√-1, are Show that the set of functions on(x) = exp(inx)/√2, where I and i = orthonormal in [—^, π] with an inner product defined with weight function w(x) = 1.
√-1, are Show that the set of functions on(x) = exp(inx)/√2, where I and i = orthonormal in [—^, π] with an inner product defined with weight function w(x) = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![√1,
=
d) Show that the set of functions on(x) = exp(inx)/√2, where n I and i
orthonormal in [-, π] with an inner product defined with weight function w(x) = 1.
are
e) For the space of functions on the interval [-1,1], a suitable orthonormal basis is ₁(x) =
exp(innx)/√2, where n € I. In the next part of this question you will find the expansion
f(x) = -x non(1). Here you will find conditions on the coefficients:
n=-∞
i) Show that for a real function f(x), Cn = C²_n.
ii) Additionally, show that for an odd function in a symmetric range, cn = -_n-
iii) Hence what can be said about the expansion coefficients of a real, odd function in a
symmetric range?
f) Using the basis specified in the previous part, calculate the coefficients {c} for the func-
tion f(x) = x in [-1,1], and check to see if the expansion coefficients satisfy the above
conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffade91a1-6997-42f9-8725-9969e19df335%2F80c73d75-c24c-4cb3-8431-79068eadda4c%2Fln96vzb_processed.png&w=3840&q=75)
Transcribed Image Text:√1,
=
d) Show that the set of functions on(x) = exp(inx)/√2, where n I and i
orthonormal in [-, π] with an inner product defined with weight function w(x) = 1.
are
e) For the space of functions on the interval [-1,1], a suitable orthonormal basis is ₁(x) =
exp(innx)/√2, where n € I. In the next part of this question you will find the expansion
f(x) = -x non(1). Here you will find conditions on the coefficients:
n=-∞
i) Show that for a real function f(x), Cn = C²_n.
ii) Additionally, show that for an odd function in a symmetric range, cn = -_n-
iii) Hence what can be said about the expansion coefficients of a real, odd function in a
symmetric range?
f) Using the basis specified in the previous part, calculate the coefficients {c} for the func-
tion f(x) = x in [-1,1], and check to see if the expansion coefficients satisfy the above
conditions.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
