2
votes
1answer
82 views

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 ...
7
votes
1answer
102 views

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 $z=\pm1$, ...
2
votes
1answer
187 views

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 ...
3
votes
2answers
54 views

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 ...
1
vote
0answers
85 views

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 ...
0
votes
2answers
195 views

$e^x-x-4$equating with zero

I want to find out the values of x where the $f(x) = e^x-x-4$ will equal zero. My problem by solving this myself is that I cannot use logarithm natural (ln) because I have a normal x: $f(x) = e^x - ...
5
votes
3answers
389 views

Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$

I have a simple question for which I am looking for a closed form expression (If there exits one). In other words, given: $$y=W(e^{ax+b})-W(e^{cx+d})+zx$$ where $W$ is the Lambert $W$ function and ...
0
votes
0answers
176 views

Can one find the inverse function for a combination of imcomplete gamma functions?

The original function was defined as $f(z)=y=\left( \Gamma \left( k,{\frac {z-b}{\theta}} \right) -\Gamma \left( k,{\frac {z-a}{\theta}} \right) \right) \left( \Gamma \left( k \right) \right) ...
5
votes
1answer
407 views

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 ...