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Complex analysis: Sketch the region in the complex plane

  • Thread starter Rubik
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  • #1
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Homework Statement


Sketch:
{z: [itex]\pi[/itex]?4 < Arg z ≤ [itex]\pi[/itex]}

Homework Equations





The Attempt at a Solution


Is it right to assume

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

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 [itex]\pi[/itex]/4 by rotating to the right in a clockwise rotation for [itex]\pi[/itex]/4 radians for the first condition and then rotating [itex]\pi[/itex] 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.
 

Answers and Replies

  • #2
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And I am also unsure if my radius is in fact a or am I missing an important step?
 
  • #3
tiny-tim
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Hi Rubik! :smile:

do you mean {z: [itex]\pi[/itex]/4 < Arg z ≤ [itex]\pi[/itex]} ?
… by rotating to the right in a clockwise rotation for [itex]\pi[/itex]/4 radians for the first condition and then rotating [itex]\pi[/itex] 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 :confused:

(and i don't understand where radius comes into it)
 
  • #4
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Oops yep I meant [itex]\pi[/itex]/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?
 
  • #5
tiny-tim
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{z : |z - 3 + i| < 4} Does this mean that z0 = (-3,i), and the radius = 4?
no, the centre is 3 - i
 
  • #6
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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?
 
  • #7
tiny-tim
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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? :confused:
 
  • #8
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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 [itex]\pi[/itex] going anti-clockwise from (0,0) and then another line in the direction of [itex]\pi[/itex]/4 from (0,0) Is that how the region is suppose to look?
 
  • #9
tiny-tim
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… I have a line in the direction of [itex]\pi[/itex] going anti-clockwise from (0,0) and then another line in the direction of [itex]\pi[/itex]/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:​
 
  • #10
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Oh okay thanks so much for all your help and sticking with me through all my confusion!! I appreciate it :D
 

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