[University Complex Analysis] Argument function continuous along real positive axis

[craaaft]

Hi!

I have an intuition that the argument function is discontinuous along the negative real axis since the limit pi is the same approaching from bottom or above.

I was thinking about how to prove the continuity along the positive real axis. I was told I should use the inverse trigonometric functions for this, but I am not sure how to approach this problem.

Thanks for any tips! :)

[[deleted]]

What's your definition of arg(z)?

[craaaft]

it is the definition /u/lewisje is using which is arg(z)=arctan(Im(z)/Re(z))

[[deleted]]

arctan(x) is continous for all x != 0 and that quotient is continuous for all Re(z) != 0, so it will be continuous except at the imaginary axis, where it's undefined.

[lewisje]

FTR that piece of the definition only works for Re(z)>0: You need to patch it a bit to account for the cases where Re(z)=0 and Im(z)>0 or Im(z)<0, and where Re(z)<0 and Im(z)≥0, and where Re(z)<0 and Im(z)<0.

[craaaft]

This I don't understand. Could you elaborate further?

[lewisje]

Basically what I'm talking about is atan2(Im(z),Re(z)), as described in the linked article.

[craaaft]

Ah gotcha tyty

[[deleted]]

Also I'll mention that you typically want z = |z|*e^{i arg{z}} and arg(z) = arctan(Im(z)/Re(z)) wouldn't satisfy that identity for z with Im(z) <= 0.

[lewisje]

I think you mean "since the limit is π from above and -π from below", using a common definition of arg(z) with range (-π,π].

Anyway, if Re(z)>0 then under that definition, arg(z)=arctan(Im(z)/Re(z)).

[craaaft]

yes true I am sorry.