Show that zn= rn(cos nθ + i sin nθ) by induction

1. HW2

(1) In this exercise, we are going to review some basic concept about complex numbers from high school.

(a) Show that the binomial theorem holds for complex numbers, i.e.

(z + w)ⁿ=

n

X

k=0

n k



z^kw^n−k for any complex numbers z, w ∈ C.

(b) (De Moivre’s Formula) Let z = r(cos θ + i sin θ) for r > 0 and 0 ≤ θ ≤ 2π. Show that zⁿ= rⁿ(cos nθ + i sin nθ) by induction.

(c) Let x = r cos θ and y = r sin θ. Assume that z = x + iy. Denote Cn(x, y) = Re(zⁿ) and S_n(x, y) = Im(zⁿ). Find the formula for C_n(x, y) and S_n(x, y) in terms of polynomial in x, y. For example, C2(x, y) = x²− y² and S2(x, y) = 2xy, and C₃(x, y) = x³− 3xy² and S₃(x, y) = 3x²y − y³.

(d) In fact,

Cn(r cos θ, r sin θ) = rⁿcos nθ, Sn(r cos θ, r sin θ) = rⁿsin nθ

according to the De Moivre’s Formula. Using exercise (c), write cos nθ and sin nθ in terms of polynomial in cos θ and sin θ respectively.

(2) Let D be any open subset of C. Define two partial differential operators on C¹(D, C) by

(1.1) ∂f

∂z = 1 2

 ∂f

∂x− i∂f

∂y

 , ∂f

∂z = 1 2

 ∂f

∂x+ i∂f

∂y

 .

(a) Recall that if a function f ∈ C¹(D, C) is holomorphic on D, then

∂f

∂z = 0.

Verify that the function f : C → C defined by x+iy 7→ x−iy is not holomorphic on C.

(b) Recall that the Laplacian of a C²-function u on D is defined to be

∆u = ∂²u

∂x² +∂²u

∂y². Show that

∆f = 4 ∂²f

∂z∂z, f ∈ C²(D; C).

Here we define ∆f = ∆u + i∆v when f (z) = u(z) + iv(z) for u, v ∈ C¹(D).

(c) Suppose g ∈ C²(D) and g is harmonic on D. Let f : U → D be a holomorphic function on an open set U of C whose image lies in D. Write f (z) = u(z) + iv(z) with u, v ∈ C²(U ). Define G = g ◦ f : U → R. Show that G is a harmonic function on U.

(3) (Check exercise 2.4 in Boas) Let D be the open disk {(x, y) : x²+ y²< 1}. The polar coordinate on D is given by

(1.2) x = r cos θ

y = r sin θ , where 0 ≤ r < 1 and 0 ≤ θ ≤ 2π.

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2

(a) Assume that u, v are real-valued C¹-functions on D, i.e. u, v ∈ C¹(D). Suppose that u, v satisfy the Cauchy-Riemann equation:

(1.3) ∂u

∂x = ∂v

∂y, ∂u

∂y = −∂v

∂x,

for (x, y) ∈ D. Define U (r, θ) = u(r cos θ, r sin θ) and V (r, θ) = u(r cos θ, r sin θ).

Show that

(1.4) ∂U

∂r = 1 r

∂V

∂θ , ∂V

∂r = −1 r

∂U

∂θ.

(b) Assume that u ∈ C²(D). U (r, θ) = u(r cos θ, r sin θ) be defined as above. Show that

u_xx+ u_yy = 1 r

∂

∂r

 r∂U

∂r

 + 1

r²

∂²U

∂θ².

(4) The space of polynomials in variable x, y with coefficients in R is denoted by R[x, y].

The Laplace operator ∆ defines a linear operator

∆ : R[x, y] → R[x, y].

A harmonic polynomial is a polynomial P (x, y) ∈ R[x, y] so that ∆P = 0. The space of harmonic polynomials with coefficients in R is denoted by V = ker ∆. A polynomial P (x, y) with real coefficients is homogeneous of degree d if P (λx, λy) = λ^dP (x, y) for all λ 6= 0. Let V_n be the set of all real homogeneous harmonic polyno- mials.

(a) Show that V_n forms a two dimensional real vector space. (Hint: let P (x, y) be a homogeneous harmonic polynomial of degree n. Denote P (r cos θ, r sin θ) = rⁿψ(θ). Use ∆P = 0.)

(b) Show that V =L∞

n=0Vn, where V0= R.

(5) (Dirichlet Problem on the unit disk D.) Solve the Dirichlet problem

(1.5) ∆u = 0 on D and u|∂D= ψ.

with the following given boundary conditions ψ and find their harmonic conjugates.

Also find holomorphic functions f (z) on D so that Re(f (x + iy)) = u(x, y).

(a) ψ(θ) = 1

2 + 3 cos θ − sin θ + 2 cos 4θ + 6 sin 5θ, 0 ≤ θ ≤ 2π. Express the solution u(x, y) in terms of polynomials {1, C_n(x, y), S_n(x, y) : n ≥ 1}.

(b) ψ(θ) = |θ| for −π ≤ θ ≤ π. (Extend ψ to a continuous period function on R.

Compute the Fourier expansion of ψ with respect to the interval [−π, π].) (6) Let V = C[0, 2π] be the space of real-valued functions on [0, 2π]. For any f, g ∈ V,

define

hf, gi = 1 π

Z 2π 0

f (x)g(x)dx.

(a) Show that (V, h·, ·i) forms a real inner product space.

(b) Show that {1, cos nx, sin nx, n ≥ 1} forms an orthogonal set in V (and hence it is a linearly independent set.)

(7) Let (X, d) be a metric space. Fix a ∈ X. For each p ∈ X, we define f_p(x) = d(x, p) − d(x, a), x ∈ X.

(a) Show that |f_p(x)| ≤ d(p, a) for all x ∈ X. Therefore f_p is a bounded function on X.

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(b) Let V the set of bounded real-valued functions on X. On V, define kf k_∞= sup

x∈X

|f (x)|.

Show that (V, k · k∞) is a real Banach space¹. (c) Show that for any p, q ∈ X

kf_p− f_qk∞= d(p, q).

(d) Let Φ : X → V be the function defined by p 7→ fp. The previous exercise shows that Φ is an isometry (and hence it is a homeomorphism from X onto Φ(X)).

Let Y be the closure of Φ(X) in V. Show that Y is complete. In other words, if we identity X with its image Φ(X), then X is dense in Y. Hence Y is the completion of X.

1A complete normed space over R (or C) is called a real (complex) Banach space