Complex Analysis: Finding Arg(z)

[Iron Eagle]

Hello everyone,

I am trying to solve this follow problem, but don't quite know how to go about getting Arg(z).

z = 6 / (1 + 4i)

I got that lzl is sqrt((6/17)^2+(-24/17)^2) but am stuck with finding Arg(z). It told me to recall that -pi < Arg(z) <= pi

Can you guys teach me how to go about finding arrrggg... this Arg(z)?

TIA.

 

[Hurkyl]

Is there any sort of number for which you know how to compute Arg?

 

[Iron Eagle]

This Arg(z) thing is new to me, so no. I'm trying to find out what it is. Thanks.

 

[Iron Eagle]

I found this on Google:

The argument of a complex number is the angle between the positive x-axis and the line representing the complex number on an Argand diagram. It is denoted arg (z)

So Arg(z) is pretty much just the angle theta that r makes with the x-axis?

Assuming that arg(z) is not case sensitive, then is it simply arctan(-4) in this case?

 

[Hurkyl]

In this case, yes, Arg(z) = Arctan(-4). (but be wary; for some z, Arg(z) cannot equal the Arctangent of anything at all! You have to make sure you take the right value for the arctangent)

Arg/arg is case sensitive, just like other complex functions. arg is a "multi-valued function", whereas Arg picks out a specific value.

 

[Iron Eagle]

Thank you so much Hurkyl... your help is much appreciated! In the meantime, I also found http://scholar.hw.ac.uk/site/maths/topic11.asp?outline=no" page, which led me to believe that I was right. Your reply, however, gave a positive confirmation of my guess. Thanks again - now I have a much better grasp of this concept.

Yang