Complex analysis: Sketch the region in the complex plane

[Rubik]

Homework Statement

Sketch:
{z: \pi?4 < Arg z ≤ \pi}

Homework Equations

The Attempt at a Solution

Is it right to assume

z₀ = 0 ; a = a (radius = a) ; and taking \alpha = \pi/4 ; \beta = \pi

And now in order to sketch the problem after setting up the complex plane is it correct to to plot z₀ at the origin and then from the origin plot \pi/4 by rotating to the right in a clockwise rotation for \pi/4 radians for the first condition and then rotating \pi to the left from the origin (anti-clockwise rotation) for the second condition and then using a solid or dashed line according to the strictly < or ≤ conditions and this gives me the correct region?

Basically I am confused as to how to rotate the angle in terms of clockwise or anti-clockwise according to the conditions given.

 

[Rubik]

And I am also unsure if my radius is in fact a or am I missing an important step?

 

[tiny-tim]

Hi Rubik!

do you mean {z: \pi/4 < Arg z ≤ \pi} ?

… by rotating to the right in a clockwise rotation for \pi/4 radians for the first condition and then rotating \pi to the left from the origin (anti-clockwise rotation) for the second condition …

no, everything is always anti-clockwise

i'm worried why you thought it wasn't ​

(and i don't understand where radius comes into it)

 

[Rubik]

Oops yep I meant \pi/4.. I was worried asking it haha it has been a long time since I have had to work with complex numbers.. Another thing I have just come across is the region {z : |z - 3 + i| < 4} Does this mean that z₀ = (-3,i), and the radius = 4?

 

[tiny-tim]

{z : |z - 3 + i| < 4} Does this mean that z₀ = (-3,i), and the radius = 4?

no, the centre is 3 - i

 

[Rubik]

With the first part from your first reply I said radius = a because I am trying to sketch the particular region covered by these angles or is that wrong?

 

[tiny-tim]

With the first part from your first reply I said radius = a because I am trying to sketch the particular region covered by these angles or is that wrong?

I still don't understand this at all.

What is a, and what has the radius to do with anything?

 

[Rubik]

Well I am not sure I just took it as an assuption.. See if I try and sketch this region I draw both these angles taking them anti-clockwise from the origin, which leaves a region in the 1st and 2nd quadrants and I am just confused as I thought I was suppose to be left with a closed region but is this not the case? I am sorry if this still makes no sense it is hard to explain a drawing in words. :/ So currently I have a line in the direction of \pi going anti-clockwise from (0,0) and then another line in the direction of \pi/4 from (0,0) Is that how the region is suppose to look?

 

[tiny-tim]

… I have a line in the direction of \pi going anti-clockwise from (0,0) and then another line in the direction of \pi/4 from (0,0) Is that how the region is suppose to look?

Yup. There's no restriction on |z|, so the region goes to infinity.

Goodnight! :zzz:​

 

[Rubik]

Oh okay thanks so much for all your help and sticking with me through all my confusion! I appreciate it :D