MAT 542  Assignment No. 4  Due: March 4, 2002

(1) Let γ be the boundary of the square with vertices at z = 0, z = 1, z = 1 + ı, z = ı oriented counter-clockwise. Evaluate

a) ∫_{γ} πe^{πz}dz,

b) ∫_{γ} {(y − x) − 3x^{2}ı} dz.

(2) Suppose

γ(θ) = 2e^{ıθ}, 0 ≤ θ ≤ π.

Evaluate ∫_{γ} \frac{(z+2)}{z}dz.

(3) Show that if γ is a continuous path in an open set D and ω_{1} and ω_{2} are closed forms in D, then

∫_{γ}(Aω_{1} + Bω_{2}) = A ∫_{γ} ω_{1} + B ∫_{γ} ω_{2},

where A and B are complex numbers.

(4) Show that the integral of (z +1)^{−2} along the imaginary axis oriented in the positive y-direction can be defined as an improper integral and evaluate it.

(5) Let γ be a piecewise differentiable path whose range is contained in an open set D. Let w = Q(z) be holomorphic in D with values in an open set V of the w-plane. Show that Γ = Q ∘ γ is a piecewise differentiable path in V, and if f is continuous in V then

∫_{Γ} f(w)dw = ∫_{γ} f (Q(z)) Q′(z)dz.

(6) Suppose γ_{0} and γ_{1} are continuous closed paths in an open set D and are homotopic as closed paths in D. Show that if ω is a closed form in D, then ∫_{γ_{0}} ω = ∫_{γ_{1}} ω.

(7) A function f on a domain D is called \underline{anti-holomorphic} if \overline{f(z)} is a holomorphic function of z on D. Determine all the functions f on D that are both holomorphic and anti-holomorphic.

(8) Let D be a non-empty connected open set in ℂ. Assume that whenever z ∈ D, so does z̄.

a) Show that D must contain a non-empty open interval I of ℝ.

b) Let g(z) = \overline{f(z̄)}. Show that f is holomorphic on D if and only if g is. c) Show that a holomorphic function f on D satisfies \overline{f(z̄)} =

f(z) for all z ∈ D if and only if f is real on D ∩ ℝ.

(9) a) Let f be a holomorphic function on the annulus A = {z ∈ ℂ; r_{1} < |z| < r_{2}} with 0 < r_{1} < r_{2} < ∞. Assume that ∫_{γ} f(z)dz = 0, where γ(t) = \frac{r_{1}+r_{2}}{2} e^{2πıt} , t ∈ [0, 1]. Show that there exists a holomorphic function F on A such that F′ = f.

b) Let A = algebra of holomorphic functions on A, and

A_{o} = {f ∈ A; there exists an F ∈ A with F′ = f} .

Compute dim_{ℂ}A/A_{o}.

c) Can you generalize b) to domains of finite connectivity? State the result and say why it should hold (complete proof not required).