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
equals the angle between the corresponding curves
and the sense of the angles are also preserved, then the
transformation is called conformal at
Transformations that preserve the magnitudes of angles but
not necessarily the sense of the angles are called
)be a transformation. At points where
exists and is nonzero, this transformation is conformal
=x(t)+y(t), a≤t≤b represent an arc C.
slope of the curve is
So at any point where z(t)≠0,
is tangent to C and arg z
measures the angle that the tangent makes with the
the angle between the directed tangent to C at
and the directed tangent to the image of C at the
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
equations of particular transformations
where a is a complex constant.
is an angle. If
the rotation is in the counter-clockwise direction and if
the rotation is in the clockwise direction.
where a is a positive number. If a>1, the
transformation is stretching while for 0<a<1,
the transformation is a contraction.