Definition.  Let z=x+iy.  Then the analytic function w=f(z)=u(x,y)+iv(x,y) defines a transformation between the xy-plane and the uv-plane.  If the transformation is such that   the angle between two intersecting curves and  at equals the angle between the corresponding curves and  at and the sense of the angles are also preserved, then the transformation is called conformal at .

Related note: Transformations that preserve the magnitudes of angles but not necessarily the sense of the angles are called isogonal.

1-    Theorem   

Let  w=f(z )be a  transformation. At points where exists and is nonzero, this transformation is conformal .

Proof.  Let z=z(t) =x(t)+y(t), a≤t≤b  represent an arc C.  Thus the slope of the curve is .  So at any point where z(t)≠0, z(t) is tangent to C and arg z (t)  measures the angle that the tangent makes with the x-axis.  Now

Suppose≠ 0,  ≠0.  Then w()≠0 and

arg w()=arg f ()+arg z ().  That is,

arg f()= arg w()-arg z() 

which is the angle between the directed tangent to C at   and the directed tangent to the image of C at the image of  .  In other words, the directed tangent to any arc through is rotated by an angle arg f () that is independent of the choice of  arg z (through . In particular, if two arcs intersect at , the angle of intersection –in magnitude and sense- is preserved by the transformation and so this transformation is conformal.

2-          General equations of particular transformations

2.1     Translation

w=z+a  where a is a complex constant.

2.2     Rotation

where is an angle.  If >0, the rotation is in the counter-clockwise direction and if <0, the rotation is in the clockwise direction.

2.3     Stretching and contraction

w=az  where a is a positive number.  If a>1, the transformation is stretching while for 0<a<1, the transformation is a contraction.

2.4     Inversion

2.5     Linear Transformation

az+