Complex Analysis 7 | Cauchy-Riemann Equations Examples

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This is my video series about Complex Analysis. I hope that it will help everyone who wants to learn about complex derivatives, curve integrals, and the residue theorem. Complex Analysis has a lof applications in other parts of mathematics and in physics.

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00:00 Intro
00:18 Quick recap
02:42 Example 1: Identity function
04:28 Example 2: Complex conjugate function
05:38 Example 3: Complex polynomial

#ComplexAnalysis
#Analysis
#Calculus
#Mathematics
#curveintegral
#integration

(This explanation fits to lectures for students in their first or second year of study: Mathematics, Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[brightsideofmaths]

I didn't write it down again but for the first equivalence in the video to hold, the functions u and v should be (totally) differentiable (everywhere) as we have discussed it in part 6. I simplified it here for better access but I guess that it could be confusing. Therefore I added this comment now :)

[andrewtaylor]

Love the style of these videos. I think it would be helpful to explain the geometric meaning of the Cauchy-Riemann equations. There's clearly something interesting going on: The 2 x 2 matrix takes the form of a linear transformation (from |Rx|R to |Rx|R) that performs a fixed rotation and fixed multiple of magnitude; that is, for any (real) 2-vector, the 2x2 matrix always rotates it by a fixed amount, and changes its magnitude by a fixed amount. The complex derivative exists at a point z if the linear transformation is of this type. This makes sense since the derivative is defined by a limit that must be constant no matter which direction z is approached from.

[actualBIAS]

So here I am, preparing to dig into quantum mechanics... Watching your videos again... Sir, you play a huge role in my journey for knowledge.

[PunmasterSTP]

Cauchy-Riemann equations? More like "Come one, these are really amazing connections!" Thanks for making videos on all these intriguing topics.

[NewDeal]

00:00 Intro
00:18 Quick recap
2:42 Example 1. Identity function
4:28 Example 2. Complex conjugate function
5:38 Example 3. Complex polynomial

[adrianschmidt]

Another nice example is f(z) = z^n.
Using the binomial theorem you can also show that the Cauchy-Riemann-Equations are true for f.
(I was just able to do it with one case for even n and another case for odd n)

[alicej]

These videos are super helpful. I’ve learnt more from this video than I have from my lectures at uni

[chenliou]

I want to ask which books you refer to

[navierstokes]

Books that you like for complex analysis?

[adammustapha]

Good evening the Bright Side of Mathrmatics. I'm having issue with complex analysis in class, my lecturer isn't helping at all. Please how can you help me with it