mathmatical physics

1. NPTEL – Physics – Mathematical Physics - 1
Lecture 41
Conformal Mapping
In this section we shall introduce the concept of conformal mapping. Mainly we shall
focus on the connection between such mappings and harmonic functions. We shall
conclude our discussion with a short note on the application of conformal mappings on
physical problems.
Mapping of functions: Mappings that preserve angles and their directions, but not
necessarily the scale are called conformal mapping. Formally we can explain in
the following way. Consider the set of equations,
𝑢 = 𝑢(𝑥, 𝑦) (1)
𝑣 = 𝑣(𝑥, 𝑦)
Where 𝑢 and 𝑣 are functions of 𝑥 and 𝑦 . The above equations represent mapping
of points between (𝑢, 𝑣) and (𝑥, 𝑦) planes. An inverse transformation can trivially
be defined as,
𝑥 = 𝑥(𝑢, 𝑣) (2)
𝑦 = 𝑦(𝑢, 𝑣)
It can happen that each point in the plane corresponds to each point in the 𝑥𝑦 plane or a
set of points in one plane correspond to a set in the other plane. In this case each point or
the set of points is said to be images of each other.
As per the transformation defined by Eqs. (1) and (2) a region or a curve in the uv plane
can be mapped into region or a curve respectively in the 𝑥𝑦 plane. The Jacobian for the
transformation in Eq. (1) is given by
𝜕𝑢 𝜕𝑢
𝜕(𝑢, 𝑣)
𝜕(𝑥, 𝑦)
= |
𝜕𝑥 𝜕𝑦
𝜕𝑣 𝜕𝑣
𝜕𝑥 𝜕𝑦
|
=
𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑣
𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥
− (3)
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2. NPTEL – Physics – Mathematical Physics - 1
By transformation defined by Eq. (1) let a point (𝑥0, 𝑦0) in the 𝑥𝑦 plane is
mapped onto a point (𝑢0, 𝑣0) in the uv plane and further two curves 𝐶1 and 𝐶2 in 𝑥𝑦
plane map onto two other curves 𝐶̃1and 𝐶̃2 in the 𝑢𝑣 plane. Let us assume that the curves
meet at (𝑥0, 𝑦0) and (𝑢0, 𝑣0) and further the angles surrounded by the respective set
of curves are same in both magnitude and direction as shown in the figure.
In this case, the mapping is said to be conformal at (𝑥0, 𝑦0). In some other
mapping, where only the magnitude of the angle is preserved but not the direction
is called as isogonal mapping. The results can trivially be extended to the case of
complex variables, where it can be stated as in the following.
Let a curve 𝐶 be presented by a parametric equation,
𝑧 = 𝑧(𝑡) (𝑎 ≤ 𝑡 ≤ 𝑏)
And 𝑓(𝑧) be a function defined at all points z on 𝐶.
Then
(4)
𝑤 = 𝑓[𝑧(𝑡)] (5)
is the equation of the image 𝑐̃ of 𝐶 under the transformation 𝑤 = 𝑓(𝑧) (in the
same sense of Eq. (1)). This transformation is said to be conformal at a point 𝑧0 if
𝑓(𝑧) is analytic there and 𝑓′(𝑧) ≠ 0. Also this transformation is conformal
at each point in the neighborhood of 𝑧0.
Example 1 : Mapping of a curve
The mapping 𝜔 = 𝑒𝑧 is conformal throughout the entire z plane since 𝑓′(𝑧) =
𝑒𝑧 ≠ 0 for all 𝑧. Now consider two lines 𝑥 = 𝐶1 and 𝑦 = 𝐶2 in the z plane as
shown in figure. The images of these curves 𝐶1 and 𝐶2 can be obtained as
follows.
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3. NPTEL – Physics – Mathematical Physics - 1
𝜔 = 𝑒𝑧 can be written in polar form as
𝜌𝑒𝑖𝜑 = 𝑒𝑥+𝑖𝑦 with 𝜔 = 𝜌𝑒𝑖𝜑
and 𝑧 = 𝑥 + 𝑖𝑦
thus 𝜌 = 𝑒𝑥 and 𝜑 = 𝑦
The image of a point 𝑧 = (𝑐, 𝑦) on a vertical
line 𝑥 = 𝐶1 has the following polar
coordinates
𝜌 = 𝑒𝑐1
𝜑 = 𝑦 in the y- plane
So the amplitude is constant and the phase
angle increases from 0 to 2𝜋. Thus the image
of the line 𝑥 = 𝐶1 in the uv plane is a circle of
radius 𝑒𝑐1 moving counterclockwise along the
circle as shown in fig. (2)
Each point on the circle is the image
of an
infinite number of points, spaced 2𝜋 units
apart alone the line.
Example 2 : Mapping of an area
Let us take the shaded region in the upper half plane as shown in the figure (Fig (iii)).
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4. NPTEL – Physics – Mathematical Physics - 1
Let 𝑧0 be an arbitrary point P in the upper
half of the z- plane denoted by R
The transformation 𝜔 = 𝑒𝑖θ0 (𝑧−𝑧0
)
𝑧−̅𝑧̅0̅
Maps onto the interior of a region 𝑅′ of unit circle |𝜔| = 1. Each point on the x axis, A,
B, C, D, E, F etc is mapped onto the boundary of the circle as shown in the figure
The angle 𝜃0 is determined by making one particular point on the 𝑥 − axis correspond to
a given point on the circle. Again as in example 1, 𝜔 moves anticlockwise along the unit
circle.
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