# Tagged Questions

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### Closed form for $\int z^n\ln{(z)}\ln{(1-z)}\,\mathrm{d}z$?

Problem. Find an anti-derivative for the following indefinite integral, where $n$ is a non-negative integer: $$\int z^n\ln{\left(z\right)}\ln{\left(1-z\right)}\,\mathrm{d}z=~???$$ My attempt: ...
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### Prove: $\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3$

I'd like to evaluate the following definite integral: $$\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3.$$ ...
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The integral \begin{align} I_{4} = \int_{0}^{1} \ln(1-x) \ \ln^{2}\left( \ln\left(\frac{1}{x}\right) \right) \ \frac{dx}{x} \end{align} can be expressed as \begin{align} I_{4} = \zeta^{''}(2) - ...
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### Convexity of polylogarithms

I want to prove the following proposition: The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$. And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for ...
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### Solve $-B \ln y -A y \ln y + A y- A =0$ for $y$

I would like to know if there is a (preferably closed-form) solution for $-B \ln y -A y \ln y + A y- A =0$ for $y$ Where $A, B \in \mathbb{R}^{+}$. I have reasons to think there isn't a closed form ...
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### Find $\lim_{x\to \infty} \ln(\exp(\operatorname{LmW}(x))+1)(\exp(\operatorname{LmW}(x))+1) - x - \ln(x)$

Find $\lim_{x\to \infty} \ln(e^{\operatorname{LambertW}(x)}+1)(e^{\operatorname{LambertW}(x)}+1) - x - \ln(x)$ Where the $LambertW$ function is defined here : http://en.wikipedia.org/wiki/Lambert_W ...
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### Two $\psi$ functions

This is either a notation/history question or a point of confusion. In (for example) Ramanujan's proof of Bertrand's postulate, he uses the following notation: $\log [x]!$ means $\log ([x]!),$ in ...
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### Triangular exponentation logarithm and inverse

The generalized formula of triangular exponentation on real numbers field is $x ^ {\triangle y} = \frac {1} {y \cdot B (x, y)} = \frac {\Gamma(x + y)} {\Gamma(x) \cdot \Gamma(y + 1)}$ It's my ...
### Inverse function of $\operatorname{li}(x)$ over $x>\mu$?
How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$? Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...