Complex number epsilon-delta problem

[jjr]

Homework Statement

Prove that [itex] \lim_{z\rightarrow z_0} Re\hspace{1mm}z = Re\hspace{1mm} z_0 [/itex]

Homework Equations

It is specifically mentioned in the text that the epsilon-delta relation should be used,

[itex] |f(z)-\omega_0| < \epsilon\hspace{3mm}\text{whenever}\hspace{3mm}0<|z-z_0|<\delta [/itex].

Where [itex] \lim_{z\rightarrow z_0}f(z) = \omega_0 [/itex]
Other equations that might be useful are
[itex] |z_1+z_2| \leq |z_1| + |z_2|\hspace{3mm}\text{(Triangle inequality)} [/itex]
and perhaps
[itex] Re\hspace{1mm}z \leq |Re\hspace{1mm} z| \leq |z| [/itex]

The Attempt at a Solution

Here [itex] \omega_0 = Re\hspace{1mm}z_0[/itex] and [itex] f(z) = Re\hspace{1mm} z [/itex],
so we want to find a delta [itex] |z-z_0| < \delta [/itex] such that [itex] |Re\hspace{1mm}z - Re\hspace{1mm}z_0| < \epsilon [/itex].

I am honestly not sure how to approach this. Any clues would be very helpful.

Thanks,
J

 

[Svein]

Start with the fact that lim_z→z0z = z₀.

 

[Dick]

Homework Statement

Prove that [itex] \lim_{z\rightarrow z_0} Re\hspace{1mm}z = Re\hspace{1mm} z_0 [/itex]

Homework Equations

It is specifically mentioned in the text that the epsilon-delta relation should be used,

[itex] |f(z)-\omega_0| < \epsilon\hspace{3mm}\text{whenever}\hspace{3mm}0<|z-z_0|<\delta [/itex].

Where [itex] \lim_{z\rightarrow z_0}f(z) = \omega_0 [/itex]
Other equations that might be useful are
[itex] |z_1+z_2| \leq |z_1| + |z_2|\hspace{3mm}\text{(Triangle inequality)} [/itex]
and perhaps
[itex] Re\hspace{1mm}z \leq |Re\hspace{1mm} z| \leq |z| [/itex]

The Attempt at a Solution

Here [itex] \omega_0 = Re\hspace{1mm}z_0[/itex] and [itex] f(z) = Re\hspace{1mm} z [/itex],
so we want to find a delta [itex] |z-z_0| < \delta [/itex] such that [itex] |Re\hspace{1mm}z - Re\hspace{1mm}z_0| < \epsilon [/itex].

I am honestly not sure how to approach this. Any clues would be very helpful.

Thanks,
J

Think about it. Which is larger ##|z-z_0|## or ##|Re(z)-Re(z_0)|##?

 

[jjr]

Well, the distance between two complex points is definitely larger or equal to the distance between the real parts of the same points...

 

[Dick]

Well, the distance between two complex points is definitely larger or equal to the distance between the real parts of the same points...

Ok, so how small shall we make ##|z-z_0|## to make sure that ##|Re(z)-Re(z_0)| \lt \epsilon##?

 

[jjr]

Hmm. I suppose that if we set [itex] |z-z_0| [/itex] smaller than [itex] \epsilon [/itex], then [itex] |Re(z)-Re(z_0)| [/itex] would have to be smaller than [itex] \epsilon [/itex]. I.e. [itex] \delta = \epsilon[/itex]?

 

[Dick]

Hmm. I suppose that if we set [itex] |z-z_0| [/itex] smaller than [itex] \epsilon [/itex], then [itex] |Re(z)-Re(z_0)| [/itex] would have to be smaller than [itex] \epsilon [/itex]. I.e. [itex] \delta = \epsilon[/itex]?

Yes.

 

[jjr]

Thank you. Not too hard to see really, but I'm still not sure how to show it. Is the statement I made sufficient? I feel like it's lacking some rigor

 

[Dick]

Thank you. Not too hard to see really, but I'm still not sure how to show it. Is the statement I made sufficient? I feel like it's lacking some rigor

You just need to show ##|Re(z)| \le |z|##.

 

[jjr]

Okay, I think I've got my head wrapped around it. I have another one, if you would care to have a look:

To be shown:
##\lim_{z\rightarrow z_0} \bar{z}=\bar{z_0}##

So
## |\bar{z}-\bar{z_0}|<\epsilon \to |z-z_0|<\delta##

My work:
I'm trying to find some relation between ##z,z_0## and their conjugates, so I tried setting
##\bar{z}=z-2Im(z)##
##\to |\bar{z}-\bar{z_0}| = |z-z_0-2Im(z)+2Im(z)|##.
Here I tried invoking the triangle inequality, but the expression I ended up with didn't help much.
Suggestions?

Thanks,
J

edit: typo

 

[Svein]

Hint: [itex]\lvert z-z_{0}\rvert ^{2}=(z-z_{0})(\bar{z}-\bar{z_{0}})=\lvert \bar{z}-\bar{z_{0}}\rvert ^{2} [/itex]

 

[jjr]

Thank you. I suppose one could take the square root from both sides and obtain ##|z-z_0| = |\bar{z}-\bar{z_0}|##, which in turn enables us to set ##\delta = \epsilon##? Thinking about this again I realize that this makes sense, that the distance between two points ##z, z_0## should be the same as the distance between the same two points mirrored about the imaginary axis.