1. ## Conformality

1) Can anyone think of a function of a complex variable which is conformal at some points but not at others? Please indicate the point/s at which it is not conformal and the points at which it is.

2) Consider a function of a complex variable and a point a in its domain. Can anyone think of a mapping which preserves angles between two particular curves intersecting at a but for which angle preservation does not occur for all pairs of curves intersecting at the same point a ?

2. Originally Posted by Ian Moore
1) Can anyone think of a function of a complex variable which is conformal at some points but not at others? Please indicate the point/s at which it is not conformal and the points at which it is.
$f(z)=z^2$.

2) Consider a function of a complex variable and a point a in its domain. Can anyone think of a mapping which preserves angles between two particular curves intersecting at a but for which angle preservation does not occur for all pairs of curves intersecting at the same point a ?
Same function works.

3. In "1) Can anyone think of a function of a complex variable which is conformal at some points but not at others? Please indicate the point/s at which it is not conformal and the points at which it is " I should have been more precise. I am after a non-analytic function satisfying the requirements previously outlined.

In "2) Consider a function of a complex variable and a point a in its domain. Can anyone think of a mapping which preserves angles between two particular curves intersecting at a but for which angle preservation does not occur for all pairs of curves intersecting at the same point a ? I should have added here, Can you give an example of a point and of a pair of curves for which angle preservation holds as well as an example of a pair of curves through the same point where angle preservation does not occur.

4. In that case use f(z) = |z|^2

5. Thankyou Perfect Hacker.