**
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+* |