Fourier Series
December 8, 2016
Categorised in: 1st Year Maths 2
Fourier Series
f(x) = $a_o/2 + ∑↙{n =1 }↖∞ (a_n cos nx + b_n sin nx)$
$a_o, a_n, b_n$ are fourier coefficients
Note (Useful Results)
(i) $∫↙{c}↖{c + 2π} cosnx dx = [{sin nx}/{n}]^{c + 2π}_c$ = 0 (n ≠ 0)
(ii) $∫↙{c}↖{c + 2π} sinnx dx = [-{cos nx}/{n}]^{c + 2π}_c$ = 0 (n ≠ 0)
(iii) $\table∫↙{c}↖{c + 2π} sin mx cosnx dx,=, 1/2 ∫↙{c}↖{c + 2π} [sin(m + n)x + sin(m-n) x] dx;,=,1/2 [- {cos(m+n)x}/{(m+n)} – {cos (m-n)x}/{(m-n)}]_{c}^{c + 2π} = 0 (m≠n) $
If m=n,
$∫↙{c}↖{c + 2π} sinmx cosnx dx = 1/2 ∫↙{c}↖{c + 2π} sin2nx dx$ = 0 (m = n)
Hence,
$∫↙{c}↖{c + 2π} sinmx cosnx dx$ = 0, for all m and n
(iv) $\table∫↙{c}↖{c + 2π} cosmx cosnx dx,=, 1/2 ∫↙{c}↖{c + 2π} [cos(m+n)x + cos(m-n)x] dx;,=,1/2 [{sin(m+n)x}/{(m+n)} + {sin(m-n) x}/{(m-n)}]_c^{c+2π} = 0 (m≠n)$
if m = n,
$∫↙{c}↖{c + 2π} cosmx cosnx = ∫↙{c}↖{c + 2π} cos^2 nx dx = 1/2[x + {sin2nx}/{2n}]_c^{c+2π} = π$ (m = n)
Hence, $∫↙{c}↖{c + 2π} cosmx cosnx$ = 0 (m≠n) or π (m=n)
(v) $\table∫↙{c}↖{c + 2π} sinmx sinnx dx,=, 1/2 ∫↙{c}↖{c + 2π} [cos(m-n) x – cos(m+n) x] dx;,=, 1/2 [{sin(m-n)x}/{(m-n)} – {sin(m+n) x}/{(m+n)}]_c^{c+2π} = 0 (m≠n)$
if m = n,
$∫↙{c}↖{c + 2π} sinmx sinnx dx = ∫↙{c}↖{c + 2π} sin^2 nx dx = 1/2 [x – {sin2nx}/{2n}]_c^{c+2π}$ = π (m=n)
Hence, $∫↙{c}↖{c + 2π} sinmx sin nx dx$ = 0 (m≠n) or π (m=n)
Euler’s Formulas
$a_o = 1/π ∫↙{c}↖{c + 2π} f(x) dx$
$a_n = 1/π ∫↙{c}↖{c + 2π} f(x) cosnx dx$
$b_n = 1/π ∫↙{c}↖{c + 2π} f(x) sinnx dx$
Even and odd functions
Even: f(-x) = f(x)
Odd: f(-x) = – f(x)
$\table ∫↙{-a}↖{a} f(x) dx,=, 2 ∫↙{0}↖{a} f(x) dx \text"f(x) is even";,=,0 \text"f(x) is odd"$
Expansion of even and odd periodic functions
When f(x) is even function
$a_o = 2/π ∫↙{0}↖{π} f(x) dx$
$a_n = 2/π ∫↙{0}↖{π} f(x) cosnx dx$
$b_n $ = 0
When f(x) is odd function
$a_o = 0$
$a_n = 0$
$b_n = 2/π ∫↙{0}↖{π} f(x) sin nx dx$
Note:
For any integer n,
sin(nπ) = 0, sin(2nπ) = 0
cos(nπ) = $(-1)^n$, cos(2nπ) = 1
sin(n±1)π = 0, cos(n±1)π = – cos(nπ)
Functions having arbitrary period (change of interval)
The fourier expansion of f(x) in interval c≤x≤c+2L is given by,
f(x) = $a_o/2 + ∑↙{n=1}↖∞ (a_n cos{nπx}/{L} + b_n sin {nπx}/{L})$
where,
$a_o = 1/L ∫↙{c}↖{c + 2L} f(x) dx$
$a_n = 1/L ∫↙{c}↖{c + 2L} f(x) cos {nπx}/{L} dx$
$b_n = 1/L ∫↙{c}↖{c + 2L} f(x) sin {nπx}/{L} dx$
Even and odd functions in interval -L ≤ x ≤ L
Even Function:
$a_o = 2/L ∫↙{0}↖{L} f(x) dx$
$a_n = 2/L ∫↙{0}↖{L} f(x) cos {nπx}/{L} dx$
$b_n = 0$
Odd function:
$a_o = 0$
$a_n = 0$
$b_n = 2/L ∫↙{0}↖{L} f(x) sin {nπx}/{L} dx$
Half range cosine expansion
$f(x) = a_o/2 + ∑↙{n=1}↖∞ a_n cos {nπx}/{L}$
where,
$a_o = 2/L ∫↙{0}↖{L} f(x) dx$
$a_n = 2/L ∫↙{0}↖{L} f(x) cos {nπx}/{L} dx$
Half range sine expansion
$f(x) = ∑↙{n=1}↖∞ b_n sin{nπx}/{L} dx$
where,
$b_n = 2/L ∫↙{0}↖{L} f(x) sin {nπx}/{L} dx$
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