Thank you so much for doing such an amazing video. It is very helpful, by the way, I wondered why you chose c2 as the opposite direction of the c1? Is that ok to choose the same direction with the c1? Thank you! :)
In the "Why it's true" slide, you apply the Cauchy-Goursat Theorem to a curve that is not simple (see the bridge)... You should do two bridges and two curves instead, right?
Hi,in the last example I don't inderstand why in parametrization we have taken t from [0..2pi] because what I know is that the angle that we use is the one between the segment that match the origin whith any point from the circle and the x axis . so in this case it would be different fom 0 to 2pi? if I can have more explanation about this parametrization please :)
what if I take f(z)= 1/z and the positively oriented contour C is the square with vertices 1+i, i-1,-1-i,i-1? I could not get a result equal to 2pi*i if I break the square into four lines and integrate them individually and sum the four integrals to get 2pi*i
I'm a little bit confused because from my knowledge the direction of integration on line integrals do not change the sign of the result. So how do the two paths on the bridge cancel out?
Yeah I actually did the calculations wrong. In a complex contour integral, the orientation does affect the sign of the result. When I did a u-sub u --> L - u and du --> -du i didn't remember to negate du. If that made any sense at all. anyways the direction does matter on these kinds of integrals.
Benjamin Rutkowski i see thanks. A question -- in real path integrals the orientation didnt matter right? Or am I mistaken? I can't remember as i last did it a while ago. Iirc it was the integral with integrand f (u (t))*sqrt( (dx/dt)^2 + (dy/dt)^2)) dt if that makes sense
Yeah. I looked back in my old calc book. because the differential (ds) in real line integrals is the small change in length, the direction doesn't change the sign of the length. However, in normal riemann integrals DELTAx --> dx is x(k) - x(k-1) where the order does affect the sign.
@@arafathislam2900 Oh man, I took complex analysis so long ago! It was a correspondence course, and from what I remember it went well! I tutor people in math and science, which is why I come back to the educational side of UA-cam every now and then. How are you doing?
Deformation of contours? More like "Your videos are some of the all-stars", of educational videos on UA-cam. Thanks again for making them!
You are a great man. You make things so easy to understand not like others who go through this and that and even than we can't understand
Thank you for clarifying this for me!
Thanks a lot this is an amazing lecture!
very clear explanation, thank you.
Thank you so much for doing such an amazing video. It is very helpful, by the way, I wondered why you chose c2 as the opposite direction of the c1? Is that ok to choose the same direction with the c1? Thank you! :)
If C2 is in the same dirn as C1 then it will intersect at the gap between the bridges
That will not make it a simply closed curve
In the "Why it's true" slide, you apply the Cauchy-Goursat Theorem to a curve that is not simple (see the bridge)... You should do two bridges and two curves instead, right?
Hi,in the last example I don't inderstand why in parametrization we have taken t from [0..2pi] because what I know is that the angle that we use is the one between the segment that match the origin whith any point from the circle and the x axis . so in this case it would be different fom 0 to 2pi? if I can have more explanation about this parametrization please :)
what if I take f(z)= 1/z and the positively oriented contour C is the square with vertices 1+i, i-1,-1-i,i-1? I could not get a result equal to 2pi*i if I break the square into four lines and integrate them individually and sum the four integrals to get 2pi*i
Silly question here. Is the t in Goursat silent?
How this connecting method is not against the definition of the curve must not intercect itself?
That's why you take C2 as -C2 direction so it doesn't intersect itself
I'm a little bit confused because from my knowledge the direction of integration on line integrals do not change the sign of the result. So how do the two paths on the bridge cancel out?
Benjamin Rutkowski I'd also like a clarification here.
Yeah I actually did the calculations wrong. In a complex contour integral, the orientation does affect the sign of the result. When I did a u-sub u --> L - u and du --> -du i didn't remember to negate du. If that made any sense at all. anyways the direction does matter on these kinds of integrals.
Benjamin Rutkowski i see thanks. A question -- in real path integrals the orientation didnt matter right? Or am I mistaken? I can't remember as i last did it a while ago. Iirc it was the integral with integrand f (u (t))*sqrt( (dx/dt)^2 + (dy/dt)^2)) dt if that makes sense
Wait actually if the integral doesn't matter on orientation, how did the bridges cancel out?
Yeah. I looked back in my old calc book. because the differential (ds) in real line integrals is the small change in length, the direction doesn't change the sign of the length. However, in normal riemann integrals DELTAx --> dx is x(k) - x(k-1) where the order does affect the sign.
Thank you so much
thanks in 2023!
My course is going to be so much easier from finding this.
I just came across your comment, and I was curious if indeed your course was easier. 🙂
@@PunmasterSTP I just came across your comment as well and am wondering how your course went.
@@arafathislam2900 Oh man, I took complex analysis so long ago! It was a correspondence course, and from what I remember it went well! I tutor people in math and science, which is why I come back to the educational side of UA-cam every now and then.
How are you doing?