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

so the polygamma function can be written in different ways?

@General12th

Жыл бұрын

Was that a poly joke?

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

@jimjam623

Жыл бұрын

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

Just seeing Dr Peyam's smile makes my day ❤️

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

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

You are amazing

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

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

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

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?

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

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 ??

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

Gamma derivatives!

Wow genial

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

What was that constant again?

@anupamamehra6068

11 ай бұрын

euler-mascheronni constant approx 0.577

@unfulfilling_entertainer

3 ай бұрын

Oily macaroni

Gama..wow

@azzteke

Жыл бұрын

Gamma.

@General12th

Жыл бұрын

​@@azzteke Gramma.

@loicetienne7570

Жыл бұрын

@@General12th Grandma ?

Can you factor the answer

@ruffifuffler8711

5 ай бұрын

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

Over your head? Nonsense!

"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

Жыл бұрын

"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

Жыл бұрын

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

@Exchromer

Жыл бұрын

@@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

Жыл бұрын

@@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

Жыл бұрын

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