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.

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 $\displaystyle a^2+y^2\ge r^2\text{ and }x^2+b^2\ge r^2$.

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?