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

LINK TO CANVAS

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.

► 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

SOURCES:
Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics: Pearson New International Edition.

HASHTAGS
#TheMathCoach #ComplexAnalysis #Complex Analysis Playlist

[TheMathCoach]

Long-time no see, thank you all for the supportive messages during my hiatus (I completed my master thesis and have worked one year at my first job)! This video is dedicated to all of you, but also specific @Sarah _ that have waited an awful amount of time for me to complete this one.

Conformal mapping is one of the hardest concepts to grasp in complex analysis and I also struggled with it a fair bit in school but I think this video will be able to show you that it is easier than one thought!

[Setsugekwa]

Seriously, you just saved my day. I had a hard time while understanding fluid flow on object with conformal mapping, and this video grasped concept for me. thx a lot.

[蔡沂倫-eh]

i spent over 5 hours not knowing my professor's explanation, but after your awesome video, im totally speechless. thank you so much!!!!

[randomideas]

I just came here to understand definition of conformal mapping but couldn't stop myself to grasp the whole concept..credit goes to you. Keeping helping us man

[PunmasterSTP]

Another truly incredible masterpiece! Sadly I have reached the end of the playlist, but I will eagerly look forward to any and all new content in the future. Thank you again, so so much!

[augustoriedinger]

You are a wonderful human being. Thank you.

[jackjonathan]

well i dont even know how to thank you.you just explained to me what no one else could.i am really greatful for your dedication

[quanle]

Dear TheMathCoach,

I hope this message finds you well. I am writing to express my heartfelt gratitude for the enlightening lecture you delivered on [specific topic or date]. Your ability to convey complex concepts with clarity and enthusiasm is truly remarkable and has greatly enhanced my understanding of the subject.

The depth of knowledge you shared, along with your engaging teaching style, has left a lasting impression on me. I particularly appreciated the practical examples and real-world applications you included, which made the material much more relatable and easier to grasp.

Thank you once again for your dedication and effort in providing such a valuable learning experience. I look forward to attending more of your lectures in the future.

Warm regards

[timothyang]

So elegantly explained. Thank you for the effort and I hope you can find the time to continue making content. Perhaps a series on undergrad abstract algebra.

[minhnam]

The video is really great. The teacher didn't teach us, she let us read the textbook and the written textbook was very difficult to understand, but after watching your video, I understood the problem! Thank you so muchh

[andreaLA]

In class we were just given definitions and formulas, not a single drawing, and I didn’t get what Möbius Transformations were. Now I understand everything. Thank you so much!!

[TruthWillSF]

Any one can solve problems, but only genius can reveal the concept 👍🏻👍🏻👍🏻👍🏻❤️

[AA-xbmx]

I love how well you explain everything, even when it may seem obvious.

In my prof's lecture, he just said that a conformal mapping is a function that preserves "the angle" and pointed to two intersecting curves on his slides. But then I thought "how do you even get an angle between curves????" And then he proceeded to show more complicated mappings with circles and I just zoned out. Thanks for clarifying that it's the angle between the tangents at the point lol

[stimulantdaimamld]

simply a thrill.words will fall short of the description.

[Aixur]

I watched so many of your videos and you saved my life so thank you. i still might not pass tomorrow but the odds for success are so much greater now.

[sandracordoba]

I can’t express how grateful I am. Great video

[ketyaportela]

One of the first times I directly subscribed to a YouTube channel. Quality video, thank you for the additional resources such as the course map, the different methods and the links to go further. Very good compromise between rigor and simplicity.

[luanmartins]

congrats for the amazing video. I'm a physics student and it helps me a lot.

[ashishkumarsarma]

Thank you very much sir 🙏 I am currently an engineering first year student and u r video helped me a lot today! Greetings from India 🇮🇳

[mariaazam]

I was trying hard to grasp the idea of conformal maps... You helped me alot... Thanks 🙂