#gammafunction — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #gammafunction, aggregated by home.social.
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All of this rage just because I asked an LLM for Π(𝑧) but it gave me Γ(𝑧) just relabelled to Π, without removing the off-by-one.
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I can never remember if I'm supposed to add one or subtract one to make it go the way I want it to.
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I can never remember if I'm supposed to add one or subtract one to make it go the way I want it to.
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Think of all the off-by-one errors we could have saved if Legendre had just left well enough alone.
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Think of all the off-by-one errors we could have saved if Legendre had just left well enough alone.
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Let \(\varpi=\dfrac{\Gamma^2\left(\frac14\right)}{2\sqrt{2\pi}}=2.62205755\ldots\) be the lemniscate constant. Then,
\[\Large\displaystyle\sum_{n=1}^\infty\dfrac{1}{\sinh^4(\pi n)}=\dfrac{\varpi^4}{30\pi^4}+\dfrac{1}{3\pi}-\dfrac{11}{90}\]#Series #Sum #InfiniteSum #LemniscateConstant #GammaFunction #Lemniscate #LemniscateOfBernoulli #Bernoulli #Math #Maths #InfiniteSeries #HyperbolicSines
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A lattice sum with a surprising closed form:
\[\Large\displaystyle\sum_{\substack{(m,n)\in\mathbb Z^2\\(m,n)\neq0}}\dfrac{1}{(m+ni)^4}=\dfrac{\varpi^4}{15}=\dfrac{\Gamma^8\left(\frac14\right)}{960\pi^2}\]
#LatticeSum #ClosedForm #LemniscateConstant #Series #Sum #GammaFunction #Pi -
A lattice sum with a surprising closed form:
\[\Large\displaystyle\sum_{\substack{(m,n)\in\mathbb Z^2\\(m,n)\neq0}}\dfrac{1}{(m+ni)^4}=\dfrac{\varpi^4}{15}=\dfrac{\Gamma^8\left(\frac14\right)}{960\pi^2}\]
#LatticeSum #ClosedForm #LemniscateConstant #Series #Sum #GammaFunction #Pi -
A small project from yesterday, plotting Gamma(x) = Gamma(y). Wolfram Alpha (left) is awful. Desmos is OK but not great. My custom code fills in the gaps, but is that really what a graph should show, or should it highlight all the discontinuities?
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A small project from yesterday, plotting Gamma(x) = Gamma(y). Wolfram Alpha (left) is awful. Desmos is OK but not great. My custom code fills in the gaps, but is that really what a graph should show, or should it highlight all the discontinuities?
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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
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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