Need some help with basic complex variables (no proofs)

[nocheesie]

need some urgent help with basic complex variables (no proofs)

Hi:
can someone give me examples of the following? (no proofs needed)

1. a non-zero complex number z such that Arg(z^2) "not equal to" 2 Arg z
2. a region in C which is not a domain
3. a non-empty subset of C which has no accumulation points
4. a continuous function f: C -> C which is not differentiable anywhere
5. an entire function which is not a polynomial.

thanks a lot in advance!

 

[mikeu]

1. a non-zero complex number z such that Arg(z^2) "not equal to" 2 Arg z

Arg z is defined to be the angle between -pi and pi which is equivalent to the actual argument of z. So if z=e^{i\pi}\implies z^2=e^{2i\pi} then Arg z = arg z = pi, but arg z² = 2pi so Arg z² = 0.

2. a region in C which is not a domain

If I recall correctly a domain has to be connected, so something like \left\{z\in\mathbb{C}:|z-2|<1\right\}\cup\left\{z\in\mathbb{C}:|z+2|<1\right\} (two discs of radius 1 centred at 2 and -2 on the real axis) would qualify.

3. a non-empty subset of C which has no accumulation points

I think an accumulation point only makes sense with respect to a sequence... Off the top of my head I can't think of a set on which no sequence could have an accumulation point, since there is always the possibility of a constant sequence. I could be wrong though...

4. a continuous function f: C -> C which is not differentiable anywhere

Technically f(z)=Re(z) qualifies, although it is only onto \mathbb{R}, not \mathbb{C}. But since \mathbb{R}\subset\mathbb{C} it can be thought of as a function f:C->C which is just not surjective.

5. an entire function which is not a polynomial.

I'm pretty sure that e^z is entire.

 

[nocheesie]

wow thank you!
i still have no idea how to do the accumulation one. i looked up the definition of accumulation point and it says:
A point z0 is said to be an accumulation point of a set S if each deleted neighborhood of z0 contains at least one point of S
i'm not sure if i understand that... any ideas?

 

[matt grime]

A z is an accumulation point of S if every open disc centred on z contains some element of S. Every point of S is an accumulation point of S. S is closed if it contains all its accumulation points

 

[shmoe]

A z is an accumulation point of S if every open disc centred on z contains some element of S. Every point of S is an accumulation point of S. S is closed if it contains all its accumulation points

nocheesie has a different definition- the 'deleted disc' part is important or the question is false.


Equivalent definition (worth proving equivalence if you want a better handle on accumulation points): z is an accumulation point of S if and only if there is a sequence in S minus z that converges to z.

I can't think of a good hint without giving it away, think very simple sets.

 

[matt grime]

So it corrects to "contains some element of S distinct from z".

 

[nocheesie]

so would the set of say, {1} work? i don't know if I'm understanding this correctly... if you have {1} then there would be no accumulation points right? since there's only 1 itself?

 

[matt grime]

Any disc centred on 1, would not contain any of the other points of S, since there is none. And if z is any other point not equal to 1, then |z-1| is strictly positive, say it equals r. Then the disc of radius r/2 about z does not contain 1 (the only point in S) so it is not an accumulation point of S. Thus S has no accumulation points. Is that what you were thinking?

 

[nocheesie]

Yes! that's what i was thinking =) since there's only 1 then it can't have any points in the set other than 1, thus no accumulation points! i think I'm finally getting this... thanks everyone very very much!

 

[Hurkyl]

Ok, you learned your lesson! The other thread is here.

https://www.physicsforums.com/showthread.php?t=64106