3. You are given the function f(:c) = a: for a: 6 (0,2). Extend it periodically to the full real line (ie to all real numbers :10) in the following manner: Set f (51;) = :r on (2, 0]; consider the base period to be the interval (2, 2), which is centered about a: = 0 and which has length 2L : 4; extend the function periodically in x with period 4 to f (x) = :1: 4 on (2,6), f(x) = a; + 4 on (6, 2), etc... (Note that the extended, periodic function has jump discontinuities.) Find the Fourier series for f (x) for all :1: on [0,2], and for each a: in this interval state the value to which the Fourier series converges. 02 b 2 ) . xox Uzx dv= sin 2 ) do duc edx - 2 2 12 V 2 na cos (2 ) Z X. -2 1) - HEE Cos / hoEy - - 2 x2-2 ) , nut cos( 2 )dx by integration by parts I 2 hy Cos ( not ) - 14 nye cos(- hat ) + 2 - 4 1X=- 2 not cos (hyt ) + 2 (0 0 ) 2 Time Cos ( net ) 100 - 4 Cos (nut ) sin(nix) 2