Help with Describing Image of |z-1| ≤ 1 in Complex Plane

[simick1712]

Hi everyone - I'm sure there's somebody here who can help with such a trivial question.

It's not a homework question before that's assumed - it's from a past exam paper, which I'm using for revision, sans answers.

It asks to describe, in the complex plane, the image of:

[tex] |z - 1| \leq 1 [/tex]

under the transformation

[tex] w = \frac{1}{z} [/tex]

Now, I found the answer to be:

[tex] Re(w) \leq \frac{1}{2} [/tex]

Is this correct?

Thanks for any help,

Simon.

 

[HallsofIvy]

Are you sure about the direction of that inequality? z= 1 itself satisfies |z-1|=0< 1 and z= 1 is mapped into w= 1 which has real part 1> 1/2.

 

[simick1712]

Oops - no, I'm not sure of that at all - what I actually meant (of course!) was

[tex] Re(w) \geq \frac{1}{2} [/tex]

 

[HallsofIvy]

Yep. If z= x+ iy, then [itex]w= \frac{1}{z}= \frac{1}{x+iy}\frac{x-iy}{x-iy}= \frac{x-iy}{x^2+y^2}[/itex] which has real part [itex]\frac{x}{x^2+ y^2}[/itex].

If z satisfies [itex]|z-1|\le 1[/itex] then [itex]|z-1|= \sqrt{(x-1)^2+ y^2}\le 1[/itex] so [itex](x-1)^2+ y^2= x^2+ y^2- 2x+ 1\le 1[/itex].
That is, [itex]x^2+ y^2\le 2x[/itex] and therefore [itex]\frac{x}{x^2+ y^2}\ge \frac{1}{2}[/itex].

 

[simick1712]

Excellent - thanks very much for your help.

Simon.