Conformal mapping w=1/z - question.

[peripatein]

Hi,

Homework Statement

I'd like to show that the mapping w=u+iv=1/z tranforms the line x=b in the z plane into a circle with radius 1/2b and center at u=1/2b

Homework Equations

The Attempt at a Solution

z*w=1=(b+iy)(u+iv)
→ 1=|(bu-yv)+i(bv+yu)|
→ u²+v²=1/(b²+y²)
Now, a circle with radius 1/2b and center at u=1/2b in the w plane would have the following form:
(u-1/2b)²+v²=1/4b²
→ u²+v²=u/b
My problem is now explicitely showing that
1/(b²+y²)=u/b
Which I am unable to do :(.
Any advice? (By the way, I have tried using polar coordinates too, to no avail. I happen to prefer this approach.)

 

[Dick]

Hi,

Homework Statement

I'd like to show that the mapping w=u+iv=1/z tranforms the line x=b in the z plane into a circle with radius 1/2b and center at u=1/2b

Homework Equations

The Attempt at a Solution

z*w=1=(b+iy)(u+iv)
→ 1=|(bu-yv)+i(bv+yu)|
→ u²+v²=1/(b²+y²)
Now, a circle with radius 1/2b and center at u=1/2b in the w plane would have the following form:
(u-1/2b)²+v²=1/4b²
→ u²+v²=u/b
My problem is now explicitely showing that
1/(b²+y²)=u/b
Which I am unable to do :(.
Any advice? (By the way, I have tried using polar coordinates too, to no avail. I happen to prefer this approach.)

I'm not sure where you are really going there. If w=1/z and z=b+iy and w=u+iv, then w=1/(b+iy). Split that into real and imaginary parts to get your u and v.