a f(z) = % + Em (an cos nga + bn sin nga) n=1 3 After writing the Fourier representation with an = 1/L²³²3. f(x) cos dx, n = 0, 1, 2, ... bn = ²³3 f(x) sin ª dx, NAX 3 n=1,2,... use the exponential forms eio te-io COSA = sin 0 eio 2 2 2i of the cosine and sine functions to put that representation in exponential form: f(x) = Σ An exp (i) where Ao ao NTX 3 An an-ibn 2 = - A-n i0 an+ibn 2 (n = 1, 2, ...). b) An = ³3 f(x) exp(-i) dx (n = 0, 1, 2, ...) 6 Then use expression of an and bn to obtain a single formula
a f(z) = % + Em (an cos nga + bn sin nga) n=1 3 After writing the Fourier representation with an = 1/L²³²3. f(x) cos dx, n = 0, 1, 2, ... bn = ²³3 f(x) sin ª dx, NAX 3 n=1,2,... use the exponential forms eio te-io COSA = sin 0 eio 2 2 2i of the cosine and sine functions to put that representation in exponential form: f(x) = Σ An exp (i) where Ao ao NTX 3 An an-ibn 2 = - A-n i0 an+ibn 2 (n = 1, 2, ...). b) An = ³3 f(x) exp(-i) dx (n = 0, 1, 2, ...) 6 Then use expression of an and bn to obtain a single formula
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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Question
Transcribed Image Text:a
f(x)
After writing the Fourier representation
=90 +1 (an cos + bn sin )
=
with
ao
2
an =
bn = ½ ſ²³3 ƒ (x) sin "™ª dx,
3
use the exponential forms
Cos 0 = e²0 te-io
sin =
2
2
2i
of the cosine and sine functions to put that representation in exponential form:
ƒ(x) = Σ‰… An exp (in)
where
Ao =
²³3 f (x) cos nx dx, n = 0, 1, 2, ...
-3
3
n=1,2,...
ηπα
3
An
an-ibn
=
2
eio
-i0
A_n = an+ibn
2
(n = 1, 2, ...).
b)
An = ³₂ f(x) exp (-i) da (n = 0, ±1, ±2, ...)
Then use expression of an and bn to obtain a single formula
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