As an alternative approach we can consider the integral of z^a/(1+z^3). After making the substitution z^3=t we get Beta-function, which is simplified by using Euler reflection formula. Finally we differentiate the result and put a=1 to get the same answer as in the video. Evaluating this integral via contour integration is beautiful too :)
@@xinpingdonohoe3978 i mean i dont know any complex analysis but its just a suggestion if its possible to explain atleast the concept of this in under 15 minutes
@@k_wlMan I dont think anyone can explain all the concepts needed for this question in under 15 mins. Better suggest Dr PK to make a separate video only about contour method
What's the confusion? In the video it's ln(z), because that's what the question is, but to evaluate it he uses a similar function along the contours which instead has ln²(z). That's the method he's using. He explains it all at 2:49.
It is a very similar story with the integral of ln(x) / (1+x^2) from 0 to infinity. You will notice that we can get the result by either: i) Integrating ln(z) / (1+z^2) directly along the full real axis. ii) Integrating (ln z)^2 / (1+z^2) around a keyhole. You can check for yourself why, but integrating ln(z) / (1+z^2) around a keyhole fails to determine the target integral, precisely because the target integral cancels out due to the branch cut. Same reason in this video
As an alternative approach we can consider the integral of z^a/(1+z^3). After making the substitution z^3=t we get Beta-function, which is simplified by using Euler reflection formula. Finally we differentiate the result and put a=1 to get the same answer as in the video. Evaluating this integral via contour integration is beautiful too :)
Thats cool my friend! Thanks for sharing haha👍👍👍
The best video by the best math youtuber and professor
Thanks a lot my friend for your support👍👍👍
Very tough but you explain and work this like the best
Thanks for your support my friend haha👍👍👍
Like I said, he is the best math professor Ive ever seen🎉
Thanks a lot my friend for your support👍👍👍
Another great video with very challenging integral
Thanks a lot my friend haha👍👍👍
The best one prof.
Thanks a lot haha👍👍👍
Can you do a video about contour integration ?
Residue theorem pls
For sure my friend! Nice suggestion. Will make one soon👍👍👍
sir i think it would be nice if you explained what the method is before using it, it gets confusing when i dont know the methods
Believe this video is for those who know the contour method
To what level of detail? How much complex analysis do you know, because this would require a lot of setup to explain?
@@xinpingdonohoe3978 i mean i dont know any complex analysis but its just a suggestion if its possible to explain atleast the concept of this in under 15 minutes
@@k_wlMan I dont think anyone can explain all the concepts needed for this question in under 15 mins. Better suggest Dr PK to make a separate video only about contour method
@iqtrainer true
Certainly isn't the hardest integral on the internet, but its good nevertheless
Thanks a lot for your support my friend👍👍👍
Super sir!!
Thanks a lot for your continued support my friend👍👍👍
@drpkmath12345
Welcome sir!!
I’m very confused, is it ln^2 or ln?
I ask because I’m the thumbnail it says ln but in the video it’s ln^2
where in the video says it is ln^2? On the whiteboard, it says ln.
ln^2 was explained later as to why he put it like that
What's the confusion? In the video it's ln(z), because that's what the question is, but to evaluate it he uses a similar function along the contours which instead has ln²(z). That's the method he's using. He explains it all at 2:49.
It is a very similar story with the integral of ln(x) / (1+x^2) from 0 to infinity. You will notice that we can get the result by either:
i) Integrating ln(z) / (1+z^2) directly along the full real axis.
ii) Integrating (ln z)^2 / (1+z^2) around a keyhole.
You can check for yourself why, but integrating ln(z) / (1+z^2) around a keyhole fails to determine the target integral, precisely because the target integral cancels out due to the branch cut. Same reason in this video
@@xinpingdonohoe3978 I saw the video again and it all makes sense now
1/2
Ooops😢