Conformal Mapping Calculator

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Example

Created on 2024-06-20Asked by James Johnson (Solvelet student)

Find the image of the unit circle

|z| = 1

under the conformal map

f(z) = z^2

Solution

Step 1: Identify the conformal map

f(z) = z^2

. Step 2: Represent the unit circle in the complex plane as

z = e^{i\theta}

, where

\theta

ranges from 0 to

2\pi

. Step 3: Apply the map

f(z)

z = e^{i\theta}

f(z) = (e^{i\theta})^2 = e^{i(2\theta)}.

Step 4: Describe the image under the map. The points

e^{i(2\theta)}

still lie on the unit circle

|w| = 1

because the magnitude remains 1. Step 5: Conclusion. The image of the unit circle

|z| = 1

under the map

f(z) = z^2

is the unit circle

|w| = 1

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Emma Jackson on Solvelet

1. Find the conformal map that transforms the upper half-plane onto the interior of the unit disk 2. Determine the conformal map that transforms the region inside the unit circle onto the region outside the unit circle.,

DefinitionConformal mapping is a mapping that preserves angles locally, that is, it keeps the local form of objects in the complex plane. It's a one-to-one mapping, that keeps the angles, but not always the distances. Conformal mappings are used in the complex analysis,fluid dynamics and cartography to represent complex geometries and to solve boundary value problem. They are a versatile tool for representing complex functions and models of the physical world. For example, the mapping w=ez maps straight lines parallel to the real axis in the complex z-plane to concentric circles in the complex w-plane. It also preserves the angles between intersecting curves.

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