1. The Fourier series to represent a function f(x) in the interval -L SX SL, nxxx nxx 1(x)=+ Σan cos +b sin L can be written in complex form, inxx f(x) = Σ che l with complex Fourier coefficients inxx Сп f(x)e L dx. Show that the complex Fourier series satisfies the following properties. (a) co= Co ao for n=0 [2] 2 (b) C = (an ibn) for n=1,2.... (c) c = (a + ib. n) for n=-1,-2.... inxx (d) The complex basis functions, e L are orthogonal in the sense that L inxx imxx Sele e Le L dx= where m is an integer. 0 if m⭑n 2L if m=n
1. The Fourier series to represent a function f(x) in the interval -L SX SL, nxxx nxx 1(x)=+ Σan cos +b sin L can be written in complex form, inxx f(x) = Σ che l with complex Fourier coefficients inxx Сп f(x)e L dx. Show that the complex Fourier series satisfies the following properties. (a) co= Co ao for n=0 [2] 2 (b) C = (an ibn) for n=1,2.... (c) c = (a + ib. n) for n=-1,-2.... inxx (d) The complex basis functions, e L are orthogonal in the sense that L inxx imxx Sele e Le L dx= where m is an integer. 0 if m⭑n 2L if m=n
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Katz, Debora M.
Chapter34: Maxwell’s Equations And Electromagnetic Waves
Section: Chapter Questions
Problem 76PQ
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![1. The Fourier series to represent a function f(x) in the interval -L SX SL,
nxxx
nxx
1(x)=+
Σan cos
+b sin
L
can be written in complex form,
inxx
f(x) = Σ che l
with complex Fourier coefficients
inxx
Сп
f(x)e L dx.
Show that the complex Fourier series satisfies the following properties.
(a) co=
Co
ao
for n=0 [2]
2
(b) C = (an ibn) for n=1,2....
(c) c = (a + ib. n) for n=-1,-2....
inxx
(d) The complex basis functions,
e L
are orthogonal in the sense that
L
inxx imxx
Sele
e Le L dx=
where m is an integer.
0 if m⭑n
2L if m=n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9dea7f9-b5df-44eb-b722-5a6f233189b7%2F37cd9e92-c92d-4315-8e83-254b26dcf9cc%2Fum1om3_processed.png&w=3840&q=75)
Transcribed Image Text:1. The Fourier series to represent a function f(x) in the interval -L SX SL,
nxxx
nxx
1(x)=+
Σan cos
+b sin
L
can be written in complex form,
inxx
f(x) = Σ che l
with complex Fourier coefficients
inxx
Сп
f(x)e L dx.
Show that the complex Fourier series satisfies the following properties.
(a) co=
Co
ao
for n=0 [2]
2
(b) C = (an ibn) for n=1,2....
(c) c = (a + ib. n) for n=-1,-2....
inxx
(d) The complex basis functions,
e L
are orthogonal in the sense that
L
inxx imxx
Sele
e Le L dx=
where m is an integer.
0 if m⭑n
2L if m=n
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