I really liked the appendix at the end :) Question out of curiosity : is it possible to extend the definition of complex integral to non differentiable lines (e.g. the sample path of a Brownian Motion) ?
I don't know the terminology. But the coefficients a and b (a + ib) of the complex numbers along the contour in the examples are positive, so how does the integral yield a negative coefficient?
is it right to say that γ' is there because differential dt is only defined on the real line, and the integral can only be indirectly mapped into [a,b] through γ?
Why? There is a study called "Sympletic geometry" to find geometric meaning, and there are many studies that need to be preceded in order to study it. If you're still an undergraduate, you don't need to know yet.
Thank you for this great series.
Thanks!
Thank you very much :)
@@brightsideofmaths Your content is pristine and extremely helpful personally, thank you indeed.
This video was contourrific! 👍
Great video! Thank you!
Thanks for this greatly helpful job🙏🏻
I really liked the appendix at the end :)
Question out of curiosity : is it possible to extend the definition of complex integral to non differentiable lines (e.g. the sample path of a Brownian Motion) ?
A little extension is possible: one need a rectifiable curve
Yeah this was a nicer version of an already nice video!
Glad you enjoyed it
Which program/software do you use to create the images on the thumbnails?:)
LaTeX and Inkscape :)
I’m confused on how in the examples, the integral yields a negative real number (-1) when all of the values along the contour are positive
What does "positive" mean for complex numbers?
I don't know the terminology. But the coefficients a and b (a + ib) of the complex numbers along the contour in the examples are positive, so how does the integral yield a negative coefficient?
@@kinwasalad268 i times i = -1
fascinating :)
is it right to say that γ' is there because differential dt is only defined on the real line, and the integral can only be indirectly mapped into [a,b] through γ?
We definitely need γ for this mapping. And the derivative comes naturally in.
Not sure why you try to explain the γ’ in such way. It simply comes from z=γ(t), dz=(dγ/dt)dt
That is not an explanation ;)
Do I have déjà vu?
PS
13:05 - Now I know how much you like the idea of a cliffhanger! ;)
th-cam.com/video/_6IlET7ZMAw/w-d-xo.html
I still don't have a very clear idea what does int(f(z)dz) geometrically mean
Why? There is a study called "Sympletic geometry" to find geometric meaning, and there are many studies that need to be preceded in order to study it. If you're still an undergraduate, you don't need to know yet.