# Tagged Questions

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### What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$\frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0$$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
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### Inverse of the Modified Bessel function

Is there any chance of having a formula or approximation to inverse the Modified Bessel function of the first kind? I mean to solve $I_M(x)$ with respect to x: $I^{-1}_M(x)$? Thanks in advance
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### Calculating $\text{erf}^{-1}(z)$ for $z\in\mathbb{C}$

All the information I found about inverse error function $\text{erf}^{-1}(z)$ was about $z\in\mathbb{R}$. Also I found some Taylor expansions for it, but as the function is unbounded near $z=\pm1$, ...
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### Inverse function of $x\mapsto \sqrt[x]x$ on $\left[0,e^{-1}\right]$

Why is it, that the inverse of $\sqrt[x]x$ is given by the infinite power tower in $x\in[\frac1e;e]$, but not in $x\in[0;\frac1e]$? I know that the power tower diverges on that interval, but even if ...
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### Can $\Phi^{-1}(x)$ be written in terms of $\operatorname{erf}^{-1}(x)$?

Can the inverse CDF of a standard normal variable $\Phi^{-1}(x)$ be written in terms of the inverse error function $\operatorname{erf}^{-1}(x)$, and, if so, how? This seems like an easy question, but ...
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### General solution for $M^{\circ -1 }(y)=x$ when $g(x)e^{f(x) }=y$

Reading this question $e^{C/x }-1=D/(x + a)$, i found my self completely unable to do anything. This is much more hard for me than my easy exercises about Lambert $W$-function. So I probably need ...
### 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 ...