How to visualize complex functions.

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[MichaelPennMath]

At the moment we only have Linear Algebra videos posted on @mathmajor. Complex Analysis videos should start going up this week.

[Ivan-mpff]

many authurs like 3 brown1 blue do not bother with numerical examples during the clip, you stand out and have made everything so intuitive and clear. Can't thank you enough.

[mathflipped]

Visualizing math objects is always challenging. But when you can pull it off, then the resulting insight is incredible! I'm starting to work on a series of videos on Visual Group Theory, which should change the way group theory is taught. This will be done in collaboration with other mathematicians, and, hopefully, some consulting help from 3blue1brown.

[pwmiles]

A rather important map is the general Moebius transformation, z->(az+b)/(cz+d) with a, b, c and d any complex numbers with ad-bc non-zero. This maps circles and lines to circles and lines.

[gniedu]

For the homework

z = t + (t+1)*i where t is a parameter that goes from -oo to +oo

z^2 = (-2t - 1) + (2t^2 +2t)*i

One possible way to think is callng, as Michael did, the real part x and the imaginary part y. So we will have

x = -2t - 1
t = -(x+1)/2

Since y = 2t^2 + 2t

Then y = (x^2 - 1)/2

This will make a parabola in the co-domain, passing through -1 (the image of i in the domain), 1 (the image of -1 in the domain) and also passing through -(1/2)i (the image of -1/2 - (1/2)i, which is also in the given line from the domain).

[ElMalikHydaspes]

going up to elliptical functions yet sometimes it still gets to me how the circle i.e. chubbiness shows up on the z to f(z) transformation aka how a simple line goes to deformed circles ... this simple video cleared that up rather nicely! 😂

[manucitomx]

This was a very instructive explanation. Visualizing complex transformations is always challenging.
Thank you very much, professor.

[dunai]

Excellent topic! I've been searching this for a long time but found none.

[isabel]

Great video! Currently taking college level Complex Analysis and this has helped tremendously.

[synaestheziac]

You used the brown to write stuff! It was a little dimmer than the rest but still legible

[sinecurve]

Mathemaniac's channel has some excellent videos on complex functions.

[yoav]

For the homework: i got smiling "parabola" with the vertex at (0, -0.5) ant the intersection with the real axis at (1, 0) and(-1, 0)

[furutapark]

A great way to visualize further is to do multiple lines at a time so that you get a kind of grid. It allows you to visualize local parameterizations of C, as well as approximate any point within the neighborhood.

[lucachiesura]

Every complex line has the form z+Az1 = B where z = a+ib, z1=a-ib; z+Az1 = B and z+Cz1 = D are parallel lines if C=D. The inversion by a circle of centre m and radius r is in the form f(z) = m + r^2/(z1-m1), but it's possible the linear approssimation by mobius matrix...

[as-qhqq]

For those wondering if the curves are parabolas, they indeed are - use the correspondence x+i y == (x, y) where the latter is a point. Keep in mind that the parabolas might be translated and/or rotated

[goodplacetostop]

15:10 Homework
15:51 Good Place To Stop

[keithphw]

Just wondering what this transform operation is applied to in engineering or physics? Does it give the forces on a particle moving in a straight line through a magnetic field? Great video

[adityansingla]

Homework solution:
If we treat the line as a parametric form, we can square it and get a new parametric equation
If we solve to set the x coordinate function be t, we get y coordinate function to be-
(t^2 -1)/2
which is a parabola and a quadratic polynomial and is symmetric along the imaginary i.e y axis
Pretty neat
Similarly, we can show that the curve obtained in this video is a parabola too

[mattcarnevali]

Amazing how the graph of lines in Z squares maps to a parabola just like the graph of y squared in the reals.

[jimgolab]

Good video. I like the pace. I have struggled with this kind of visualization for years.