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There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta function $B(z; a, b)$, the Lerch transcendent $\Phi(z, s, a)$, the Weierstrass elliptic function $\wp(z;g_2,g_3)$, hypergeometric-family functions, etc.

Is it possible to express each (or at least some) of these functions as a composition of several analytic functions of 1 or 2 complex variables?

Or, if we restrict their domain to reals, it is possible to express them as a composition of several inifinitely differentiable (with respect to each variable) functions of 1 or 2 real variables?

The same question applies to functions of 2 variables (e.g. polylogarithms, incomplete elliptic integrals, Hurwitz zeta function, Bessel-family functions, etc.): Is it possible to represent them as a composition of several infinitely differentiable functions of 1 variable and the single fixed function of 2 variables $(x,y)\mapsto x+y$?


To give an example when the answer to the last question is positive, consider the complete beta function $B(a,b)$. It can be represented as $$B(a,b)=\exp\big((\ln\Gamma(a)+\ln\Gamma(b))+(-\ln\Gamma(a+b))\big)$$ that is a composition of the 2-variable sum function and several infinitely differentiable 1-variable functions $x\mapsto\exp(x)$, $x\mapsto\ln\Gamma(x)$ and $x\mapsto -x$.

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  • $\begingroup$ Are you familiar with Kolmogorov's superposition theorem? It's very close to what you're asking if I interpret your question correctly. The only difference is it requires a sum over $2n+1$ and $n$ terms if your function's domain is $\mathbb{R}^n$. Page 2: cca-net.de/vasco/publications/kolmogorov.pdf $\endgroup$ – Chester Jul 13 '15 at 3:03
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    $\begingroup$ As far as I know, the solution given in Kolmogorov's superposition theorem doesn't use analytic functions, see the middle of page 2. $\endgroup$ – Vladimir Reshetnikov Jul 13 '15 at 3:46
  • $\begingroup$ Re-posted at Mathoverflow. $\endgroup$ – Vladimir Reshetnikov Feb 21 '16 at 18:18
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I found an answer for essentially the same question asked earlier at MathOverflow.

TL;DR

There exist analytic functions $f$ of 3 variables such that they cannot be represented as a composition of continuously differentiable functions of two variables. The general problem in the class $C^\infty$ seems to be still open.

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