Derivation of Poisson's Formula for of Laplace's Equation on the Unit Disk: Complex Fourier Series!

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[@suboptimallife]

The enjoyment he derived from the derivation of Poisson's Formula is typically what makes these videos so much fun to watch. Genuinely infectious. It reminds me of a maths lecturer at uni who stood back after deriving Newton's Quotient, took a moment and whispered "beautiful", more to himself than his audience. Quite humbling to see.

[@SalvatoreIndelicato]

A years later I realize that at university nobody had ever explained to me in such a clear and brilliant way the Fourier series and the Laplace equation. Congratulations to the prof. Pavel from Italy

[@georgesadler7830]

Professor MathTheBeautiful, thank you for a solid Derivation of Poisson's Formula for of Laplace Equation on the Unit Disk and their impact on Complex Fourier Series. This is an error free video/lecture on TH-cam TV.

[@MathTheBeautiful]

Glad it was helpful!

[@NotJames1]

that was brilliant!

[@hebaomar7535]

Thanks for the useful inspiration Prof. Grinfeld. How can I find accurate value for integration constant in this example ?

[@0Navin0]

brilliant

[@onesoftware1766]

sir how to solve this integral on unit cirle

[@luisrodriguez-nd6wx]

How would you find the Poisson integral formula if you had Neumann BC?

[@relike868p]

I am confused: why do you take the Fourier coefficients of the boundary condition?
Is it because you have set u(1,theta) = f(theta)?

[@onesoftware1766]

the equation which circle at end

[@danielvolinski8319]

Prof. Grinfeld: the sum of the auxiliary series whose result is 1/(1-x) converges only when |x|

[@aeroscience9834]

Daniel Volinski yea, because x in this case is r e^i(...), so |x|=r, however this is the unit disk, so r

[@danielvolinski8319]

Prof. Grinfeld: I use this formula with the boundary condition: f(\theta)=cos(2*\theta) as in the previous video, but I get the solution: u(r,\theta)=r^2*cos(2*\theta)/2-cos(2*\theta)/(2*r^2) which is not the same as before.

[@MathTheBeautiful]

Send me a screenshot of your work. The expression you typed equals zero, but with the plus sign it looks like it would be correct.

[@danielvolinski8319]

This is a PDF with the result created out of wxMaxima CAS.
Please take a look at the bottom of page 6 where the Poisson formula is used.
The result is different than the previous video in the series on the same problem.
1drv.ms/b/s!Avfq2HpP4hNagg7Ktm1OC3etf1fz
Let me know if you can access the file

[@MathTheBeautiful]

There's definitely a discrepancy between the two lines. The integral is finite when r=0 but the evaluated expression is infinite at r=0. But I'll try to think about why that is.

[@MathTheBeautiful]

My guess is that the software chose the wrong branch on the complex log. With enough assumptions to avoid these issues, Mathematica produces the right result:
$Assumptions =
0 < \[Theta] && \[Theta] < \[Pi]/4 && 0 < r && r < 1; 1/(2 \[Pi])
Integrate[(Cos[2 \[Alpha]] (1 - r^2))/(
1 - 2 r Cos[\[Theta] - \[Alpha]] + r^2), {\[Alpha], -\[Pi], \[Pi]}]
Output: r^2 Cos[2 \[Theta]]

[@MrFizzynoah]

I think theres a factor of r missing in your thumbnail for this video!

[@ares12265]

And we come to the birth of the concept of algorithm. There are many many much respected al-Khwarizmi, I beg you, do not start spinning in the coffin, everything is in order. And now when we have an algorithm, we open ourselves the opportunity to come to the concept of "Call" and in particular to make "Call the Force"