This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Homework assignment 2 ∗ Due date: Monday February 25, 2008. 1. (# 3.2.2 (g)) Sketch f ( x ), the Fourier series of f ( x ) and calculate the Fourier coefficients when f ( x ), x ∈ [ − L,L ] is given by: f ( x ) = braceleftBigg 1 , x < 2 , x > 2. (slight modification of # 3.3.2 (c)) Let f ( x ), x ∈ [0 ,L ] be defined as f ( x ) = braceleftBigg , x < L 2 x, x > L 2 Sketch both Fourier sine series and Fourier cosine series of f ( x ), and calculate their coeffi- cients. 3. (3.3.18) Let f ( x ) be piecewise smooth and continuous. (a) Under what conditions are f and its Fourier series equal for all x ∈ [ − L,L ]. (b) same question for Fourier sine series but x ∈ [0 ,L ]. (c) same question for Fourier cosine series and x ∈ [0 ,L ]. 4. For integer N ≥ 1 we defined the Dirichlet kernel D N ( x ) = sin ( ( N + 1 2 ) x ) 2 π sin ( x 2 ) if x negationslash = k 2 π, k ∈ Z 1 π ( N + 1 2 ) if x = k 2 π, k ∈ Z • Prove that if x negationslash = k 2 π , k ∈ Z...
View Full Document
- Summer '06
- Fourier Series, Periodic function, Partial differential equation, Fourier sine series