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According to Maple: so:=solve( f(x)-exp(( x - f(x)) /B ),f(x));latex(so); $$ f(x) =B{\it LambertW} \left( {{\rm e}^{{\frac {x}{B}}}}{B}^{-1} \right) $$ And for the title: so:=solve( f(x)-A …

Let $K(x)$ be the complete elliptic integral of the first kind (the argument is the parameter $m = k^2$). Let $$ A = \int_0^1 \arcsin(K(x)) dx$$ With precision $1000$ decimal digits $\Re A = \frac{\ …

Too long for comment, here is partial answer for $n=5$ per R B's request. According to Maple: n:=5:q:=(1-p)*sum(t^i,i=0..n)-p*sum( (1-t)^i,i=0 .. n):so:=solve(q,t): so[1] and so[2] are complex and …

For questions like this searching for the first few terms in OEIS might help 3, 5, 8, 10, 14, 16, 20, 23, 27, 29, 35, 37, 41 This is A006218. There are a lot of references and bounds for the sum. I …

With the help of wolfram alpha we got very long closed form for $\pi$ in terms of algebraic numbers, logarithms of algebraic numbers and $cot^{-1},coth^{-1}$ which can be expressed as logarithms. From …

For natural $A$ define $$ f(A)=\sum_{n=1}^\infty \frac{1}{A^n}\left(\frac{1}{An+1}- \frac{1}{An+A-1}\right)$$ $f(A)$ is BBP (Bailey-Borwein-Plouffe) formula and allows digit extraction in base $A$. Fo …

Wolfram Alpha claims there is no closed form in terms of standard funcions for $\int \lfloor x \rfloor dx$ but we believe we found simple closed form agreeing with experimental data. Define $i_1(x)=x …

Related to this question Q1 Is there real or complex analytic function $f(x)$ such that its positive real zeros are the primes and it is given in closed form of compositions of already named function …

Let $a$ be an element of some ring or field, possibly finite. Is there closed form for $\sum_{i=1}^n{a^{i^2}}$? sage and wolframalpha couldn't solve it. If $a$ is primitive n-th root of unity this is …

High precision numerical computations suggest: $$c_1=73/5760 $$ $$c_2=3625/580608$$ $$c_3=5233001/1393459200$$

Working with precision 500 decimal digits, mpmath in sage computes: $$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$ We believe the LHS of \eqref{1} d …

Conjecture: $$ I(3,3)=-2\, \left( {\it arcsinh} \left( 1/2 \right) \right) ^{3}-3/2\, \left( {\it arcsinh} \left( 1/2 \right) \right) ^{2}\sqrt {5}+{ \frac {3}{20}}\,{\pi }^{2}$$ This holds to at …

Edited Maple's $\psi$ disagrees with Wolfram Alpha and your integral, so here are some conjectures with both: According to Maple -- your equality fails with this definition of psi. $$ 24 \Im{\psi^{( …