ss..fourier_series_formulaes.pdf

FOURIER SERIES 1. A function f(x) can be expressed as a Fourier series in (0, 2π) as a0 ∞ f ( x) = + ∑ [a n cos nx + bn sin nx] 2 n=1 2π 1 Where a 0 =...

FOURIER SERIES 1. A function f(x) can be expressed as a Fourier series in (0, 2π) as a0 ∞ f ( x) = + ∑ [a n cos nx + bn sin nx] 2 n=1 2π 1 Where a 0 = π ∫0 f ( x ) dx 2π 1 an = π ∫ f ( x) cos nxdx 0 2π 1 bn = π ∫ f ( x ) sin nxdx 0 2. A function f(x) can be expressed as a Fourier series in (-π, π) as a0 ∞ f ( x) = + ∑ [a n cos nx + bn sin nx] 2 n=1 π 1 Where a0 = π ∫π f ( x ) dx − π 1 an = π − ∫π f ( x ) cos nxdx π 1 bn = π − ∫π f ( x ) sin nxdx 3. A function f(x) can be expressed as a Fourier series in (0, 2L) as a0 ∞  nπx nπx  f ( x) = + ∑ an cos + bn sin 2 n=1  L L  2L 1 Where a 0 = L ∫ 0 f ( x ) dx n πx 2L 1 an = L ∫ 0 f ( x ) cos L dx n πx 2L 1 bn = L ∫ f ( x ) sin 0 L dx 4. A function f(x) can be expressed as a Fourier series in (-L, L) as a0 ∞  nπ x nπ x  f ( x) = + ∑ an cos + bn sin 2 n=1  L L  L 1 a0 = L −L ∫ f ( x ) dx nπx L 1 an = ∫ f ( x ) cos dx L −L L n πx L 1 bn = L ∫ −L f ( x ) sin L dx 5. If a function f(x) is an even function in (-π, π) then its Fourier series expansion contains a0 ∞ cosine terms only, i.e. f ( x) = + ∑ [an cos nx ] 2 n=1 π 2 Where a0 = π ∫ 0 f ( x ) dx π 2 an = π ∫ f ( x ) cos nxdx 0 6. If a function f(x) is an odd function in (-π, π) then its Fourier series expansion contains ∞ sine terms only, i.e. f ( x) = ∑ [ bn sin nx] n=1 π 2 bn = π ∫ f ( x ) sin nxdx 0 7. If a function f(x) is an even function in (-L, L) then its Fourier series expansion contains a0 ∞  nπ x  cosine terms only, i.e. f ( x) = + ∑ an cos 2 n=1  L  L 2 a0 = L ∫ 0 f ( x ) dx nπx L 2 a n = ∫ f ( x ) cos dx L0 L 8. If a function f(x) is an odd function in (-L, L) then its Fourier series expansion contains ∞  nπx  f ( x) = ∑  bn sin L  sine terms only, i.e. n=1  n πx L 2 bn = L ∫ f ( x ) sin 0 L dx 9. A function f(x) can be expressed as a half range Fourier cosine series in (0, π) as a0 ∞ f ( x) = + ∑ [an cos nx ] 2 n=1 π 2 Where a0 = π ∫ 0 f ( x ) dx π 2 an = π ∫ f ( x ) cos nxdx 0 10. A function f(x) can be expressed as a half range Fourier sine series in (0, π) as ∞ f ( x) = ∑ [ bn sin nx] n =1 π 2 bn = π ∫ f ( x ) sin nxdx 0 11. A function f(x) can be expressed as a half range Fourier cosine series in (0, L) as a0 ∞  nπ x  f ( x) = + ∑ an cos 2 n=1  L  L 2 a0 = L ∫ 0 f ( x ) dx nπx L 2 L ∫0 an = f ( x ) cos dx L 12. A function f(x) can be expressed as a half range Fourier sine series in (0, L) as ∞  nπx  f ( x) = ∑  bn sin n=1  L  n πx L 2 bn = L ∫ 0 f ( x ) sin L dx 13. Parseval’s identity in (0, 2π) [ ] 2π  a0 2 ∞ 2  ∫ f ( x) dx = π  + ∑ an + bn  2 2 0  2 n =1  14. Parseval’s identity in (-π, π) [ ]  π  a0 2 ∞ ∫ f ( x) dx = π  + ∑ a n + bn 2 2 2 −π  2 n =1  15. Parseval’s identity in (0, 2L)  a0 2 2  [ ] 2L ∞ ∫ f ( x) dx = L  + ∑ an + bn  2 2 0  2 n =1  16. Parseval’s identity in (-L, L)  a0 2 2  [ ] L ∞ ∫ f ( x) dx = L  + ∑ an + bn  2 2 −L  2 n =1  17. Parseval’s identity in (0, π) provided half range cosine series [ ]  π π  a0 ∞ 2 ∫ f ( x) dx = + ∑ an 2 2  0 2 2 n=1  18. Parseval’s identity in (0, π) provided half range sine series [ ]  π π ∞ ∫ f ( x) dx = ∑ bn 2 2 0 2  n=1  19. Parseval’s identity in (0, L) provided half range cosine series L  a0  [ ] L 2 ∞ ∫ f ( x) dx =  + ∑ an  2 2 0 2 2 n =1  20. Parseval’s identity in (0, L) provided half range sine series L ∞ 2  [ ] L ∫ f ( x) dx =  ∑ bn  2 0 2  n=1 