Conformal Mapping

Conformal mapping is a function defined on the complex plane which transforms a given curve or points on a plane, preserving each angle of that curve. If f(z) is a complex function defined for all z in C, and w = f(z), then f is known as a transformation which transforms the point z = x + iy in z-plane to w = u + iv in w-plane. If this transformation preserves the angles between curves in both magnitude and sense (clockwise or counterclockwise), then the mapping is called conformal mappings.

If there is a transformation such that the magnitude of the angles between the curves is identical, but the sense (clockwise or counterclockwise) is opposite. In that case, the transformation is called isogonal.

Definition of Conformal Mapping

Consider the transformations u = u(x, y) and v = v(x, y), which maps a point P(x_o, y_o) in the z-plane to a point P’(u_o, v_o) in the w-plane. Let curves C₁ and C₂ intersect at point z_o = (x_o, y_o) in z-plane is mapped into curves C₁’ and C₂’ in w-plane intersecting at w_o = (u_o, v_o).

If the transformation is such that the angle between C₁ and C₂ at z_o is equal both in magnitude and sense to the angle between the curve C₁’ and C₂’ at w_o, it is known as conformal mapping at z_o = (x_o, y_o).

Conditions for Conformal Mapping

The sufficient condition for a transformation w = f(z) to be a conformal mapping is:

The necessary condition for a transformation being a conformal mapping is:

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• Complex Numbers
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• Integration
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Solved Examples on Conformal Mappings

Example 1:

Given transformation w = f(z) = z², which lies in the area in the first quadrant of the z-plane bounded by the axes and circles |z| = a and |z| = b, where (a > b > 0). Discuss the transformation in the w-plane and check whether it is a conformal mapping.

Solution:

The given transformation w = z²

Let z = re^i𝜃 and w = Re^iϕ. Then we have

Re^iϕ = r²e^i2𝜃 that is,

R = r² and ϕ = 2𝜃

Now, the circle |z| = r = a, 0 ≤ 𝜃 ≤ 𝜋/2 in the z-plane is transformed into a semicircle

|w| = R = a², 0 ≤ ϕ ≤ 𝜋 (since R = r² and ϕ = 2𝜃)

in the w-plane.

Similarly, the quadrant |z| = r = b, 0 ≤ 𝜃 ≤ 𝜋/2 in the z-plane is transformed into a semicircle

|w| = R = b², 0 ≤ ϕ ≤ 𝜋 (since R = r² and ϕ = 2𝜃)

in w-plane.

This shows that the annular region between the circles |z| = a and |z| = b in the first quadrant of z-plane is transformed into the annular region between |w| = a² and |w| = b² in the upper half-plane of the w-plane as shown in the figure.

Let us check for the conformality for the given transformation.

Differentiating both sides of w = z² with respect to z, we have

dw/dz = 2z ≠ 0 for any z in the given region.

Thus, the given transformation is a conformal mapping.

Example 2:

Show that the transformation w = (2/√z) – 1 transforms the region outside the parabola

y² = 4(1 – x) into the interior of the unit circle in the w-plane.

Solution:

The given transformation w = (2/√z) – 1 ⇒ (w + 1)² = 4/z ….(1)

If |w| = 1 be the unit circle in the w-plane, the let w = e^iϕ. Thus by the given transformation (1) we have,

(e^iϕ + 1)² = 4/re^i𝜃 where z = re^i𝜃

Or, 1 + e^iϕ = (2/√r)e^-i𝜃/2 ……….(2)

Now, e^iϕ = cos ϕ + i sin ϕ and e^-i𝜃/2 = cos 𝜃/2 – i sin 𝜃/2

Equating real and imaginary parts of both sides in (2)

1 + cos ϕ = (2/√r) cos 𝜃/2 and sin ϕ = –(2/√r) sin 𝜃/2

cos² ϕ + sin² ϕ = [(2/√r) cos 𝜃/2 – 1]² + (4/r) sin² 𝜃/2

⇒ 4/r – (4/√r) cos 𝜃/2 = 0

That is, 1/r = cos² 𝜃/2 or 2/r = 1 + cos 𝜃 ………..(3)

The equation (3) represents a parabola in the z-plane.

Now for the interior points of the unit circle |w| = 1, we have w = ae^iϕ (a < 1). Substituting this in (1), we get

(ae^iϕ + 1)² = 4/re^i𝜃 or 1 + ae^iϕ = (2/√r)e^-i𝜃/2

Again, equating the real and imaginary parts on both the sides

1 + a cos ϕ = (2/√r) cos 𝜃/2 and a sin ϕ = –(2/√r) sin 𝜃/2

Then, a² = [(2/√r) cos 𝜃/2 – 1]² + (4/r) sin² 𝜃/2

Or, 4/r – (4/√r) cos 𝜃/2 = a² – 1 < 0 as a < 1

Or, 1/√r < cos 𝜃/2

Or, r > 2/(1 + cos 𝜃)

which shows that the point (r, 𝜃) lies outside the parabola. Hence, the exterior of the parabola corresponds to the interior of the unit circle in the w-plane.