4. A 2π-periodic function f(x) has the Fourier series expansion f(x) =1+2 Σε cos(nx). h=1 The complex Fourier series expansion of f(x) is then given by: α (A) f(2) ~ Σ n=0 e-n einr α (B) f(x)=1+ Σ engine Σ ==0 n#0 α (C) f(x) = Σ e pins H== + (D) f(x) = Σ 81118 - -|n| ping (E) f(x) = 1/2 + Σ e-tmlpina 8=18 n#D Solution. α f(x) =1+2 Σ n=1 = 1+ Σ n=l =1+Σ e-spine +Σ n=l n=1 = Σ e h== e einx einx +e-inr 2 -|n|gina p=n -1 + Σ h==0 en eina ?
4. A 2π-periodic function f(x) has the Fourier series expansion f(x) =1+2 Σε cos(nx). h=1 The complex Fourier series expansion of f(x) is then given by: α (A) f(2) ~ Σ n=0 e-n einr α (B) f(x)=1+ Σ engine Σ ==0 n#0 α (C) f(x) = Σ e pins H== + (D) f(x) = Σ 81118 - -|n| ping (E) f(x) = 1/2 + Σ e-tmlpina 8=18 n#D Solution. α f(x) =1+2 Σ n=1 = 1+ Σ n=l =1+Σ e-spine +Σ n=l n=1 = Σ e h== e einx einx +e-inr 2 -|n|gina p=n -1 + Σ h==0 en eina ?
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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Please explain very simply. I have no clue how they got the last lines (the ones by the red question marks). Thank you sm.
Transcribed Image Text:4.
A 2-periodic function f(x) has the Fourier series expansion
(A) f(x) ~ e-n einz
The complex Fourier series expansion of f(x) is then given by:
n=0
00
(B) f(x)=1+ Σ e-n einz
3=-∞
n#0
(C) f(x) = Σ
n=-00
en einx
→ (D) f(x)~ e-In einz
72=18
DO
(E) f(x)~1/2+ Σ e-\n\ einx
f(x)=1+2e cos(nx).
82=18
n#0
n=1
Solution.
f(x)=1+2e
n=1
=1+Σ
=
n=1
=1+Σ
n=1
-12
e-n pinz
einx te inz
2
-n
e einx +Σ
n=1
Č Σ en ein
inx
12=-∞0
∞o
e
-inx
+ Σ en eina
n=-00
J?
J?
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