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1. A function f(x) is said to be periodic with period a if f(a + x) = f (a), €.g. sin x, cos x are periodic with period 2n., , 2. (a) A function f (x) is said to be even if f (-x) = f (x) e.g. cos x,, SEC X, x are even functions., Graphically an even function is symmetrical about the y-axis., , (b) A function f(x) is said to be odd if f(—x) = — f(x) e.g. sin x, cosec x,, are odd functions., , Geometrically an odd function is symmetrical about the origin., y y, , Fig. 6.3 (a) : Even Function Fig. 6.3 (b) : Odd Function, (c) A function can be neither even nor odd e.g. &%, 10%, x°-x, are neither, odd nor even., , 3. We also need the following results, where nis an integer, (i) sin nm = 0 (ji) sin 2nm = 0 (iil) cos mm = (-1)” (iv) cos 2nn = 1

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Applied Mathematics - Ill (6-4) Fourier Series, (Clvil/ Const. / Chem. / Prod.), (v) cos (n+ 1) m= cos nn cos tn ¢ Sin nn sin n = - cos m, (vi) sin (n+ 1) m= sin mm cos 7 + Cos Mm Sin tt = 0., (vii) sin (2nz + x) = sin x, (viii) cos (2n nm + x) = cos x, 4. We also need the following results, 2 sin Acos B= sin (A + B) + sin (A— B), 2 cos Asin B= sin (A+ B) —sin (A- B), 2 cos Acos B= cos (A + B) + cos (A— B), -2 sin Asin B= cos (A + B) - cos (A- B)., 5. Ifo (x)= (x)- g(x) then 6 (x) will be even or odd according to the following, , , , , , , , , , , , table, F(x) E E 0) 0, g(x) E 0 E, $ (x) = F(x) - g (x) E 0 0) E, , , , , , , , , , , , 6. We need the following theorem, , , , Theorem des f(x) dx = at f(x). dx if f(x) is even., =0 if f(x) is odd, _, , 7. Wealso need the ‘following.two integrals often in Fourier series., , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ax, 4. e* sin bx dx = —° asin bx — bcos bx, J a + b? [ ], 2. J e™ cos bx dx = [acos bx - bsin bx], a* + b?, 8. We also need the following results, 2 c+20, ET “cosmx dx =| Sin =0 (n# 0), c n lc, 2n C+27, a. ts sin nx dx = - SOR nx =0 (n#0), c n lc, c+2n | 2, 3. I sin mx cos nx dx =" “[sin(m +n) x+sin(m—n)x}ax, , 2 s|- cos (m+n) x _ seston ait Chen, , 2 m+n m-n, , ic