I would appreciate if someone could explain Conformal Mapping using Complex Analysis using an example. I get the rough idea but have no clue how complex analysis comes into the picture.
Conformal mapping is a technique used in complex analysis to map one complex plane onto another in a way that preserves angles and shapes. This means that the mapping does not distort the shape of the original figure, and the angles between any two intersecting curves are preserved after the mapping.
Conformal mapping has various applications in engineering, physics, and mathematics. It is used in the study of fluid dynamics, electromagnetism, and quantum mechanics. It also has applications in image processing, computer graphics, and cartography.
Conformal mapping is different from other types of transformations, such as affine or projective transformations, because it preserves angles and shapes. Other transformations may change the size and shape of the original figure, but conformal mapping maintains these properties.
Some commonly used conformal mappings include the exponential mapping, the logarithmic mapping, the Mobius transformation, and the bilinear transformation. These mappings are often used in solving complex analysis problems and have different properties and applications.
While conformal mapping is a powerful tool in complex analysis, it has some limitations. It can only be applied to regions that are simply connected, meaning there are no holes or gaps in the region. Additionally, conformal mapping cannot preserve all properties of a figure, such as area or perimeter.