a mathematically stunning formula

a mathematically stunning formula. I find the formula for the second derivative of the gamma function at 1 which gives you pi^2/6 plus the euler mascheroni constant squares. This gives a beautiful identity that relates two mysterious constants. The gamma function is a generalization of the factorial which is used to count permutations and combinations. This is related to nice integrals using logarithmic and exponential functions. A must see for calculus students and anyone who is interested in math and sciences. It involves the Hurwitz and the Riemann zera functions and might be related to the Riemann hypothesis
Integral e^-x ln(x) • Integral of e^-x ln(x)...
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[@GeoffryGifari]

so the polygamma function can be written in different ways?

[@General12th]

Was that a poly joke?

[@jimjam623]

Love the enthusiasm you bring to math - that alone makes challenging stuff so much more enjoyable to learn. Thank you 😊 🙏

[@jimjam623]

And yes - it’s definitely a close second to e^(i*pi) = -1 👌😎

[@cycklist]

Just seeing Dr Peyam's smile makes my day ❤️

[@rgerk]

Next video: gamma'''(1) = -EulerGamma^3 - (EulerGamma Pi^2)/2 - 2 Zeta[3]

[@alienmoonstalker]

I liked the part when you used Polygamma potion to get to Digamma alley.

[@brendanlawlor2214]

the Basel result of pi^2/6 also arises in the Variance of the extreme value distribution namely exp ^- ( e ^- x ) which becomes the digamma namely the gamma integrand times a logarithm . My work in finance studies the Integral of the gamma Not it's Differential as here but there is no closed form as the logarithm appears on the bottom . Any ideas here Doc ? really appreciate some help . Integral of the Gamma Function ??

[@General12th]

Triple cool! I've never liked the gamma function compared to the version without the offset input. Is the polygamma function one of those times it's so much easier to use the "weird" gamma function instead of the "natural" pi function?

[@soyoltoi]

What is the nth derivative of Γ at 1? Do the values converge as n blows up?

[@ruceblee969]

What was that constant again?

[@anupamamehra6068]

euler-mascheronni constant approx 0.577

[@unfulfilling_entertainer]

Oily macaroni

[@garyhuntress6871]

Over your head? Nonsense!

[@tgeofrey]

Gama..wow

[@azzteke]

Gamma.

[@General12th]

​@@azzteke Gramma.

[@loicetienne7570]

@@General12th Grandma ?

[@nicolascamargo8339]

Wow genial

[@davidramos4707]

Can you factor the answer

[@ruffifuffler8711]

Useful as the ergative of the imaginary underlying ellipse?, stretching the truth, but still in relative vendiction??

[@mikelolis3750]

I've been writing my undergraduate thesis on Analytic Number Theory and as a part of it I had to develop the theory of the Gamma function as a meromorphic function. I learnt many awesome things about it and one of them was the formula for its logarithmic derivative, so I decided to try solving this problem with it but without seeing the video and I indeed got the thumbnail's answer. Pretty cool :P

[@Noam_.Menashe]

I used this once to solve an integral. I think it was integral from 0 to infinity of sinx/x*(lnx)^2dx

[@thomasborgsmidt9801]

Why did You stop there? Well when you take ln to anything, then e is involved in one way or the other. This means there is a connection between e, pi and gamma. Personally I am messing around (in my limited mental capacity!) with the inflation, interest rate in national economics. The basic problem in economics relevant to this conjecture is: A certain amount of inflation is desireable: There is a constant rationalisation in economics meaning that costs are being reduced by (at least) volume production, because on the production line there is a constant improvement - omitting certain unnecessary processes and checks, further more some beneficial changes are being made that fights scrapping. Example. When you produce microprocessors, you make a silicone disk with about 50 processors. In the beginning you get 1 (maybe 2) processors that pass all the tests. Gradually you learn to produce the disk in such a way, that more of the processors are sound, which lowers the cost pr. processor, simply by eliminating scrapping - and some of the tests are unnecessary, because the errors they are supposed to find never occurs. Is there something that cost labour it is testing. When you produce an astronomical mirror - by far the most labour involve is in testing and checking that you polish the right places to eliminate irregularities. This means the producer will invest in these improvements and the workers that are able to improve will get a pay raise. But that means that the cost will lead to a higher profit and higher wages. Meaning the average wage level is improved.

[@Jack_Callcott_AU]

Hey Dr Peyam. When I saw that final result I swooned and nearly fainted. 🧠

[@anushkakar1923]

Find the number of solutions of equation 2^x+3^x+4^x−5^x=0 ....how to solve this question...pls help

[@The1RandomFool]

The recurrence relation of the polygamma function can be used to derive its series representation.

[@SamuelAndradeGTutos]

You are amazing

[@frozenmoon998]

Gamma derivatives!

[@dgrandlapinblanc]

I agree with you for final. Thank you very much.

[@MarcusCactus]

"Prime" means "first" (cf. primal, premier, prime minister,...). What in heaven does "double prime" mean? Something doubly singular? Actually that is the second derivative, so one says "F second" or here "Gamma second". Also, remember that zeta is Z, so the tail turns to the right, not the left (which makes it more of a J).

[@Exchromer]

"Prime" here is not referring to "first", it is referring to the symbol (') which is called a prime mark. Eg. " Take a point A' " is read "Take a point A prime". Simmilarly, the symbol ('') consists of two prime marks, hence the second derivative of gamma would be shortened to gamma double-prime. This way of describing derivatives is incredibly common.

[@jameeztherandomguy5418]

@@Exchromer Yes but the reason it is called "prime" is because it is the first derivative. The second derivative would be second etc

[@Exchromer]

@@jameeztherandomguy5418 the (") symbol's name is double prime as is called by typographers and linguists... '' is read double-prime. ''' is read triple prime, etc. If you like saying f-second go for it, but saying f-double-prime is just as correct as the former, it is simply referring to notation rather than the meaning. Same as you read 3+5 as "three plus five" instead of "sum of 3 and 5". While the latter is the meaning of the expression, the former is the way it is written.

[@MarcusCactus]

@@Exchromer This is true in typography, not in maths. (Not in physics either. 23" is 23 seconds, not «double primes», be it time or geography.) Just look at the other languages. Those that I know (check?) express "prime" as "first" (derivative) and then "second", "third", etc. (Maybe Dr. Peyam is using a German way of saying?) Even more so that you do not write "quadruple prime" but "fourth": ƒ to the IV, not IIII. Do you say "F prime V"? No, of course.

[@almightyhydra]

??? They're just marks, not roman numerals. Fourth derivative is written f''''