Image of a line

I think arg(z)=0 means z is a positive real number, and Im(z)=0, so instead of the usual z=x+yi, z=x in this case. Then you should be able to simplify it and match it to u+iv.

Yeah, and I expect after simplifying the e bit and separating the whole thing into the real and imaginary parts you'll have something to work with

I'd expand and match the real parts to u (real part of w) and match the imaginary parts to v (imaginary part of w) and see if I can form some relationship.

I'd expand and match the real parts to u (real part of w) and match the imaginary parts to v (imaginary part of w) and see if I can form some relationship.

Hi guys!

I have a question, if a complex function corresponds to a positive rotation about the point i by pi/4 radians. I get that the overall transformation is
(z-i)*(e^pi/4*i)+i

But I don't understand how to find the image of the line arg(z)=0 under f(z)?

What method can I use to do it algebraically?
I'm getting the answer as Re(z)

But honestly not sure with what I'm doing!

If anyone could help that you be great!

Can you not just use the transformation directly
z = x*e^i0
So
(z-i)*(e^pi/4*i)+i = z*(e^pi/4*i) + i-ie^pi/4*i =
x*e^pi/4*i + offset

Oh thank you, I see. But how would I get the image of the line from this?
Sorry this question I've been working on all day so I have been going round in circles

x is the non-negative modulus (real component of original line). The transformed line is
offset + half line at 45 degrees
So just work out the offset (rotation of the orign).

Would it be pi/4?
As working it out to get i=ie^pi/4i

Would what be pi/4? You can do simple geometry to get the rotation of the origin by 45, or use the expression a couple of posts ago. It should be straightforward either way.

The angle of rotation?
Ah I'll keep on working on it!
Thanks for the help

Not sure what you don't understand?

I just don't get how to get the image of the line when arg(z)=0
My answers have been all over the place hahaha

Copied from #9,

Can you not just use the transformation directly
z = x*e^i0
So
(z-i)*(e^pi/4*i)+i = z*(e^pi/4*i) + i-ie^pi/4*i = x*e^pi/4*i + offset

Which part don't you understand? I've left the offset (rotation of origin) for you to calculate.

From calculating the offset
I get ie^pi/4i=i
Do I then solve this?
To get -1+root2

ie^pi/4i=i
How?

ie^pi/4i = e^i3pi/4
As i = e^ipi/2