# Complex analysis r-neighbourhood problem

• Nov 1st 2012, 08:58 AM
paulluap1991
Complex analysis r-neighbourhood problem
hi, i need some help with the following problem:

Let

z = a + ib , w = x + iy be two complex numbers that belong tothe neighbourhoodN(0, r).

Show that at least one of
a + iy; x + ib also lies in N(0, r).

i'm not sure where to begin with solving this problem, any help would be appreciated.
• Nov 1st 2012, 09:31 AM
Plato
Re: Complex analysis r-neighbourhood problem
Quote:

Originally Posted by paulluap1991
Let
[LEFT][B]
z = a + ib , w = x + iy be two complex numbers that belong tothe neighbourhoodN(0, r).

Show that at least one of
a + iy; x + ib also lies in N(0, r[FONT=CMR12]).

From the given you know that $\displaystyle a^2+b^2<r^2\text{ and }x^2+y^2<r^2$.

Now you want to show that $\displaystyle a^2+y^2<r^2\text{ OR }x^2+b^2<r^2$.

Suppose that $\displaystyle a^2+y^2\ge r^2\text{ and }x^2+b^2\ge r^2$.
• Nov 1st 2012, 09:32 AM
HallsofIvy
Re: Complex analysis r-neighbourhood problem
I would start by drawing a circle, representing the neighborhood, in the complex plane and look at pairs of points. Given points (a, b) and (x, y) the points (a, y) and (x, b) together with them form a rectangle. Can you find points (a, b) and (x, y) so that not both (a, y) and (x, b) are in N?