How to prove this?!🤔

\[\displaystyle\int_\tfrac{1}{2}^1\dfrac{\psi(x)}{1+\Gamma^2(x)}dx=\ln\left(\sqrt{\dfrac{1+\pi}{2\pi}}\right)\]

where \(\Gamma(x)\) and \(\psi(x)\) are the gamma and digamma functions respectively.

#logarithm #pi #digammafunction #gammafunction #definiteintegral #integral

Last updated 3 years ago

lagomoof · @lagomoof
2 followers · 10 posts · Server mastodon.social

Multiply 20527686 by its natural logarithm. The result differs from an by less than 1/20527686(!)

[Modified from OP birdsite 20131027]

#integer #math #maths #logarithm #nearinteger

Last updated 3 years ago

MarcNo · @MarcNo
360 followers · 626 posts · Server mastodon.cloud
· @mrdk
28 followers · 195 posts · Server mathstodon.xyz

@mittelwertsatz @chris_sangwin To express that idea more formally, define functions \[ L_n(x) = \int_1^x \xi^{n-1} x\xi. \]
Then \( L_0(x) = \ln x \) and \( L_n(x) = \frac{x^n - 1}n \) otherwise. But all \( L_n \) obey the functional equations
\[ L_n(x y) = y^n L_n(x) + L_n(y), \quad 𝐿_n(1)=0. \]
My question now is: Do these "generalized logarithms" occur somewhere naturally?

#functionalequation #integration #logarithm

Last updated 3 years ago

mkdukin · @mkdukin
18 followers · 131 posts · Server masto.ai

Ah, the new solution is to let her rip and then blame the victims that they didn't apply the choice policies correctly.

How very .

#socialmedia #censorship #logarithm #libertarian

Last updated 3 years ago

Blaergh Blaerghuson · @Blaergh
7 followers · 21 posts · Server qoto.org

a basic guide to reading and notation

#exponent #logarithm

Last updated 3 years ago

biggh · @biggh
0 followers · 4 posts · Server mas.to

preparation: : When log5 = p. Express the following in terms of “p”.

#logarithm #olevels #igcse

Last updated 3 years ago