I could never find any documentation on this derivation and the history! I'm sooo glad and thankful you made this video! Your explanations made perfect sense and you went over every step, thanks!
Thanks. I’m glad you enjoyed the video. Yes, for some reason most sources I have encountered just present the integral form of the Gamma Function without providing any background on how it was discovered. That might be because the derivation is tricky and took some educated guesses on Euler’s part. I guess this video is mainly of historical interest, but it’s exciting to follow Euler’s genius.
@@FireSermon83 yeah it's very exciting to see how Euler figured this out, and going over the derivations I often gain new skills and intuition 😊 so thank you for this incredible video
thank you, thank you, thank you youtube algorithm for recommending this channel. i've been looking for just this type of content. i don't understand pure abstractions. i learn by following historical developments. i need to know what the inventor was thinking when they invented whatever it is being presented. this style of presentation is perfect for me and i thank you from the bottom of my heart for making and sharing it. overjoyed to subscribe and will be binging the other 11 videos. (please make more).
Thanks for the positive feedback. I could never understand why textbooks don’t derive the Gamma Function- maybe it’s because Euler’s derivation required a lot of intelligent guesswork. But it’s very rewarding to see how he came up with it!
Loved the explanation, which I never saw in full before. Note, though, that you didn't prove that the integral form gives the same results as the infinite product form for all x. There are infinitely many expressions that match the values of the factorial function for non-negative integers but which differ for the rest of the real numbers. The infinite product and the integrals you developed so nicely agree FOR ALL REALS except negative integers, where both functions are undefined. I've read 😊lp
...continued: The integral and infinite product forms meet an additional criterion that forces them to agree. That condition is for the factorial curve to be log-convex for positive reals.
Thanks for your comments, which are well taken. When I get around to revising the videos I will try to make these points more clear. I certainly could have mentioned that the integral form diverges if we let “n” equal a negative integer. Because this video is already rather long, I do have a separate video on my channel where I show that the infinite product form and integral form of the gamma function are equivalent. I’m just an amateur, and my videos are not meant to serve as a textbook or rigorous treatment of the material- which is why I avoided a discussion of issue of the gamma function being log-convex. It feels like too much detail for an introductory video, and honestly, is beyond my own level of mathematical sophistication.
@DoctorPeregrinus since posting my comment, I saw you made a subsequent video showing equivalence of product and integral forms, so you already answered that part. I agree that the log-convex thing would require WAY too much background to fit into one video. Your explanation is the best I've seen!
Great explanation, and very nice combination of writing and digital manipulation. Have you considering using more animation to show expressions moving from one step to the next? That could shorten the video by reducing the time spent watching you write out the same expressions again and again.
I'm so thankful that you posted this! I've been looking forever for a derivation of the integral form of the gamma function but haven't found any until now! As much as I dislike the way you phrase and write certain things it didn't take away from the actual derivation! I wish I had my pencil but I lost it (and am too autistic to use another one that isn't of the same type) so unfortunately I couldn't take notes. I'll probably reference this video again in the future, then lol. Thanks again for this!
Thanks for your comments. Are there any specific parts of the video you feel could be phrased or written out better? I plan to revise and improve these videos in the future based on viewer feedback.
@@FireSermon83 personally I dislike writing the fractions out front rather than putting the integral inside the fraction or vice versa but that’s just my brain wanting to compact things. I understand why people do it (and in the case of your video it was actually really helpful), but it’s a thing I can’t get over. A bit of ambiguous notation I say you write was e^lnx^n or sometrhing like that, where it was hard to tell if you were supposed to do the e^lnx first or the lnx^n (without context, at least). Most mathematics youtubers I see write parentheses around the e^lnx for clarity, so I would recommend adopting that also, even if context would clear it up. You say “the integral from 1 to 0” when the lower bound was 1 and upper bound was 0 which to me has always been backwards. I don’t know if you’re necessarily incorrect there but it bugged me. Not enough to turn my attention off but enough my mind made note of it. I think the way you did u-substitution was a little bit weird (adding an extra step of du/dx={stuff} and then multiplying by dx) but it was fine. Everything else was really good! Those are all I really remember objecting to mentally. The first is stupid so probably don’t go out of your way to change that bit of notation, it’s just on my mind enough that I thought I should mention it. I like your style, very reminiscent of Sal Khan. As much as I enjoy the likes of 3blue1brown, your video was also very good. Overall I think the second and third things I mentioned are the most important criticisms I had, the others were either personal brain stuff or minor nitpicks that amount to maybe a few seconds of extra time spent which didn’t add up to much.
Thanks for your helpful feedback. I will try to answer in a general way that I hope will be helpful. L’Hopital’s rule tells how to evaluate limits when we have an expression that equals either “0/0” or “infinity/infinity.” The rule says that to find the limit, we just have to differentiate the numerator and denominator of the expression repeatedly until we can differentiate no further, and then take the limit. So here we have (1-x^0)/0. As long as x itself does not equal zero, which we will have to assume here, x^0 always equals 1. So (1-x^0)/0 = 0/0. That expression is undefined unless we treat it as a limit to be evaluated! The way to make sense of this expression then is to rewrite it as (1-x^z)/z and take the limit as z goes to zero and use L’Hopital’s rule to evaluate.
I could never find any documentation on this derivation and the history! I'm sooo glad and thankful you made this video! Your explanations made perfect sense and you went over every step, thanks!
Thanks. I’m glad you enjoyed the video. Yes, for some reason most sources I have encountered just present the integral form of the Gamma Function without providing any background on how it was discovered. That might be because the derivation is tricky and took some educated guesses on Euler’s part. I guess this video is mainly of historical interest, but it’s exciting to follow Euler’s genius.
@@FireSermon83 yeah it's very exciting to see how Euler figured this out, and going over the derivations I often gain new skills and intuition 😊 so thank you for this incredible video
thank you, thank you, thank you youtube algorithm for recommending this channel. i've been looking for just this type of content. i don't understand pure abstractions. i learn by following historical developments. i need to know what the inventor was thinking when they invented whatever it is being presented. this style of presentation is perfect for me and i thank you from the bottom of my heart for making and sharing it. overjoyed to subscribe and will be binging the other 11 videos. (please make more).
Thanks for the positive feedback. I could never understand why textbooks don’t derive the Gamma Function- maybe it’s because Euler’s derivation required a lot of intelligent guesswork. But it’s very rewarding to see how he came up with it!
Materials presented are top of the class, and care enough to share. Thank you indeed.
Thanks!
You got 1 subscribe from me man! You more than deserve it
Thanks!
Bro you are superhero of the 🌎
At the end perfect video does exist actually 🔥🔥
Thanks! I’m glad you liked it!
This is truly insightful! And thank you so much!
I’m glad you liked it. Thanks for your positive comments!
Loved the explanation, which I never saw in full before. Note, though, that you didn't prove that the integral form gives the same results as the infinite product form for all x. There are infinitely many expressions that match the values of the factorial function for non-negative integers but which differ for the rest of the real numbers. The infinite product and the integrals you developed so nicely agree FOR ALL REALS except negative integers, where both functions are undefined. I've read 😊lp
...continued: The integral and infinite product forms meet an additional criterion that forces them to agree. That condition is for the factorial curve to be log-convex for positive reals.
Thanks for your comments, which are well taken. When I get around to revising the videos I will try to make these points more clear. I certainly could have mentioned that the integral form diverges if we let “n” equal a negative integer. Because this video is already rather long, I do have a separate video on my channel where I show that the infinite product form and integral form of the gamma function are equivalent. I’m just an amateur, and my videos are not meant to serve as a textbook or rigorous treatment of the material- which is why I avoided a discussion of issue of the gamma function being log-convex. It feels like too much detail for an introductory video, and honestly, is beyond my own level of mathematical sophistication.
@DoctorPeregrinus since posting my comment, I saw you made a subsequent video showing equivalence of product and integral forms, so you already answered that part. I agree that the log-convex thing would require WAY too much background to fit into one video. Your explanation is the best I've seen!
Thanks for the supportive comments. I’m glad you enjoyed the video!
Great explanation, and very nice combination of writing and digital manipulation. Have you considering using more animation to show expressions moving from one step to the next? That could shorten the video by reducing the time spent watching you write out the same expressions again and again.
I'm so thankful that you posted this! I've been looking forever for a derivation of the integral form of the gamma function but haven't found any until now! As much as I dislike the way you phrase and write certain things it didn't take away from the actual derivation! I wish I had my pencil but I lost it (and am too autistic to use another one that isn't of the same type) so unfortunately I couldn't take notes. I'll probably reference this video again in the future, then lol. Thanks again for this!
Thanks for your comments. Are there any specific parts of the video you feel could be phrased or written out better? I plan to revise and improve these videos in the future based on viewer feedback.
@@FireSermon83 personally I dislike writing the fractions out front rather than putting the integral inside the fraction or vice versa but that’s just my brain wanting to compact things. I understand why people do it (and in the case of your video it was actually really helpful), but it’s a thing I can’t get over.
A bit of ambiguous notation I say you write was e^lnx^n or sometrhing like that, where it was hard to tell if you were supposed to do the e^lnx first or the lnx^n (without context, at least). Most mathematics youtubers I see write parentheses around the e^lnx for clarity, so I would recommend adopting that also, even if context would clear it up.
You say “the integral from 1 to 0” when the lower bound was 1 and upper bound was 0 which to me has always been backwards. I don’t know if you’re necessarily incorrect there but it bugged me. Not enough to turn my attention off but enough my mind made note of it.
I think the way you did u-substitution was a little bit weird (adding an extra step of du/dx={stuff} and then multiplying by dx) but it was fine.
Everything else was really good! Those are all I really remember objecting to mentally. The first is stupid so probably don’t go out of your way to change that bit of notation, it’s just on my mind enough that I thought I should mention it.
I like your style, very reminiscent of Sal Khan. As much as I enjoy the likes of 3blue1brown, your video was also very good.
Overall I think the second and third things I mentioned are the most important criticisms I had, the others were either personal brain stuff or minor nitpicks that amount to maybe a few seconds of extra time spent which didn’t add up to much.
Thanks again. I’ll go over your comments again and make a note of those suggestions when I update the video in the future.
@@FireSermon83 of course! No problem!
I think we need some kind of justification to take the limit under the integral. That doesn't seem hard, though.
The L'Hopital section was unclear for me. But great video'
Thanks for your helpful feedback. I will try to answer in a general way that I hope will be helpful.
L’Hopital’s rule tells how to evaluate limits when we have an expression that equals either “0/0” or “infinity/infinity.” The rule says that to find the limit, we just have to differentiate the numerator and denominator of the expression repeatedly until we can differentiate no further, and then take the limit.
So here we have (1-x^0)/0. As long as x itself does not equal zero, which we will have to assume here, x^0 always equals 1. So (1-x^0)/0 = 0/0. That expression is undefined unless we treat it as a limit to be evaluated! The way to make sense of this expression then is to rewrite it as (1-x^z)/z and take the limit as z goes to zero and use L’Hopital’s rule to evaluate.