Have you done the integral of ln(gamma(x)) dx, with limits 0 and 1/2? It's a fun integral, and if you have never presented the Glaisher-Kinkelin constant on your channel, here's a perfect opportunity. There are quite a few very funky integrals and products involving that constant.
en.wikipedia.org/wiki/List_of_mathematical_functions?wprov=sfti1#Piecewise_special_functions Wikipedia page listing pretty much all of the special functions you’ll ever encounter. I couldn’t find a table showing how they’re related so you’re gonna have to look at each article individually
Before calculation the derivative of the ratio of two gamma functions it is convenient to make logarithm and then to differentiate, i.e. (log(Г(u)/Г(v)))'= (log(Г(u)))' - (log(Г(v)))'= psi(u)•u'- psi(v)•v' Because psi(x)=(log(Г(х)))'=Г'(х)/Г(х) by definition
Around 6:10 when you take the derivative with respect to s, wouldn’t you get an extra factor of ½ from the chain rule with (s+1)/2 and (s+2)/2 inside the gammas?
Ah it’s fixed around 13:40 when you say the target integral was half of this I’(-1/2). It didn’t seem to be originally defined that way, but it gives the necessary extra factor of ½ to make the final result correct.
@@maths_505 honestly I’d love to see a video about the beta function in general and you can prove all the equivalent integral definitions of the beta function as well as if there are series or product representations and also the relationship with the gamma function
Considering how much you use the beta function I think it would be a good addition the same way you have videos proving properties of the gamma function
No chain rule when differentiating gamma function? Derivative of gamma(2x) for example , its not gamma(2x) times derivative of 2x which is 2? Edit: nevermind i saw a correction in comments
My idea.. 1)x÷sint,perveliminare la radice al denominatore.. 2)feyman I(a) ={Sint} ^a.. con I=I'(-1/2)..3).noto che I(a) la posso esprimere in beta function..ok I=-√(π/2)(Γ(1/4))^2/4
I would use a combination of feynmann and contour integration according to my old friend gamelin taught me... I miss you gamelin... "Analyse par Jean Marie Monnier" got nothing on you :(.
Viewing your videos I am very much left with the impression that you are someone who is very much steeped in the theory of special functions. I studied mathematic over 50 years ago ( mathematical physics to be exact) and naturally we covered all the special functions that come up so often in your solutions. But I can not say I have any of them in the forefront of my thinking. I am ok on the basic functions but can not see Gamma of Beta functions hiding inside integrals the way you can. I wonder if perhaps you are someone who managed to absorb the first half of Whittaker and Watson’s Modern Analysis. Its call Modern Analysis but must be getting on for being ninety years old, but still modern in my opinion.
I actually studied special functions in detail in the course of solving integrals. It's something I'm quite passionate about which is why I'm always on the hunt for opportunities to invoke them. I'll add Modern Analysis to my (growing) list of books to read.
@@maths_505 There are a great many topics I hoped to master but never in the end got round to studying. I guess in the end I was just too lazy. One of these things was the Wiener Hopf method for singular integral equations. I found it all rather difficult. Now if by some chance you could give me the understanding I lack in a twenty minute video then I would end my days ( I am well into my seventies) more satisfied that would be the case otherwise.
@@maths_505 This is a tall order so please don’t feel yourself under any pressure on my behalf. There is a classical book by Muskhelishvilli ( he has other book). In the seventies I borrowed the book from the library, but I never made it passed the first chapter. So it all went on the rather long list of things I failed to achieve.
@@tomasstride9590 thank you my friend. I've just started grad school and that's been taking quite a bit of my time but I will definitely try to squeeze some time out for such content.
@@maths_505 search "jee simplified subject" U will get very good questions (in Hindi but don't listen to soln just see the soln instead ...or u can just search "mathsmerizing integrals questions".. English videos...and it has previous year questions of jee advanced as well
Have you done the integral of ln(gamma(x)) dx, with limits 0 and 1/2? It's a fun integral, and if you have never presented the Glaisher-Kinkelin constant on your channel, here's a perfect opportunity. There are quite a few very funky integrals and products involving that constant.
I like that when you say "ooookaayy cooll! " 😂😂
I think next time I'll count them, together with the "terribly sorry about that" , promised 😉
this is not just amazing it is "Kamaal"! love your videos!
Just waited for your video till late night
Thanks for improving our math interest
bro i tried to be awake but slept😂
Very smart solution. Thanks.
do you know where i could find a table of all the gamma, digamma ,etc functions and all their relations with eachother?
Just follow me on Instagram you'll find lots of cool stuff I use in my videos there 😂
you could try the NIST handbook of mathematical functions
en.wikipedia.org/wiki/List_of_mathematical_functions?wprov=sfti1#Piecewise_special_functions
Wikipedia page listing pretty much all of the special functions you’ll ever encounter. I couldn’t find a table showing how they’re related so you’re gonna have to look at each article individually
Before calculation the derivative of the ratio of two gamma functions it is convenient to make logarithm and then to differentiate, i.e.
(log(Г(u)/Г(v)))'= (log(Г(u)))' - (log(Г(v)))'=
psi(u)•u'- psi(v)•v'
Because psi(x)=(log(Г(х)))'=Г'(х)/Г(х) by definition
Thank you for this one! It’s a really elegant one.
Around 6:10 when you take the derivative with respect to s, wouldn’t you get an extra factor of ½ from the chain rule with (s+1)/2 and (s+2)/2 inside the gammas?
Ah it’s fixed around 13:40 when you say the target integral was half of this I’(-1/2). It didn’t seem to be originally defined that way, but it gives the necessary extra factor of ½ to make the final result correct.
@@MK_Search classic case of two mistakes cancelling out 🤣🤣
That's a nice one.
Please make a video on the constant which you used at the end of video
Could you check if at 13:45 it’s actually 1/2 I can’t seem to understand where that 1/2 is coming from?
Very cool integral also
Where do you find all of these constants and their equivalent forms? Is there like a table of them?
The lemniscate constant was pointed out by a viewer and that's how I found out about it.
@@maths_505 The leminiscate biscuit
Why (***)/4√2 (***) in the minute 14:05 ...????
Is there a way to do this without the intermediate digamma steps? E.g some clever subs and trickery?
Not that I know of. But the final closed form in terms of the gamma function implies that the digamma function is pretty much unavoidable.
Can you do a video proving the relationship between the beta and gamma functions?
That's actually pretty trivial so I'll do a write up on my Instagram
@@maths_505 I don’t have instagram lol
@@skyethebi time to make one 😂
@@maths_505 honestly I’d love to see a video about the beta function in general and you can prove all the equivalent integral definitions of the beta function as well as if there are series or product representations and also the relationship with the gamma function
Considering how much you use the beta function I think it would be a good addition the same way you have videos proving properties of the gamma function
This integral should be decorated with hanging ribbons
No chain rule when differentiating gamma function? Derivative of gamma(2x) for example , its not gamma(2x) times derivative of 2x which is 2? Edit: nevermind i saw a correction in comments
Do you think this can be done without the use of the beta function?
The final result includes the gamma function so fortunately no.
@@maths_505 Thank you.
Can you tell us the writing app/software that you use to write ✍️ electronically on your computer 🖥️ screen?
@@bobbybannerjee5156 I write on my tablet using an s pen and Samsung notes
Which constant is that ω? Some article about? Thankk you!!
Wikipedia should suffice. Search lemniscate constant.
It's asking for a trigonometric substitution, but the log on top ruins it!!
My dear friend....the logarithm yearns for trigonometric arguments.
My idea.. 1)x÷sint,perveliminare la radice al denominatore.. 2)feyman I(a) ={Sint} ^a.. con I=I'(-1/2)..3).noto che I(a) la posso esprimere in beta function..ok I=-√(π/2)(Γ(1/4))^2/4
I would use a combination of feynmann and contour integration according to my old friend gamelin taught me... I miss you gamelin... "Analyse par Jean Marie Monnier" got nothing on you :(.
For some reason I keep getting I(s)=1/2*B(s/2+1/4,1/2) and I dont see my mistake...
Viewing your videos I am very much left with the impression that you are someone who is very much steeped in the theory of special functions. I studied mathematic over 50 years ago ( mathematical physics to be exact) and naturally we covered all the special functions that come up so often in your solutions. But I can not say I have any of them in the forefront of my thinking. I am ok on the basic functions but can not see Gamma of Beta functions hiding inside integrals the way you can. I wonder if perhaps you are someone who managed to absorb the first half of Whittaker and Watson’s Modern Analysis. Its call Modern Analysis but must be getting on for being ninety years old, but still modern in my opinion.
I actually studied special functions in detail in the course of solving integrals. It's something I'm quite passionate about which is why I'm always on the hunt for opportunities to invoke them. I'll add Modern Analysis to my (growing) list of books to read.
@@maths_505 There are a great many topics I hoped to master but never in the end got round to studying. I guess in the end I was just too lazy. One of these things was the Wiener Hopf method for singular integral equations. I found it all rather difficult. Now if by some chance you could give me the understanding I lack in a twenty minute video then I would end my days ( I am well into my seventies) more satisfied that would be the case otherwise.
@@tomasstride9590 may Allah bless you with more years to come and yes I'll gladly take up that task my friend.
@@maths_505 This is a tall order so please don’t feel yourself under any pressure on my behalf. There is a classical book by Muskhelishvilli ( he has other book). In the seventies I borrowed the book from the library, but I never made it passed the first chapter. So it all went on the rather long list of things I failed to achieve.
@@tomasstride9590 thank you my friend. I've just started grad school and that's been taking quite a bit of my time but I will definitely try to squeeze some time out for such content.
ooookkaayyy coooll 😂
There's merch on that now 😂
Can you include JEE Advance questions pls 🥺
Could you email me some good ones? I'd love to help out.
@@maths_505 I got sir.
@@maths_505 search "jee simplified subject" U will get very good questions (in Hindi but don't listen to soln just see the soln instead ...or u can just search "mathsmerizing integrals questions".. English videos...and it has previous year questions of jee advanced as well
early!