6. Let V = C[0, 2] with inner product (f.g) = f f(x)g(x) dr. Also define po(r) 1 integers k set 2k-1(2) = cos(kx) and 24 (2) = √ sin(kx). = (c) With f(x) = x, assuming that f(x) = Eko (f. 9k) 4k (1), derive Leibniz's formula [Hint: Set r= r = n/2.] √2T (a) Show that {90, 9₁, 92,...} is an orthonormal set in V. [Hint: Use the product-to-sum identities.] (b) Let f(x) = r. Find ||f|| and (f.n) for each n ≥ 0. (You don't need to give details of the integral evaluations, just the resulting values.) and for positive π 1 1 1 3 5 =1--+
6. Let V = C[0, 2] with inner product (f.g) = f f(x)g(x) dr. Also define po(r) 1 integers k set 2k-1(2) = cos(kx) and 24 (2) = √ sin(kx). = (c) With f(x) = x, assuming that f(x) = Eko (f. 9k) 4k (1), derive Leibniz's formula [Hint: Set r= r = n/2.] √2T (a) Show that {90, 9₁, 92,...} is an orthonormal set in V. [Hint: Use the product-to-sum identities.] (b) Let f(x) = r. Find ||f|| and (f.n) for each n ≥ 0. (You don't need to give details of the integral evaluations, just the resulting values.) and for positive π 1 1 1 3 5 =1--+
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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![6. Let V =
C[0, 2] with inner product (f.g) = f
= cos(kx) and (21 (2) = √ si
1
integers k set 2k-1(2) = co
f(x)g(x) dr. Also define po(z) =
sin(kx).
/2π
(a) Show that (90, 91, 92,...} is an orthonormal set in V. [Hint: Use the product-to-sum identities.]
(b) Let f(x) = 1. Find ||f|| and (f.n) for each n 20. (You don't need to give details of the integral
evaluations, just the resulting values.)
and for positive
0 (f. pk) (a), derive Leibniz's formula =1--+
1 1 1
3 5 7
k=0
π
4
ΠΑΣ1 (1) 5) and then comparing the power series coefficients of both sides.
n²
(c) With f(x) = x, assuming that f(x) =
[Hint: Set = π/2.]
(d) With f(x) = x, assuming that ||f||² = Exo (f.k)² (see problem 10 of the midterm for why this is a
reasonable statement), find the exact value of Σk=1 42
1
Remarks: The identity ||f||² = x (f.9k) is known as Parseval's identity. The problem of computing
the value of the infinite sum k=1 is known as the Basel problem. The correct value was (famously)
sin(x)
as the infinite product
k²
first found by Euler, who evaluated the sum by decomposing the function
TX](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3fb2678-363b-4b49-9fdf-72f65eb687c8%2F575ccf86-d5e6-4d85-ba2c-5999461d105f%2Ffyujm1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6. Let V =
C[0, 2] with inner product (f.g) = f
= cos(kx) and (21 (2) = √ si
1
integers k set 2k-1(2) = co
f(x)g(x) dr. Also define po(z) =
sin(kx).
/2π
(a) Show that (90, 91, 92,...} is an orthonormal set in V. [Hint: Use the product-to-sum identities.]
(b) Let f(x) = 1. Find ||f|| and (f.n) for each n 20. (You don't need to give details of the integral
evaluations, just the resulting values.)
and for positive
0 (f. pk) (a), derive Leibniz's formula =1--+
1 1 1
3 5 7
k=0
π
4
ΠΑΣ1 (1) 5) and then comparing the power series coefficients of both sides.
n²
(c) With f(x) = x, assuming that f(x) =
[Hint: Set = π/2.]
(d) With f(x) = x, assuming that ||f||² = Exo (f.k)² (see problem 10 of the midterm for why this is a
reasonable statement), find the exact value of Σk=1 42
1
Remarks: The identity ||f||² = x (f.9k) is known as Parseval's identity. The problem of computing
the value of the infinite sum k=1 is known as the Basel problem. The correct value was (famously)
sin(x)
as the infinite product
k²
first found by Euler, who evaluated the sum by decomposing the function
TX
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