Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment X.

1. Problem 1. a) Map the inside of the circle r = a cos ϕ, a > 0, on the inner part of cardioid ρ = (1 + cos θ)/2;

b) Map the inside of the circle r = a cos ϕ, a > 0, on the inner part of the right branch of the lemniscate ρ = \sqrt{cos(2θ)};

c) Map |z| < 1 onto the inner part of the cardioid ρ = A(1 + cos θ), A > 0, in such a way that w(0) = A/8, w′(0) > 0.

2. Problem 2. a) Find the image of |z| < 1 under w = R(z + mz^{2}), R > 0, 0 ≤ m ≤ 1/2. Find the image of the polar net of z-plane;

b) Find the image of the semicircle |z| < 1, ℜz > 0, under w = z + z^{2} ;

c) Find the image of |z| < 1 under w = R(z + z^{n}/n), R > 0, n ∈ N;

d) Find the image of |z| > 1 under w = R(z + 1/(nz^{n})), R > 0, n ∈ N.

3. Problem 3. a) Map 0 < arg z < πβ, 0 < β < 2, with a cut along the arc of a circle |z| = 1 from z = 1 to z = e^{iα}, 0 < α < π onto the upper half-plane;

b) Map |z| < 1, with a cut along the radius [−1, 0] and a segment [a, 1], 0 < a < 1, onto the upper half-plane;

4. Problem 4. a) Find the image of |z| < 1 under w = z/(z^{2} + 1);

b) Find the image of the sector −π/n < arg z < π/n, |z| < 1, under w = z/(1 + z^{n})^{2/n}, w(z) > 0 for z > 0; Hint: Consider z_{1} = z^{n}, z_{2} = z_{1}/(1 + z_{1})^{2}, z_{3} = z_{2}^{1/n}. c)* Map the part of the z-plane to the left of the right-hand branch of the hyperbola x^{2} − y^{2} = 1 on a half-plane; Hint: Consider on one side the mapping of the upper half of the region by w = z^{2}, on the other side the mapping of a quadrant by w = z^{3} − 3z.