Homework 3, due October 6 Math 4041, Fall 2009
1. Let u0be the function defined on the interval [0, π] by the formula
u0(x) = x(π−x).
Let ube the extension of u0to [−π, π] as an o dd function, and let fbe the extension
of uto Ras a periodic function of period 2π.
a) Find f(−1) and f(4).
b) Verfy that fis differentiable at 0.
c) Verfy that fis differentiable at 2π.
2. Let u: [0, π]→Rbe defined by
u(x) = (xif x∈[0, π/2]
π−xif x∈(π/2, π].
Write down the sine series of u. Show your work.
3. Let Wbe the space of (real- or complex-valued) C2periodic functions on Rof
period 2π. This is a vector space (over Ror C), and the set Kof elements f∈W
such that ∂2
θf=−32fis a finite dimensional subspace. Find a basis for K.
1