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Argand diagram regions

Hi,

I'm doing the following question:

maths2.png

I can do part a. Where I get a circle centre (9/8, 0), radius 3/8

I can sketch this fine, but I don't know how to plug in |z| < 3 in order to find the Region R.

I shaded in the inside of the circle but apparently its the region outside it. :frown:

Thanks,
Malc.
Reply 1
Original post by Sir_Malc
Hi,

I'm doing the following question:

maths2.png

I can do part a. Where I get a circle centre (9/8, 0), radius 3/8

I can sketch this fine, but I don't know how to plug in |z| < 3 in order to find the Region R.

I shaded in the inside of the circle but apparently its the region outside it. :frown:

Thanks,
Malc.


Literally just choose a test point that satisfies z<3|z| < 3. An obvious one is z=iz=i, this satisfies z=1<3|z| = 1 < 3. This is a point inside the circle z<3|z| <3.

Now where does this point map to? Well, w=zz+i=i2i=12w = \frac{z}{z+i} = \frac{i}{2i} = \frac{1}{2}, which is outside your ww-circle.

You've just shown that a point inside the z circle gets mapped to a point outside the w circle. It'd be nice if you could mark these points on your sketch to help you visualise what's going on.
Reply 2
Original post by Zacken
Literally just choose a test point that satisfies z<3|z| < 3. An obvious one is z=iz=i, this satisfies z=1<3|z| = 1 < 3. This is a point inside the circle z<3|z| <3.

Now where does this point map to? Well, w=zz+i=i2i=12w = \frac{z}{z+i} = \frac{i}{2i} = \frac{1}{2}, which is outside your ww-circle.

You've just shown that a point inside the z circle gets mapped to a point outside the w circle. It'd be nice if you could mark these points on your sketch to help you visualise what's going on.


Thankyou.

I tested a few other points and it works like you said.

Many thanks. :smile:
Reply 3
Original post by Sir_Malc
Thankyou.

I tested a few other points and it works like you said.

Many thanks. :smile:


Cheers! You'll only need to test one point in the actual exam. :smile:

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