Conformal Mapping | Möbius Transformation | Complex Analysis #25
Everything you need to know about Conformal Mappings in Complex Analysis. The video will show you the best method to solve Conformal Mapping problems with the help of Möbius Transformations.
The video will include concepts as:
► Definition of a Conformal Map (aka. conformal transformation, angle-preserving transformation, or biholomorphic map).
► Definition of a Möbius Transformation.
► All the main Elementary Transformations (Translation, Magnification, Rotation, and Inversion) that makes up the Möbius Transformation.
► How to determine a Conformal Mapping and how to know if it is unique.
LINK TO COMPLEX ANALYSIS PLAYLIST
• The Complete Guide to Complex Analysi...
LINK TO CANVAS
https://drive.google.com/file/d/1_gxu...
SUPPORT
Consider subscribing, liking or leaving a comment, if you enjoyed the video or if it helped you understand the subject. It really helps me a lot.
TIMESTAMPS
- Introduction 00:00
- Definition Conformal Mapping 00:40
- Definition Möbius Transformation 03:20
- Elementary Transformation 04:44
- Translation 04:54
- Rotation 05:22
- Magnification 05:43
- Inversion 06:10
- Example conformal mapping rectangle to rectangle 09:03
- Example conformal mapping circle to circle 12:52
- Example the inside of a circle to the right half-plane 14:54
- Theorem Möbius Transformation and the Cross-Ratio 20:24
- The first example again by using the Cross-Ratio 21:25
CLARIFICATION OF THE METHODS:
To determine a Conformal Mapping you can choose one of two methods. The first one can always be used while the second know can only be used by knowing how three points are mapped.
Method 1:
Start by drawing both of the figures (the figure you start with and the figure which we will get after the transformation) and then compare them. The next step is to try to transform the figure you started with so that it looks exactly like the last figure by using the four transformations (translation, rotation, magnification, and inversion).
If the figures differ in size, use magnification, if the figures differ in position, use translation and so on. Note that if you are transforming an area then you can keep track of this area by taking one point that makes up this area and see how it is mapped.
After this is done then you will probably have made a sequence of transformations f_[1], f_[2],...., f_[n] and the last thing is to combine the results from each one of them and since each transformation is depending on the result of the earlier transformation you get the following: f_n(f_[n-1](...(f_[2](f_[1](z))))).
Method 2:
If you know how three unique points are mapped (z_1 is mapped to w_1, z_2 is mapped to w_2 and z_3 is mapped to w_3) then we can use the definition of the cross-ratio and the fact that this cross-ration is not affected by the mapping.
The Conformal mapping can then be determined by solving this equation for z:
(((z-z_{2})(z_{1}-z_{3}))/((z-z_{3})(z_{1}-z_{2})) = (w-w_{2})(w_{1}-w_{3})/((w-w_{3})(w_{1}-w_{2}))
CONCEPTS FROM THE VIDEO:
► Conformal Mapping
A conformal mapping is a function f(z) that preserves local angles.
► Möbius Transformation
A möbius transformation is a function that can be written on the following form:
f(z) = (az+b)/(cz+d)
where the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0. The great thing about this function is that it is always conformal according to the following theorem.
► Theorem Conformal and Analytic Functions
An analytic function f(z) is conformal at z_0 if the derivative at this point is not equal to zero.
Proof: http://www.maths.lth.se/matematiklth/... (Theorem 47) [page removed by linköping math center]
Read about it here: https://www-users.math.umn.edu/~olver... (p. 34-35)
► Cross-Ratio
The cross-ratio of a quadruple of distinct points with coordinates z_1, z_2, z_3, z_4 is given by
(((z - z_{2})(z_{1} - z_{3}))/((z - z_{3})(z_{1} - z_{2}))
Cross-ratios are invariant (is not affected) under Möbius transformations.
SOCIAL
► Follow me on Youtube: http://bit.ly/1NQhPJ9
► Follow me on Twitter: / the_mathcoach
SOURCES:
Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics: Pearson New International Edition.
https://en.wikipedia.org/wiki/Conform...
http://mathworld.wolfram.com/Conforma...
https://commons.wikimedia.org/wiki/Fi... [Thumbnail]
HASHTAGS
#TheMathCoach #ComplexAnalysis #Complex Analysis Playlist
Watch video Conformal Mapping | Möbius Transformation | Complex Analysis #25 online without registration, duration hours minute second in high quality. This video was added by user TheMathCoach 31 August 2019, don't forget to share it with your friends and acquaintances, it has been viewed on our site 84,22 once and liked it 2 thousand people.