The beauty of the complex exponential -- Complex Analysis 4

[@Nikolas_Davis]

Essentially, the Exp and Log complex functions can be seen as transformations from Polar to Cartesian coordinates, and vice-versa. This is made clear by viewing their actions on the respective grids: the Exp function takes the x-y grid to a grid of rays & concentric circles; whereas the Log function takes us back from rays & circles to the x-y grid.

[@kozokosa9289]

I'm trying to get ahead of my complex analysis course next semester, thank you very much for making a playlist regarding this course

[@aweebthatlovesmath4220]

This is so beautiful. Thank you!

[@lexinwonderland5741]

Love the content as always! One comment, I wish in the Mathematica images, you would trace the parameter so we can see directly which parts of the curve map to which parts of the modified complex curve... otherwise, fantastic as always, so grateful you're making this so accessible!

[@YassFuentes]

That will be great!

[@sunev]

Yes. Here is an example. I Hope this will help. Manipulate[Show[ParametricPlot[{ReIm@c1@u, ReIm@f@c1@u}, {u, -2, 2}, PlotRange -> All], Graphics[{PointSize[Large], Blue, Point@ReIm@c1@t, Orange, Point@ReIm@f@c1@t}]], {t, -2, 2}]

[@MGoebel-c8e]

Thank you - this is very useful. What would have helped me for the first videos in the series is to compare to real analysis. I got confused by the “second graph” and the entire mapping procedure until I finally understood that while real analysis has dedicated input and output axes, producing one graph on one plane, complex analysis treats both axes equally and maps from a (x,y) input point to another (x2,y2) point on the same plane, so there is both an “input graph” and an “output graph”. This is quite important also for understanding the meaning of a complex integral. I guess this idea is obvious or trivial for a professional like you, but when i approached this topic as a learner for the first time this was a major roadblock in understanding - FYI.

[@YassFuentes]

I do like this series, It's like been in college again. Superb to keep the touch with such a beautiful subject. Willing the lesson con Cauchy integrals

[@fordtimelord8673]

When you introduce the Complex logarithm, I think it would be more intuitive to start off with polar coordinates, Re^(iy). Also, in teaching and using complex analysis, I think most people will find using tau instead of pi more intuitive. Thanks a million for doing this from the ground up. Great teaching!

[@pseudolullus]

Great video, loved the mappings

[@atreidesson]

So now we know that e^x can transform a set of 9 lines in a Chaos sigil

[@ibrahimcirozlar1733]

Excuse me but is the answer for the 3rd warm up question "if x=e, then Z= e + iy. LogZ = 1 + i * Pi/2 and this makes a straight line through x=1 point and parallel to the y axis." I am just learning Complex analysis and I do not have much experience. If wrong, please correct me. I appreciate your help friends.

[@الفيزياء-ب2ي]

انا اعتقد انك على صواب أيضا

[@fantiscious]

Log z = ln|z| + i Arg z for z = x + iy |z| = |x + iy| = √(x² + y²) Arg z = Arg(x + iy) = tan⁻¹(y/x) if x > 0 Therefore, Log z = Log(e + iy) = ln|e + iy| + i Arg(e + iy) = ln(√(e² + y²)) + i tan⁻¹(y/e) = ln(e² + y²)/2 + i tan⁻¹(y/e) Setting u = ln(e² + y²)/2 and v = tan⁻¹(y/e) after some work we get v = ±tan⁻¹(√(e²ᵘ⁻² - 1)) which can now be plotted

[@arkodasgupta0412]

@@fantiscious yup I also did this. I think the graph would have horizontal asymptotes at y = pi/2 and -pi/2

[@usernameisamyth]

0:43 Euler's *number :)

[@aniqa71596]

why exponential turns imaginary y maps into arg of pie in negative of real line and not on positive ??

[@aniqa71596]

no one talks about range of y is from pie to -pie

[@matthewzeits8909]

Any way to translate the Mathematica examples to gnuplot and octave?

[@matthewzeits8909]

It's been 25 years since I took numerical analysis in grad school which was the last class I took with mathematica. I can probably figure it out myself but if someone knows off hand...

[@downwithreactionaries9031]

Euler's constant !!!!==== Euler's number. I guess you are referring to "Euler's number", not "Euler's constant".

[@aniruddhvasishta8334]

What's the difference?

[@lucachiesura5191]

let's go...

[@123bluestorm1]

Anulus!!!

[@Nikolas_Davis]

An *n* ulus.

[@thenewdimension9832]

❤❤❤❤

[@artificialresearching4437]

I might be wrong, but this could work: kzbin.info/www/bejne/hqe4o5WmqL2Ngrc P.S. A piece of advice: make video 1.5 times faster, I speak very slowly)

[@Stobber1981]

Better to speak slowly and let users use the playback speed feature.