Hey BSOM, I wanted to let you know how much I appreciate this series. I watched what I could over the winter and it helped me get a 97% in my Real Analysis course this past semester as a junior. I couldn't have done it without your help man so thankyou!
I got stuck in my research and doubted my maths, so I spent ~10 hours reviewing the real analysis. Now I am a bit more confident and getting ready for your functional analysis series. Thank you so much!
Congratulations on finishing up this great series ! I'll definitely come back here from time to time to review some fundamental concepts I may have doubts on, and I'm sure many other students will do so as well !
Hello, thank you for the great video. I've got a question: when we say that for example integral on [-1, 1] of (1/x) doesn't exist, do we mean that in all possibile manners in which we approach to x = 0, the integral has no definite value? And if it is so, how it's possibile to say that: "approaching in a specific way (the symmetric one, prescribed by Cauchy) we get a definite value for the integral"? It makes not much sense to me, could you help me?
I found by accidemt something you could find interesting... if you take in Wolfram-Alfa the finite duration fourier transform between [-1; 1] of the function (1-x^2)^4 (which is an approximation of e^(-4x^2) but with compact-support), you will find a polynomial of the frecuency mixed with sinc functions (is a bit large for here), but is analytic and perfectly defined on the reals, let call it F(w)... but when you try to plot F(w) or take its absolute integral it becomes messi since numerical issues rises, like is a not-calculable function near w=0 for computers... is really weird, but if you just simple integrate it without the absolute value int_R F(w)dw = 2 pi.... hope you view it for the real analysis series... since is like a ficticious noise-like-rippling singularity that rises from nowhere and mess your analysis... only through finite terms taylor expansion you can avoid it... is a weird example where precission plays against you because of floating point issues.
Regarding the integral of 1/x from -1 to 1, one can still find meaning without using the Cauchy p.v.. In fact, making the substitution y=1/x with dx=-dy/y^2, the integral remains -dy/y with the same limits of integration. Of course, the only number that is equal to its opposite is zero.
I think there's a flaw in that reasoning: to add or subtract the result of those integrals, they need have to be properly defined, but this is not the case. It is like saying that "x+1=x" has infinite solutions because you can "divide by infinity" to obtain "0=0".
@@japedr I see what you mean, but in this case the substitution yields an equation of the form I=-I where I is the original integral. Nothing is added or substracted.
hi, I really loved the videos, they are really helpful for uni lectures. you cover same topics in much more understandable way than the professor in class!! I am also curies about, if you will ever make videos about differential equations, for example first order linear differential equations. Thank you!
Hey BSOM, I wanted to let you know how much I appreciate this series. I watched what I could over the winter and it helped me get a 97% in my Real Analysis course this past semester as a junior. I couldn't have done it without your help man so thankyou!
Nice work! :)
I got stuck in my research and doubted my maths, so I spent ~10 hours reviewing the real analysis. Now I am a bit more confident and getting ready for your functional analysis series. Thank you so much!
You are very welcome! And thank you for your support :)
Congratulations on finishing up this great series ! I'll definitely come back here from time to time to review some fundamental concepts I may have doubts on, and I'm sure many other students will do so as well !
I'm following this whole series from the beginning which helps me to figure out a lot of interesting things! Thank you for everything!
wow you truly are the best further mathematics youtube channel
Thanks a lot :)
Great work... my better understanding of real analysis is not possible with out you...Love you from India...
These are very informative and fun lectures! I'll have to crack open my old Calculus textbook and take a crack at the harder exercises and proofs.
Thank you! I am glad that you could enjoy it. I have other lectures as well if you want to dive in Complex Analysis, for example :)
Fantastic series! Thanks so much!
Thank you very much :)
Hello, thank you for the great video. I've got a question: when we say that for example integral on [-1, 1] of (1/x) doesn't exist, do we mean that in all possibile manners in which we approach to x = 0, the integral has no definite value? And if it is so, how it's possibile to say that: "approaching in a specific way (the symmetric one, prescribed by Cauchy) we get a definite value for the integral"? It makes not much sense to me, could you help me?
The integral with the usual definition does not exist. A limit of ordinary integrals, however, exists.
Brilliant series. Cheers mate.
Thank you for watching :)
I found by accidemt something you could find interesting... if you take in Wolfram-Alfa the finite duration fourier transform between [-1; 1] of the function (1-x^2)^4 (which is an approximation of e^(-4x^2) but with compact-support), you will find a polynomial of the frecuency mixed with sinc functions (is a bit large for here), but is analytic and perfectly defined on the reals, let call it F(w)... but when you try to plot F(w) or take its absolute integral it becomes messi since numerical issues rises, like is a not-calculable function near w=0 for computers... is really weird, but if you just simple integrate it without the absolute value int_R F(w)dw = 2 pi.... hope you view it for the real analysis series... since is like a ficticious noise-like-rippling singularity that rises from nowhere and mess your analysis... only through finite terms taylor expansion you can avoid it... is a weird example where precission plays against you because of floating point issues.
Regarding the integral of 1/x from -1 to 1, one can still find meaning without using the Cauchy p.v.. In fact, making the substitution y=1/x with dx=-dy/y^2, the integral remains -dy/y with the same limits of integration. Of course, the only number that is equal to its opposite is zero.
I think there's a flaw in that reasoning: to add or subtract the result of those integrals, they need have to be properly defined, but this is not the case.
It is like saying that "x+1=x" has infinite solutions because you can "divide by infinity" to obtain "0=0".
@@japedr I see what you mean, but in this case the substitution yields an equation of the form I=-I where I is the original integral. Nothing is added or substracted.
Great video :)
I tried to log in the new webpage, but I could not :(
Oh, that is strange. Maybe you can send me an email with screenshot?
@@brightsideofmaths It works from another computer. Perhaps the error is on my side.
hi, I really loved the videos, they are really helpful for uni lectures. you cover same topics in much more understandable way than the professor in class!!
I am also curies about, if you will ever make videos about differential equations, for example first order linear differential equations. Thank you!
The series already exists :)
The first videos can be found here: tbsom.de/s/ode
@@brightsideofmaths you've just become the saviour of the day :)
@@meltedwings Thanks! The series is not complete yet but some videos are in early access for supporters :)
very interesting, thanks !