y) =..., sin(r – y) = ..
(b) Using part (a) derive (explain all the steps) the trig identities
cos x cos y = ..., sinx sin y = ..., sin x cos y =...
(c) Assume that m,n are nonnegative integers (zero included!). Using part (b) evaluate the
following integrals:
sin mæ sin nædr, | a
cos mr cos nad,
sin mæ cos nadr.
Hint: You should be careful and consider different cases: 1) n = 0, m = 0, 2) n = 0, m > 0,
3) n > 0, m = 0, 4) n > 0, m > 0, m = n, and 5) n > 0, m > 0, m ± n.
(d) Let us assume that given function f can be represented on the interval [-7, 7] as a finite
linear combination of constant, sines, and cosines:
f(x) = ao +
(ar cos ka + b̟ sin kæ).
Using part (c) find the expressions for ao, ak, bk in terms of f(x).
Hint: Multiply both sides of the equality with an appropriate function and integrate from
-n to T. A lot of terms should disappear.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb71c6ab5-fa2e-44e7-8527-e1c5ce6732ff%2F976e87d8-68f3-4ba6-a2bd-491938067bac%2F4axu9yq_processed.png&w=3840&q=75)
Transcribed Image Text:Basic version of Fourier polynomials.
(a) Write down the trig identities for
cos(x + y) = ..., cos(x – y) = ..., sin(r+y) =..., sin(r – y) = ..
(b) Using part (a) derive (explain all the steps) the trig identities
cos x cos y = ..., sinx sin y = ..., sin x cos y =...
(c) Assume that m,n are nonnegative integers (zero included!). Using part (b) evaluate the
following integrals:
sin mæ sin nædr, | a
cos mr cos nad,
sin mæ cos nadr.
Hint: You should be careful and consider different cases: 1) n = 0, m = 0, 2) n = 0, m > 0,
3) n > 0, m = 0, 4) n > 0, m > 0, m = n, and 5) n > 0, m > 0, m ± n.
(d) Let us assume that given function f can be represented on the interval [-7, 7] as a finite
linear combination of constant, sines, and cosines:
f(x) = ao +
(ar cos ka + b̟ sin kæ).
Using part (c) find the expressions for ao, ak, bk in terms of f(x).
Hint: Multiply both sides of the equality with an appropriate function and integrate from
-n to T. A lot of terms should disappear.
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