#logarithm — Public Fediverse posts

#memes #dankmemes #logarithm #math #mathmemes #log #sunshine

L o g a r i t h m s

Euler–Mascheroni constant! :euler:

In fact, the last one is:
\[\large\displaystyle\int_1^{+\infty}\mathrm dx\ \left(\frac{1}{\lfloor x\rfloor}-\frac1x\right)=\gamma\approx0.5772156649\]

Equivalently,
\[\large\displaystyle\lim_{n\to\infty}\left(\sum_{k=1}^n \frac1{k}-\ln n\right)=\gamma=0.5772156649\ldots\]
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Unsolved problem in mathematics:
Is Euler–Mascheroni constant irrational? If so, is it transcendental?

#Euler #Mascheroni #EulerMascheroni #Constant #gamma #EulerConstant #EulersConstant #EulerMascheroniConstant #Irrational #Irrationality #Transcendental #Transcendence #Unsolved #UnsolvedProblem #Maths #Mathematics #Indeterminate #IndeterminateForm #IndeterminateForms #Inf #Infinity #HarmonicNumber #HarmonicNumbers #HarmonicSeries #Logarithm #Log #Logarithms #NaturalLogarithm #Integral #ImproperIntegral

My wife was working through finding the derivative of the #exponential #function #exp(x) from first principles.I was made aware that she hadn’t actually seen why the number e=2.7128… was the #base the of the function and that that’s what you need to start with. In fact, that means one must actually start by finding the first differential of a general #logarithm and find #e from there. Once you’ve find the #Derivative of #lln, the #derivative of the #ExponentialFunction is straightforward. (1/2)

An excellent general result.

If \(\Re(s)>1\),
\[\displaystyle\int_0^\infty\frac{\ln x}{x^s+1}~\mathrm dx=\frac{\pi^2}{4s^2}\left[\sec^2\left(\frac{\pi}{2s}\right)-\csc^2\left(\frac{\pi}{2s}\right)\right]\]

Special cases:
\[\displaystyle\int_0^\infty\frac{\ln x}{x^2+1}~\mathrm dx=0\]
\[\displaystyle\int_0^\infty\frac{\ln x}{x^3+1}~\mathrm dx=-\frac{2\pi^2}{27}\]
\[\displaystyle\int_0^\infty\frac{\ln x}{x^4+1}~\mathrm dx=-\frac{\pi^2}{8\sqrt2}\]
\[\displaystyle\int_0^\infty\frac{\ln x}{x^5+1}~\mathrm dx=-\frac{4\pi^2}{25}\left(\frac{2+\sqrt5}{5+\sqrt5}\right)=-\frac{(5+3\sqrt5)\pi^2}{125}\]

#Integral #Integrals #GeneralResult #GeneralResults #Result #Results #Logarithms #Logarithm #Integration #DefiniteIntegral #Calculus #IntegralCalculus

Try to prove the following two results that relate the harmonic numbers to the golden ratio. Have an excellent weekend.

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^n}=2\sqrt5\ln\varphi\]

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^nn}=\frac{2\pi^2}{15}-2\ln^2\varphi\]

where \(\varphi=\frac{1+\sqrt5}2\) is the golden ratio; and \(H_n=\left(1+\frac12+\frac13+\ldots+\frac1n\right)\) is the \(n\)-th harmonic number.

#GoldenRatio #HarmonicNumbers #HarmonicNumber #Logarithm #Pi #Summation #Math #Sum #InfiniteSum #Binomial #BinomialCoefficient #Maths #WeekendChallenge

Integral challenge!

\[\displaystyle\int_0^\infty\ln\left(1+\dfrac{\cosh\alpha}{\cosh x}\right)\ dx=\dfrac{\pi^2}{8}-\dfrac{\arccos^2(\cosh\alpha)}{2}\]

#Integral #Integrals #IntegralChallenge #HyperbolicFunction #HyperbolicCosine #Logarithm #DefiniteIntegral

Interesting integral! #Challenge
\[\displaystyle\int_0^1\dfrac{\ln (x)\ln(1-x)}{x(1-x)}\operatorname{Li}_2(x)\ dx=5\zeta(2)\zeta(3)-8\zeta(5)\]
Where \(\operatorname{Li}_2(x)\) denotes the dilogarithm (or Spence's function), and \(\zeta(x)\) denotes the Riemann zeta function.

#ZetaFunction #Zeta #Dilogarithm #SpenceFunction #Polylogarithm #Integral #DefiniteIntegral #Integration #Integrals #RiemannZetaFunction #Logarithm #Function #LogarithmicFunction

Interesting integral! #Challenge
\[\displaystyle\int_0^1\dfrac{\ln (x)\ln(1-x)}{x(1-x)}\operatorname{Li}_2(x)\ dx=5\zeta(2)\zeta(3)-8\zeta(5)\]
Where \(\operatorname{Li}_2(x)\) denotes the dilogarithm (or Spence's function), and \(\zeta(x)\) denotes the Riemann zeta function.

#ZetaFunction #Zeta #Dilogarithm #SpenceFunction #Polylogarithm #Integral #DefiniteIntegral #Integration #Integrals #RiemannZetaFunction #Logarithm #Function #LogarithmicFunction

How to prove this?!🤔

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

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

#Integral #definiteintegral #gammafunction #digammafunction #pi #logarithm