Real Analysis II.

Instructor: Dmitry Ryabogin

Assignment III.

1. Problem 1. Let T (x, y) = (f(x, y), g(x, y)) be a map from R^{2} to R^{2} defined as follows

f(x, y) = g(x, y) = \frac{x^{2}y}{x^{2} + y^{2}} , (x, y) ≠ (0, 0), f(0, 0) = g(0, 0) = 0.

a) Is T continuous at (0, 0)?

b) Do partial derivatives \frac{∂f}{∂x}(0, 0), \frac{∂f}{∂y} (0, 0) exist?

c) Is T differentiable at (0, 0)?

2. Problem 2. Complex differentiation.

A function (a map) l : C → C is called R-linear (C-linear), if

l(z_{1} + z_{2}) = l(z_{1}) + l(z_{2}) ∀z_{1}, z_{2} ∈ C, l(λz) = λl(z) ∀λ ∈ R, ∀z ∈ C

( l(z_{1} + z_{2}) = l(z_{1}) + l(z_{2}) ∀z_{1}, z_{2} ∈ C, l(λz) = λl(z) ∀λ ∈ C, ∀z ∈ C).

a) Prove that any R-linear function is of the form

l(z) = az + bz̄, a = \frac{1}{2}(α − iβ), b = \frac{1}{2}(α + iβ), α = l(1), β = l(i).

b) Prove that any C-linear function is of the form

l(z) = az, a = l(1).

c) Prove that an R-linear function is C-linear iff l(iz) = il(z).

d) Compute the Jacobian of an R-linear map and of a C-linear map. What geometric conclusions can you make from the computation?

e) Let z ∈ U ⊆ C. A function f : U → C is called R-differentiable (C-differentiable) at the point z, provided

f(z + h) − f(z) = l(h) + α(z, h),  lim_{h→0} \frac{α(z, h)}{h} = 0,

where l is an R-linear (C-linear) function. Thus,

f(z + h) − f(z) = ah + bh̄ + α(z, h),  lim_{h→0} \frac{α(z, h)}{h} = 0.

Prove that

\frac{∂f}{∂x} = a + b, −i\frac{∂f}{∂y} = a − b,

or

a = \frac{1}{2}(\frac{∂f}{∂x} − i\frac{∂f}{∂y} ), b = \frac{1}{2}(\frac{∂f}{∂x} + i\frac{∂f}{∂y} ).

f) Complex Analysis starts with the notion of C-differentiability. The above formulas show that C-differentiability yields b = 0. Prove that for f = u + iv the condition b = 0 can be written as

\frac{∂u}{∂x} = \frac{∂v}{∂y} , \frac{∂u}{∂y} = −\frac{∂v}{∂x},

the so-called Cauchy-Riemann equations.

g) Observe that f(z) = x + 2iy is nowhere differentiable in the complex sense.

3. Problem 3. Polar coordinates.

a) Let P := {x ∈ R^{k} : x = ((a − t)x_{1}, (a − t)x_{2}, ..., (a − t)x_{k−1}, t)}, (x_{1}, ..., x_{k−1}) ∈ A, t ∈ [0, a] be a pyramid in R^{k} with a base A ⊂ R^{k−1} and height a. Prove that

vol_{k}(P) = \frac{a}{k} vol_{k−1}(A).

Hint. Use Fubini.

b) Let v_{k} be the volume of the Euclidean ball (of radius 1) in R^{k}, and let σ_{k−1} be its surface area. Prove that σ_{k−1} = k v_{k}.

c) Let S^{k−1} be the unit sphere. Show that every x ∈ R^{k}, except for x = 0, has a unique representation of the form x = ru, where r is a positive real number and u ∈ S^{k−1}. Thus, R^{k} may be regarded as a product (0, ∞) × S^{k−1}.

d) Let m_{k} be Lebesgue measure on R^{k}, and define a measure σ_{k−1} on S^{k−1} as follows: if A ⊆ S^{k−1} and A is a Borel set, let Ã be the set of all points ru, where 0 < r < 1 and u ∈ A, and define σ_{k−1}(A) = k m_{k}(Ã). Prove the formula

∫_{R^{k}} fdm_{k} = ∫_{0}^{∞}r^{k−1}dr ∫_{S^{k−1}} f(ru)dσ_{k−1}(u)

for every nonnegative Borel function f on R^{k}.

Hint. If 0 < r_{1} < r_{2} and if A is an open subset of S^{k−1}, let E be the set of all ru with r_{1} < r < r_{2}, u ∈ A, and verify the formula for the characteristic function of E. Then approximate.

e) Check that the above formula coincides with familiar results when k = 2 and k = 3.

f) It is convenient to normalize integrals as above by pulling out the factor kv_{k}, and write

∫_{R^{k}} fdm_{k} = kv_{k} ∫_{0}^{∞}r^{k−1}dr ∫_{S^{k−1}} f(ru)dσ(u), σ(S^{k−1}) = 1.

Compute v_{k} using the following trick: integrate the function f(x) = exp\{−\frac{1}{2} ∑_{j=1}^{k} x_{j}^{2}\}

in both ways. This function is at once invariant under rotations and a product of functions depending on separate coordinates. Hence,

∫_{R^{k}} fdm_{k} = Π_{j=1}^{k} ∫_{R} e^{−x_{j}^{2}/2}dx_{j} = (2π)^{k/2} = kv_{k} ∫_{0}^{∞}r^{k−1}e^{−r^{2}/2}dr = v_{k}2^{k/2}Γ(k/2 + 1),

and find v_{k}.

g) It is known that

Γ(\frac{k}{2} + 1) ≈ \sqrt{2π}e^{−k/2}(\frac{k}{2})^{(k+1)/2}.

Conclude that v_{k} is roughly (2πe/k)^{k/2}. This is extremely small when k is large. What is going on?