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Here is my question:

Suppose my function is defined in terms of Riemann integral in the following form

$$z=g(y)= \int_{}^{}f(x,y)dx$$.

Is there any explicit formula of inverse function $h(z)$; $y=h(z)=h(g(y))$.

I'm sorry if this is a stupid question. Thank you.

kolobokish
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    In general there is no reason for $g$ to be invertible – Federico Nov 21 '18 at 19:41
  • Yes, i mean where $g(y)$ is strictly increasing function. – kolobokish Nov 21 '18 at 19:43
  • And even if g was invertible, the inverse may not have a clean, closed form. – rubikscube09 Nov 21 '18 at 19:43
  • And given that $g$ is expressed as an integral, $g$ itself might not have a clean, closed form. – Federico Nov 21 '18 at 19:45
  • Maybe you can provide a concrete example? – Federico Nov 21 '18 at 19:45
  • No, not a concrete example. I'm seeking expression containing $f$ and boundaries of the integral. – kolobokish Nov 21 '18 at 20:01
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    Is your integral from $0$ to $\infty$ ? I would consider first the case of $f(x,y)=\sum_{p,q=0}^{p,q=n} a_{pq}x^py^q$. – Jean Marie Apr 21 '20 at 08:22
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    @JeanMarie Something like truncated Taylor, or orthogonal polynomials with some wieghts? I want to understand where we are heading to? – kolobokish Apr 21 '20 at 08:49
  • @JeanMarie I also wanted to mention that it was not purely theoretical question, but, unfortunetaly I forgot what I was analyzing.)) – kolobokish Apr 21 '20 at 08:52
  • Maybe 1D vs. 2D cdf (cumulative distribution functions) ? – Jean Marie Apr 21 '20 at 08:54
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    @JeanMarie Not exactly. I recalled. This was connected with quantiles of martingale process. I was trying to recover "quantile process" of general martingale process, or at least find some of its properties. I conjectured that for martingales the quantile process should be non-decreasing from some level above 1/2 ($\alpha^{} > 1/2$) and non-increasing from some level below ($\beta^{} < 1/2$). But unfortunately it was of my own interest (I have not much time to deal with something I want for myself). I haven't shown this yet, but i believe it is true. )) – kolobokish Apr 21 '20 at 09:02
  • @JeanMarie I'm sorry to bother you, but I remember there was an inverse formula of Lagrange. Are there any generalizations? Yep. I found the link https://en.wikipedia.org/wiki/Lagrange_reversion_theorem – kolobokish Apr 21 '20 at 09:04
  • I had an interesting result concerning reciprocal functions but I am unable at present to find it back... Keep trying... – Jean Marie Apr 21 '20 at 09:16
  • @JeanMarie Thank you very much. I do appreciate it. Thank you. – kolobokish Apr 21 '20 at 09:20

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