Math 6720
HW7
Appl. Complex Var., Asymptc Mthds
NAME: .....................................................................
1. Consider the expansion of a T -periodic function w(t) in Fourier series over Fourier harmonics
w(t) =
∞
2π
1 X
Wk eik T t .
T k=−∞
(a) Formulate and prove the orthogonality of these Fourier harmonics.
(b) Derive formula for the Fourier coefficients Wk .
(c) Show that if w(t) is even, then Wk is also even.
(d) Show that if w(t) is odd, then Wk is also odd.
(e) Show that if w(t) is real-valued, then Wk = W−k .
Alexander Balk
due 3/9/2015
2. (a) Prove the Riemann-Lebesgue lemma: If a function w(t) is of the class L1 (−∞, ∞), then its Fourier transform
Z ∞
W (ν) =
w(t)e−i2πνt dt
−∞
has the following two properties:
i. W (ν) is a continuous function for all real ν (−∞ < ν < ∞).
ii. W (ν) → 0 as ν → ±∞.
(b) Is it true: If w ∈ L1 (−∞, ∞), then W ∈ L1 (−∞, ∞) ?
3. Consider the boundary value problem for the heat equation on the whole line −∞ < x < ∞:
∂u
∂2u
=k 2
∂x
∂x
u(x, 0) = f (x)
u → 0 as x → ±∞
(parameter k > 0 is the thermal diffusivity, f (x) is a given function, f (x) → 0 as x → ±∞).
(a) Derive the formula for its solution as a convolution of Green’s function with the initial condition f (x).
(b) Show continuous dependence on the initial condition: A small variation in the initial condition causes small
variation in the solution.
4. How Fourier transform leads to analytic functions. Consider a continuous function f (x), −∞ < x < ∞.
Suppose
|f (x)| < Ae−ax if
− ∞ < x < 0,
|f (x)| < Be−bx if 0 < x < ∞.
(A, a, B, b are real constants; A, B are positive; a, b can have any signs, but a > b.)
(a) Show that the Fourier transform
Z ∞
F (µ) =
f (x) e−iµx dx
−∞
is an analytic function in some strip. Find this strip.
(b) Show that the original can be restored by the complex inverse Fourier transform
Z iγ+∞
f (x) =
iγ−∞
F (µ) eiµx
dµ
2π
(the integration here is along an arbitrary line Imµ = γ that belongs to that strip).
Suggestion: Check that the function f (x)eγx exponentially decays at x → ±∞ for any γ ∈ (b, a). Consider the usual real Fourier
transform of this function.
5. Show that
F[xα H(x)] =
Γ(α + 1)
(iµ)α+1
(F denotes the Fourier transform, H(x) is the Heaviside function).
(a) Which branch of the complex power do we need to take?
(b) For which complex µ is this formula valid?
Suggestion: First consider µ = −iσ (σ > 0) and use Euler’s Γ-function. Then apply the uniqueness Th.
6. Integrate
Z ∞
0
√
x2
x
π
dx = √
+1
2
7. Integrate
Z ∞
1
dx
π
√
dx =
2
2
x x −1
Suggestion: Replace x by z and choose a branch with one branch cut along the integration path. Use the contour of a big circle with the
exclusion of the branch points and branch cuts.