Where did the Gamma Function come from?!?! (Full Derivation)

An intriguing and in-depth derivation of the notorious, yet spectacular, Gamma Function. Get some popcorn, this one's interesting!

[@dustydrawer4494]

Definitely a more rigorous approach to defining the gamma function. I've seen another video on this topic from linesthatconnect that has a more intuitive approach for defining a limit formula for the gamma function. It's nice seeing both angles.

[@Mathority1729]

Thanks for watching again, truly appreciate your time! And apologies for the late reply, was traveling the past couple days 🙃. I’ll definitely check out the video you’re talking about, but I agree, it’s great that there’s multiple ways to approach a solution/derivation. That’s one of the many beauties of math!

[@dustydrawer4494]

@@Mathority1729 one of the many reasons why it's a passion of mine!

[@amjadmalik7285]

When I fid the Gamma function, it was just defined, used & never explained! That was in 1981. Just grateful for a superb demo. Thankyou.

[@Mathority1729]

I’m super glad to hear the video helped you!! Thank you so much for watching, I really appreciate it! 😄

[@herbie_the_hillbillie_goat]

One of the several speculative theories I've heard is that Euler adjusted the Gamma function to be one off from the factorials when he realized that doing so gives the recursive: Γ(x+1) = xΓ(x). Others have said that it brings it in line with Euler's Beta function. I don't know how true any of these theories are, but I've always been of the camp that advocates defining things that make the most sense rather than how they make my pet formulas look. I'd prefer a Γ(n) = n!, and Tau as the circle constant.

[@fetch7312]

I was trying to derive the gamma function with the help of my calc teacher before winter break, and i thought it had something to do with pi notation, but I find this solution very interesting and elegant. Thank you for the video!

[@Mathority1729]

That’s awesome to hear! It was way back in high school (8 years ago maybe) when I figured out this derivation as a matter of fact! I was spending some time messing around with the gamma function because I thought it was so cool and really wanted to understand where it came from, and it was a beautiful revelation to me to see how it ties in with the log function. I would show some of my proofs for various things like this to my calculus teacher as well haha. He was awesome lol. Anyways, super glad to hear of your effort and adventure into the gamma function and math in general. Thank you so much for watching and for your kind comment, I really appreciate you! 😄

[@vishtoxic5928]

@@Mathority1729 you must have had a really great calculus teacher to share things with back then!

[@ChaimYosefMariateguiLeviPhD]

Thanks, i understand the gamma function after 50.years thanks to your video!

[@Mathority1729]

That makes me super happy to hear!! Thanks so much for watching, I really appreciate you! 😄

[@JustNow42]

The gamma function is more than that. There are surely an infinite number of functions that pass through n!. For all n in N. But the gammafunction is the only one that? Come on let's have it

[@Mathority1729]

@@JustNow42 look up the proven *Bohr-Mollerup Theorem* , which states that the Gamma Function is the only positive function f, with domain on the interval x>0, that simultaneously has the following three properties: • f(1) = 1, and • f(x + 1) = x f(x) for x > 0 and • f is logarithmically convex So it’s unique in that way! Very interesting stuff! Thanks for watching! Btw there are other extended definitions of the factorial function, such as Hadamard's gamma function, which defines it for all real values and no poles. Anything other than the original gamma function, we technically call pseudo-gamma functions

[@txreddy]

Same here 🙂

[@goncalofreitas2094]

I loved this explanation, thank you so much! Never thought those log integrals at the beginning would lead us right to the Gamma function, really nice!

[@Mathority1729]

Is my pleasure! Glad to hear you enjoyed it!! Thank you so much for watching! 😄

[@jhonnyrock]

I've been playing around with the gamma function recently, this was awesome to learn its derivation! Thank you!

[@Mathority1729]

So glad to hear that! Thanks a ton for watching! 😄

[@fcvgarcia]

Many many thanks for sharing. What a great explanation!

[@Mathority1729]

No problem! I’m very glad you enjoyed! Thank you so much for watching! 😄

[@tahafarah5692]

Excellent intuition , thank you ❤

[@valentinx1107]

Thank you so much for this awesome proof!

[@mujimuji6002]

Dude, without this is the simplest and slickest formulation of the gamma function i’ve ever seen! Of course i’ve seen the -ln(x)^n version but never knew how we got there; not to mention the product version of this is terrifying for someone just learning. I hope your channel blows up soon cause that was quality boss. (Also what app do you use to write? It looks cool lol)

[@Mathority1729]

Thank you so much for the kind words! I’m super glad you enjoyed the video and found value in it! I’ve always wondered why there wasn’t a more straightforward explanation for the gamma function on KZbin, one that even an entry level calculus student could grasp! There’s various derivations, but I agree that they’re not as palatable for someone who’s passionate about math but just starting their journey into it. Again, thanks a ton for the support, many more videos to come this new year! And btw the app I’m using is just Notability! Happy New Year! 🎊

[@MinecraftForever_l]

You've earned my subscribe! It was really astonishing!!!

[@Mathority1729]

Thank you so much my friend! Really glad you enjoyed the video 😄

[@MinecraftForever_l]

@@Mathority1729 how about a complex argument? Will you make such a vid?

[@Mathority1729]

@@MinecraftForever_l I’ll make note of that! Thanks for the suggestion, that’s a good idea 👍

[@undisclosedmusic4969]

I have to admit, any math video with f-bombs gets my approval

[@Mathority1729]

Hahahaha 😂, thanks so much for watching, I really appreciate it! 😄

[@devrus265]

You earned my subscription, the video was clearly explained

[@Mathority1729]

Thanks a ton! Glad to hear you enjoyed it! 😄

[@ultrametric9317]

Advice from an old head - never use n and u together :) They look so much alike it is easy to mistake them for each other. As to why gamma is so defined? The function shows up most primitively in the formulas for the volume measure of the n-ball. This has an extra 1 where it is unnatural, and therefore the "shifted gamma" without the 1 is most natural. But that's life :)

[@Mathority1729]

You’re absolutely right! I noticed after posting the video that the n and u together were a poor choice of variables haha….my apologies for that! Will be noted for future videos! 😅 Personally, I prefer the function without the shift of 1 as well! I agree the Volume of n-ball and some other great results would look more natural! However, I can already see the other side making the opposite case since technically the formula for the Surface Area of an n-ball would go from having no shift under the current accepted definition of the gamma function to having an extra -1 shift if we had our way 😆 I still do agree with you though that I would prefer to have the gamma function not shifted, simply so as to maintain parity with the factorials where Γ(n)=n! I think that’s simple and beautiful, and the way that it’s naturally derived in my opinion :) And thanks a ton for the helpful and fun comment and for watching the video, I appreciate it greatly!

[@HighKingTurgon]

Gorgeous video. Really helped me induce it along with your explanation.

[@Mathority1729]

Thank you so much for the kind words, so glad you enjoyed the video! 😄

[@vishtoxic5928]

Thanks for showing everything step-by-step and not skipping any parts. This was really fascinating to see, I always wondered how a factorial of a non-integer could be computed and the gamma function formula you derived shows how we can find the result. I'm excited for more of your videos! Edit: What side are you on, do you agree with the formula having n+1 within its parameters or just n.

[@danielc.martin1574]

I know how to motivate the definition by the laplace transform of t^n, but definitely this is simpler and better

[@Mathority1729]

Thanks so much for watching! Glad you liked the solution!

[@joetsu7109]

best proof i have seen on youtube for the gamma function thanks alot for this amazing video

[@Mathority1729]

Thanks so much for the kind words, really glad you enjoyed the video!! 😄

[@burakmuezzinoglu3730]

This is the best derivation of gamma function on KZbin cause it starts with an observation I.e. reality, I mean math itself. Nearly all other derivations out there are just some “rigorous proofs” of a some formula brought from heavens of the Lord by a genius man already knowing the formula. Please do a similar work for Lambert W

[@Mathority1729]

Hahaha thanks a ton! I’m really glad you enjoyed the video! And I agree, it’s more fun and useful to derive concepts in a way that is more straightforward and realistic, especially for the average person! I think this is how you bring people into mathematics!

[@thereisnospoon205]

This was both detailed and clear, a real matter class in how to explain complex ideas

[@Mathority1729]

I really appreciate that! Glad you enjoyed! Thank you so much for watching and for the kind words 😄

[@willyh.r.1216]

Very satisfying!

[@Mathority1729]

Glad to hear it! Thank you so much for watching!

[@manjunathchandrasekhar1830]

Hats off ! Brilliant explanation. How did the lower and upper bound of integration between 0 to 1 in the log function come about?

[@bestondivine1368]

Thank you so much... really helpful

[@Mathority1729]

Glad you enjoyed! Thanks for watching! 😄

[@Randomguyonline123]

Helpful video really benefited from it thank you :)

[@Mathority1729]

I’m very glad to hear! Thanks a ton for watching, I greatly appreciate it! 😄

[@9s572]

Thank you so much!!!

[@Mathority1729]

No problem! I’m really glad you enjoyed the video! Thank you so much for watching, truly appreciate it! 😄

[@fernandojimenezmotte2024]

amazing derivation of GAMMA´s FUNCTION , well explained and very didactic

[@Mathority1729]

Thank you so much! Really appreciate you watching! Glad you enjoyed 😃

[@rupamgudlu1312]

Wonderful video

[@Mathority1729]

Thank you so much! 😄

[@junetai]

Thank you for providing the clear explanation of gamma function

[@Mathority1729]

No problem! Thank you so much for watching! 😄

[@larzcaetano]

May God bless you and receive you in Glory. This is incredible. Thank you so much!

[@Mathority1729]

Thanks so much for watching and for the kind words! God bless you too! 😄

[@larzcaetano]

@@Mathority1729 is there a community on discord or anything for your channel? Or if there is a way to get in touch with you at all?

[@Daniel-yc2ur]

Awesome stuff

[@Mathority1729]

Thanks for watching, glad you enjoyed!

[@brendonpearl8333]

That helped me lot. Still learning

[@Mathority1729]

Super glad to hear that! Glad you benefited! Thanks so much for watching 😄

[@asakhhh]

Great video! My question is, how to get to the expansion of this integral to non-integer numbers?

[@ZahidAli-ur8lv]

Beautiful ❤️❤️

[@Mathority1729]

Thanks so much :)

[@sharthkumarshetty]

Thank you❤

[@Mathority1729]

Of course! Thanks for watching! 😄

[@MikeT10101]

Excellent video. Subscrided ; )

[@Mathority1729]

I appreciate that so much! Glad you enjoyed! Thank you for the support 😄

[@mohamedmouh3949]

thank you so much 🤩🤩🤩

[@Mathority1729]

Of course! Thank you for watching! 😄

[@lucasfranco1758]

i loved it tysm

[@Mathority1729]

So glad you enjoyed it! Thanks so much for watching! 😄

[@othila9902]

Hi, I would like to know how you can affirm that this derivation of the gamma functions applies to al real positive numbers? In this case you used the logⁿ(x), but you took n as a positive integer. I'd like to know if there's a way to affirm this works for any real number (asise from negative integers), thanks!

[@bigaltheoriginal6579]

whoa all my old math texts just state as a given never explained !

[@Mathority1729]

Exactly! That always bothered me! Many years ago when I was trying to figure out where the gamma function came from, the best I could find from different math texts was the log definition of the gamma function, but it never explained why or how to transform it into the recursive definition. I spent like a whole day way back in high school during a weekend playing around with it myself till I figured it out haha. And I’m super glad to share with all of you! Everyone should know how to derive it, it’s very satisfying. And thank you so much for watching, I really appreciate you! 😄

[@bignerd9514]

amazing video! btw what writing software do you use?

[@Mathority1729]

Thanks so much! I just use notability!

[@spicymickfool]

Another derivation, the Laplace Transform of a polynomial function yields the result when evaluated at s=1.

[@Mathority1729]

Absolutely, that’s another great way! Thanks for watching, rly appreciate it 😄

[@samcousins3204]

so why do we make the change of variables at the end, even once we've created a functional definition of n! ?

[@Mathority1729]

That’s a very good question! The reason for the change of variable is because the final equation provides a practical definition for the Gamma Function. I’ll make a video on this in the future actually, but in the meanwhile to see for yourself, try evaluating Γ(n+1) and solve the integral using integration by parts. You end up getting n times the integral definition of the Gamma Function, hence generating a recursive definition where Γ(n+1) =nΓ(n) =n(n-1)Γ(n-1) =n(n-1)(n-2)Γ(n-2) etc. Hopefully that helps at least a little bit! Definitely try it out for yourself! I will explain further in a future video And thanks so much for watching the video and for asking a great question that I’m sure many others are wondering as well! Really appreciate your time and your interest! 😄

[@samcousins3204]

@@Mathority1729 oh, I see. Thanks for the response, and great video.

[@therealist9052]

Where does the intuition / reasoning for the variable substitution starting at 18:24 come from? That's the only thing I'm confused on. Like just why that particular set of substitutions??

[@user-im7lm8op5l]

Прикольно. Спасибо за Математику

[@jack_t_r7068]

10/10 explanation, wish your u’s and n’s looked a bit more different for sake of clarity haha

[@Mathority1729]

Lol sorry about that, I noticed that too afterwards, definitely could’ve been better! Or at least should’ve chosen a different variable…. But anyways, thanks so much for watching, I really appreciate it! And I’m glad that you found the explanation enjoyable!

[@ultrametric9317]

BTW you might give the derivation of the volume measure of the n-ball. Very fun.

[@Mathority1729]

It’s incredible that you mention that! I have a fun story haha. About 9 years ago, back when I was taking my first calculus course back in school, I was doing my homework and randomly had a desire to derive the area of a circle by integration. I figured it out after a while, and immediately went on to try and derive the volume of a sphere, which I eventually figured out as well. From there I wondered if I could extrapolate the method to figure out the formula for the volume of a 4-D sphere/ball in theory, so I applied the method and came up with a result for a 4D and 5D ball. And I realized that you could recursively define the Volume of an (n)-dimensional ball based on the volume of an (n-2)-dimensional ball. Basically, wrote down a recurrence relationship for the volume of an n-ball. I had no idea if it was right, so I looked online to see if anyone had done this before, and of course, they had haha. But to my delight, my results tallied with what I found online, and I still remember how fun it was! And this recurrence relationship is how the gamma function pops up! But anyways, thanks for bringing this up, I’m super glad you reminded me of the n-ball! I had almost forgotten….but now perhaps I’ll plan a future video on the topic! Cheers and Happy New Year btw!

[@MeMyselfForWho]

Very cool and educational, thank you. Do you have a background in math?

[@Mathority1729]

Glad you enjoyed! I actually have a background in computer science, however, I’ve always been most passionate about math. Back in school, I would compete in various math competitions at the state level, and I would love to just learn new math for fun. And the beauty of pure math is something I still very much enjoy to this day :) Thanks so much for watching the video and giving the channel a shot, I greatly appreciate it! 😄

[@MeMyselfForWho]

@Mathority1729 I totally get it, I want to understand pure math in its entirety 😅

[@mihaipuiu6231]

Interesting proof of n!

[@Mathority1729]

I appreciate that, glad you enjoyed! Thanks for watching 😄

[@augustodutra3839]

One reason for the shift might be because the gamma function becomes the mellin transform applied to e^(-u).

[@tamilselvanrascal5956]

🎉🎉🎉

[@miguelf218]

Oh god thanks

[@Mathority1729]

No problem, thanks a ton for watching! :)

[@sotocsick3195]

Great video! Easy to follow and to understand :) . But why does the gamma function work for all z element C, R(z) > 0?

[@Mathority1729]

Thanks a ton for watching! Appreciate it! That’ll have to be an entirely separate video haha

[@sotocsick3195]

@@Mathority1729 can't wait to see it :)

[@mashiroelaina2177]

Hi, I’m a Chinese high-school student. I found that ur vid about gamma function is really inspiring. Is it okay if I transport your video to Chinese study platform cuz Chinese cannot get access to KZbin ( I use some special ways) and I think videos introducing gamma functions in Chinese platform are not really good (they just tell u how to apply this in tests with no process of deduction this formula). I will write a description about the fact that I transport the video from here. If you are not willing to let me transport this video, I will not transport😢

[@Mathority1729]

Thanks so much for watching! Glad you enjoyed! And I really appreciate you asking for permission, however, I don’t feel comfortable having the video moved to other platforms. I’m so sorry to disappoint you on that :/

[@mashiroelaina2177]

@@Mathority1729 thx! I will not move this video to other platforms🫡

[@helmutklocker4075]

Small problem dealing with limit at x=1

[@latifrahman3245]

Please derive in the similar way a formula for Beta function.

[@biddu2683]

👍👍👍👍👍👍

[@Mathority1729]

Thank you so much for watching! 😄

[@raghavkumar4914]

Origin of lambert W function please

[@KSM94K]

I would like to request you about something, can you upload how to solve differential equations using Fourier and Laplace transforms, I know their physical significance but still , it's just my request and your choice..... Btw love your videos, you'll be famous in math community soon

[@Mathority1729]

I will definitely plan some videos for that, those are great suggestions! And thank you so much for the kind words and support! It means a lot, I tremendously appreciate it! 😄

[@gjproductions9337]

At 18:52 why does dx contain du?

[@Mathority1729]

Since we are doing a change of variable, we need to to compute the differential of the new variable, u

[@dzukik]

Why are you evaluating the derivative of log(x) as 1/x and not 1/(xln10)? Did u mean to write lnx instead of logx?

[@diegocabrales]

In mathematics is not usual to use ln(x) as the notation of the natural logarithm. That's the notation used by engineers. It is instead used log(x) and it's the notation used here, as you can see because he said that dlog(x)/dx = 1/x. If you want to talk about the decimal logarithm, you then write log10(x). The engine Wolfram|Alpha and the programming language Scilab are another example of this notation in mathematics, since the command log is used for the natural logarithm and log10 for the decimal logarithm. The reason for this notation is that natural logarithm is more common to use than decimal one. There are various reasons for that: The exponential function f(x) = exp(x) = e^x appears almost everywhere and its inverse function is log(x). The derivative dlog(x)/dx = 1/x, while the derivative dlog10(x)/dx = [1/log(10)](1/x), with that annoying constant 1/log(10). etc.

[@herbie_the_hillbillie_goat]

The circle constant should be defined by it's radius as the circle itself is defined by its radius (all points equidistant from a central point). As are the circle (i.e. trig) functions. Tau is logically the better choice.

[@Mathority1729]

I 1000% agree! I say the same thing all the time 🤣 Thanks so much for watching!

[@helmutklocker4075]

x ln(x)=0 for x=1

[@SnowboardAddict37]

I’m sorry, but this did *not* have to be 20 minutes long. Needlessly dragging this over