Cauchy's Integral Formula with Examples | Complex Integration | Complex Analysis #14

[@Deadlybrownboy]

Hey, thanks a lot for this video. At 5:30 I believe you forgot the factor of 1/3 for the final result.

[@TheMathCoach]

My pleasure and yea damn you are right! Always something that slips through, thanks for letting me know. Is it OK if I pin your comment for others to easily see it as well? Have a great day :)

[@Deadlybrownboy]

@@TheMathCoach Of course. You too!

[@TheMathCoach]

Great, thank you!

[@TheMathCoach]

I decided to reupload it with better audio - hope you are ready to do A LOT of problems with Cauchy's Integral Formula, I might have overdone myself in this video, but I thought "hey, I wanna add this one too, and this one and ...". The next video is done and scheduled to go live next Sunday 12:00 AM CET. Hope you are having a great day!

[@connectingspyyo8159]

This is by far the best math channel I've found. You're amazing!

[@TheMathCoach]

thank you for the kind words, happy to be able to help! :)

[@wy2714]

This is my first time commenting on a KZbin video and I have to say that this channel hands down has the best explanation on complex analysis! I am thankful for all this playlist and please continue to do more topics on it! Anyway just a question, if the contour isn’t shown or given, how do we know if the singularity is inside gamma?

[@TheMathCoach]

Glad you like them and welcome to the channel! If the contour is not defined for us then we can not know if the singularity is inside of gamma!

[@PunmasterSTP]

Another phenomenal video, and thank you again so much for making "complex" topics so easy to understand!

[@TheMathCoach]

That is what I do, the word complex doesn't really convey the course correctly and might steer people away from the course. It is really only a name, but I think the content is really interesting and it all comes together quite beautifully. Thank you again!

[@PunmasterSTP]

@@TheMathCoach You are incredibly welcome, and I thank you again for making all of these videos and putting them up! I've been exploring the math side of KZbin a bit more recently, and I came across Faculty of Khan, who at first I thought was part of Khan Academy but isn't. He also has a complex analysis playlist ( kzbin.info/aero/PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB ), and I'm currently working my way through it now!

[@sleepeasyguitar9505]

This video just saved me so much time. Thank you.

[@euler0148]

It is a beautiful video where i could find all information i needed thanks you

[@TheMathCoach]

Happy to hear that!

[@herrjonna2007]

Subscribed! What an absolute legend! Video is crisp, the audio is clear, your English is awesome and your hand writing is understandable and pretty! Honestly, best videos when it comes to advanced math topics! Well done! Also, Swedish?

[@TheMathCoach]

Happy to hear that you found my content useful, welcome to the channel jonna and just let me know if you have any questions about the content and I will be sure to get back to you :) I hope that it was the handwriting that won you over! Japp, det är jag - vad gav iväg mig? xD

[@herrjonna2007]

@@TheMathCoach Oh yes, it's quite useful, the entire series! A nice angle to have when my books are lacking in explanation :) Of course, the handwriting was the cherry on top ;) Hehe som nordmann var det noe kjent med aksenten :D

[@TheMathCoach]

haha den tog mig igen alltså!

[@herrjonna2007]

@@TheMathCoach Hehe ja, men engelsken din er top notch! Jeg tror ikke det er mange som legger merke til det, eller bryr seg ;) Igjen veldig bra jobbet med videoene! You're a life saver!

[@maomi3176]

Hi Coach. You're the best Coach I ever had in my life. Please solve the two big mysteries of my life. Fourier Analysis and fast and easy solving of ODEs and PDEs. Please 🙏🏻

[@TheMathCoach]

Hello, happy to help out but that are not covered in this course! :)

[@Daniel-ti1pq]

Thank you so much. It is such a neat and detailed lecture that I'm going to use screenshots of it as a note😁

[@TheMathCoach]

check the description, the whole canvas can be found on google drive. happy to help!

[@Daniel-ti1pq]

@@TheMathCoach Thank you boss

[@ahmetertugrulbilgic1443]

That is beautiful. Of course I liked what you do here. Thank you :)

[@TheMathCoach]

Thank you very much!

[@everettcook634]

Thank you for the help, very concise and lays out the concept nicely.

[@TheMathCoach]

Thank you, I make it one of my life goals to have neat handwriting and it seems to also affect my structure of the videos :)

[@deanshalem]

You are incredible, much appreciated. Bless your soul!

[@TheMathCoach]

Happy to help!

[@gaaraofddarkness]

Thanks, this helped me a lot! you are very nice guy

[@TheMathCoach]

Glad to help, and Happy to hear that you my content was useful to you!

[@bendavis2234]

Hey quick question, I'm not sure what the 1^x written in the curly brackets is supposed to represent... can you clarify? Also is the parameter 'n' just supposed to represent the degree of the equation minus one? Thanks a lot, I had fun solving your examples in this video!

[@TheMathCoach]

(1*) is a reference to the formula on the top-right so you guys know which formula is applied at what stage in calculations, I also add the input values into the formula at the same time in the video. "n" is just the variable in the formula which in this example is the same as the power of the parenthesis minus one. Happy to help!

[@heyb0ss594]

Thanks so much! Fantastic explanation :D

[@TheMathCoach]

I'm just happy to help boss :)

[@SM-ku1jy]

Thank you for the amazing video 🙏🏻

[@TheMathCoach]

Happy to help out S M :) Best of luck with your studies! Exams soon maybe?

[@dittodoritos9458]

Thanks! I was so confused about this lol

[@gokuvegeta9500]

9:08 Why do all the curves have to be positively oriented to use Cauchy's Integral formula? 🙄 Also, one more question while I'm at it. Let's say we have a domain between 2 curves C1 and C2 and the "hole" is enclose by C2. I remembered you saying that we have to orient them in such a way that the domain would lie on our left side if we imagine ourself travelling on the curve. That means C1 would need to be anticlockwise and C2 would need to be clockwise But here, you say that all curves need to be anticlockwise for Cauchy's Integral formula to work Please enlighten me

[@TheMathCoach]

It is not necessary but we have to choose an "orientation" when doing integration and this is the orientation that the author choose to defined it by. You could go the other way around (negatively oriented contour) but then you would need to add a minus sign at the start of the formula. Don't really understand the second part of the question, sorry. Could you try to describe it again and add timestamps to relevant parts in the videos or send me image on the e-mail in the bio and I will try to get back to you. Great name btw, go dragonball!

[@amaliepalmund620]

Your videos are great, I learn from and enjoy them a lot!! :) FYI, you have a minor typo af 11.33, you write +4Pi*I but it should be (and you also say) -4Pi*i.

[@TheMathCoach]

I'm happy to hear that and thanks for the notice! I also added a correction about that in the bottom right corner, and was one of the reasons I decided to re-uploaded the video :)

[@baraaor5115]

Thank you so much, that was really helpful :) question: what would happen if the numerator has a singularity inside the contour? tan(z) for example...

[@TheMathCoach]

Hello Baraa, happy to hear that! You would not be able to use the theorem if the numerator (aka. f(z)) is not analytic everywhere inside the contour. You might be able to rewrite with the help of tan(z) = sin(z)/cos(z) but still you need to match the expression in the theorem to be able to use it in the first case. So it is not going to be applicable for all problems.

[@baraaor5115]

@@TheMathCoach Awesome! Thank you for the quick response and the clarification :D

[@LP-ht7ir]

One question when you do the cauchy integral formula And you have two poles Do you not have to take both cases? Ik you can use residue theorem But if they specify to use the integral formula Do you not consider both cases As in with both poles i was told you need to add these but the textbook says otherwise and now im confused, any help would be very much appreciated.

[@riccardofasano1040]

What if there is more than one singularity enclosed by the contour? Cauchy's formula seems to be useful for one singularity at time

[@TheMathCoach]

That is true! you will have to get a new tool for contour integrals with multiple singularities. It is introduced in the video called Cauchy Residue Theorem and is number 16 in the playlist (you can find that in the description).

[@HunterWhetstone]

beautiful, THANK YOU!!!

[@TheMathCoach]

Glad it was helpful! This is one of my favorite videos, so really happy that it can be useful to people!

[@averagegamer9513]

At 11:48, you said the write answer but wrote the wrong expression.

[@MsSam1983]

hi, in all examples of CIF, cosine is used in the numerator. The contour drawings is cutting th real axis between 1 & 2. how we know this is between 1 & 2 on real axis. ????

[@TheMathCoach]

Hello, I mentioned it at 2:50 but I just **made up** a contour that "would be really hard to parametrize" (don't even try it) and the contour I made up ended up crossing between 1 and 2 on the real axis. If you are doing problems in your textbook then the author will hopefully define the contour for you. You are not supposed to know the definition of the contour if no definition is stated.

[@AA-xb4mx]

IT ALL MAKES SENSE NOW :0

[@TheMathCoach]

RIGHT :D

[@gameplayerth1677]

At 11:32 I think that ,it should be -4πi.

[@omaridbraym5220]

Thank you you are the best

[@TheMathCoach]

I'm just happy to help omar! :)

[@omaridbraym5220]

@@TheMathCoach thank you again

[@kniinortey31]

thank you very much sir it was a great video. I have a question with q2 where you factorize 1on 3 where didn't u multiply it back to the final result

[@TheMathCoach]

Thank you for observing that and letting me know, the factor 1/3 should be included in the final result and I simply missed it when writing out the expression.

[@raffaalzenazhafirah8712]

how do we know the sketch of the contour, how do we draw it? if the sketch is from the question, but what if from the question there is no any sketch of the contour?

[@TheMathCoach]

Hopefully the question will include a description of the contour "half-circle in the upper half plane", "the unit circle" or something along those lines and then you can draw it from these descriptions. Otherwise it will be hard.

[@ranimahassen6296]

When you say it will be hard do you mean it will be a challenging question you'd be expected to work out or it is usually past the expectations of a question? Thanks.@@TheMathCoach

[@amiralidorodian1932]

i love this thank you

[@thabodjoye9789]

thank you so much!!

[@TheMathCoach]

Happy you liked it and found it useful!

[@alvaro5704]

Nice videos!!

[@TheMathCoach]

Happy that they are useful for you, next one in the series will be released this sunday :) so it is coming more content!

[@omaridbraym5220]

Can you proved the inttegral of cauchy's?

[@TheMathCoach]

The proof of Cauchy's Integral *Theorem* can be found in the previous video in this playlist and the proof of Cauchy's Integral *Formula* will be released next Sunday 12:00 AM CET :) Did I answer your question? Or is it another proof you are looking for?

[@omaridbraym5220]

Yes thanks

[@TheMathCoach]

@@omaridbraym5220 great, see ya next week!

[@omaridbraym5220]

@@TheMathCoach see you

[@TheMathCoach]

@@omaridbraym5220 It is uploaded now!