hard integral

[paper temple]

Hi I got into summation representations for constants and taylor series etz etz.. so i found this integral however I cant find a good way to approach this, this i tried:

1. IBP, but i end up with a 1/x term idk what 2 do with it. (IBP to try and recreate the euler constant as integral)

2. e^-x as limit but then i ended up with a limit resulting in infinity.

If anyonr can help or explain how to deal with this integral i'm you very grateful!!

[thorny wagon]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

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3. Feel free to nominate the person for helper of the week in #helper-nominations

[stark patio]

PDF File (apex predator)

PDF File (apex predator)

PDF File (apex predator)

[frozen willow]

this is supposed to be ln²x=that

[paper temple]

ah

[frozen willow]

lnx is just del / del s

[paper temple]

a double partial sum inside an integral, gotta look up some theory about that to use it

[frozen willow]

you can swap the integral and derivative with Leibniz' rule

[paper temple]

gotta switch up operation right he

ye

[frozen willow]

which leaves you with the integral definition for gamma(s+1)

[paper temple]

how do i state s=0 inside the integral

[frozen willow]

and the second derivative of that wrt s

[stark patio]

PDF File (apex predator)

[paper temple]

or do i call it I(s) and let I(original) = I(0)

ohhhh

cool

[frozen willow]

not supposed to be cubed on bottom

mb

[paper temple]

i understand

lol

[frozen willow]

yeah this also works

it's an element of feynmans integral trick

(really just applied Leibniz rule)

[paper temple]

ye, i know feynman so though, this should work out

brb

gotta help the other men

[frozen willow]

the requests in #1015578016606343218 are getting more complicated by the day

[paper temple]

idk where 2 put it, this integral is not from a class at school though. But im not doing a minor yet so idk

bachelor*

[frozen willow]

if you think it's uni level then it's best to post there

more people with the skillset necessary will see it

[paper temple]

alright, thx

@frozen willow gamma''(s+1) when s = 0 gives gamma''(1) but according to defenitions of the double derivative of gamma that value is undefined?

i do know -y = gamma'(1) but deriving a summation like mascheroni is something else

or derivative of integral might hmmm

thta might just me it

[frozen willow]

Gamma'' is trigamma function

which is definitely defined at 1

[frozen willow]

sorta

trigamma is the second derivative of the log gamma function

but it is definitely defined

[paper temple]

ye log gamma

i thought couldnt use it here

[frozen willow]

why not?

[paper temple]

lnF(z)

i dont see a ln in my equations,

F = gamma function 🙂

[frozen willow]

there are identities available to solve for the second derivative of the gamma function alone

[stark patio]

PDF File (apex predator)

[paper temple]

what

[frozen willow]

that's the second derivative of the gamma function

not the logarithmic gamma function

just the gamma function

[paper temple]

1,x is?

just x i suppose

[frozen willow]

see

psi(1,x) is a way of noting the trigamma function

first derivative of the gamma function

solve for gamma''

and there

@paper temple

[paper temple]

i just gotta see a derivation before i understand this

even for the first step

like where does digamma come from and why this and why that etzetz

[frozen willow]

etzetz?

the digamma function is a special case of the Hurwitz zeta function

[stark patio]

PDF File (apex predator)

[paper temple]

etc etc *

in english,

second derivative of riem zeta function?

[frozen willow]

nah

have a readhttps://en.m.wikipedia.org/wiki/Hurwitz_zeta_function

it's a zeta function of 2 parameters

[paper temple]

crazy

oh we just work it out for a & b instead of the entire z = a +bi

[frozen willow]

uh?

[paper temple]

nvm

[stark patio]

PDF File (apex predator)

[paper temple]

but oh funny, the x^(s-1) is x^s * 1/x dang now i understand that first step u did with this integral

[frozen willow]

yes

[paper temple]

someting i havent realised yet

[frozen willow]

also if you look at the definition of digamma with gamma'/gamma it makes sense in the context of log derivatives

d/dx ln(f(x)) = f'(x)/f(x)

[paper temple]

i notice f'/f integrated yes exactly

gives that

and what is an assumption? cos i dont think they just made up that ln()

[frozen willow]

what?

[paper temple]

wait lemme read some

how did mathmaticians define the digamma function?

who says u gotta take the ln of gamma

[frozen willow]

for functions with a gradient that is absolutely crazy it makes sense to take the logarithm of it to slow the growth, the properties of the log functions also help out a lot

[paper temple]

so it is an assumption?

digamma = d/dz lnF(z)

[frozen willow]

pretty much

not really an assumption as it is defined that way

[paper temple]

thats what i meant yes

[frozen willow]

and you can derive from the integral representation for gamma

[paper temple]

yes

just foudn that out :))

or just using F(x+1) = xF(x)

[frozen willow]

yes indeed

well (x+1)F(x)

[paper temple]

but

when u do F(x+1) and then IBP, u end out with x * F(x)

[frozen willow]

ohh i thought you were defining the function

[paper temple]

or do u say that if u expand to xfx + fx

[frozen willow]

not the IBP

mb

[frozen willow]

yes

[paper temple]

and the second derviative is then just d^2/dx^2 lnFx

[frozen willow]

yesa

[paper temple]

moke nice

thx for your time , i enjoyed it

gotta catch the sleep now, have a nice day 🙂

[frozen willow]

you yoo

[paper temple]

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