Conformal Mapping Homework: f(z) = 1/(z-1), c=i

[desaila]

Homework Statement

"Study the infinitesimal behavior of f at the point c. (In other words, use the conformal mapping theorem to describe what is happening to the tangent vector of a smooth curve passing through c.)"

f(z) = 1/(z-1), c=i

Homework Equations

|f'(c)| and arg f'(c)

The Attempt at a Solution

I know what |f'(c)| is, d/dz is -1/(z-1)^2, and evaluates out to 1/2i. However, I'm not sure what exactly that's saying about the behavior. Does it mean it's shrinking on the imaginary axis by 1/2 ?

Also, about the argument... this is something I can't quite wrap my head around. I've read in this math text, and the wiki entry on arguments, but I'm not quite sure I get it. The equation in this book is, the argument of z = |z|(cos(theta)+ i*sin(theta)) where |z| = sqrt(x^2+y^2).

Thanks.

 

[HallsofIvy]

Since the problem specifically says "use the conformal mapping theorem", what is the conformal mapping theorem and how does it apply to this problem?

I very much doubt that your book says that the "argument" of x. If z= a+ bi, in polar form is r (cos(\theta)+ i sin(\theta))= re^{i\theta} then the "modulus" of z is r= \sqrt{a^2+ b^2} and the "argument" of z is \theta= arctan(b/a). By the time you are working with "conformal mapping", that should be old stuff.

 

[desaila]

It is old stuff, but I didn't quite understand it then. The conformal mapping theorem, according to the book, is, "If f is analytic in the disc |z-zo|<r and if f'(zo) != 0, then f is conformal at zo."

Where zo is z with subscript 0.