An example of trans-series from Euler's equation There seems to be a generalization of the notion of the "asymptotic series" to a notion of "trans-series". One "simple" example of this seems to arise in trying to solve the Euler's equation by power-series, $$\frac {d\phi (z)} {dz } + A\phi(z) = \frac{A}{z} $$
A power series solution to this is, $$\phi_0(z) = \sum_{n=0}^{\infty} \frac{a_n } {z^{n+1}} $$ where $a_n = A^{-n}n!$. This series is an asymptotic series with zero-radius of convergence because the coefficients grow factorially in $n$. 


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*Now what I don't understand is as to how from all this it follows that one can construct a ``family of formal solutions to this ODE based on $\phi_0(z)$" as, $\phi(z) = \phi_0(z) + Ce^{-{Az}}$


The above is apparently an example of a "trans-series". 


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*I don't undertand as to how this has two small parameters $z$ and $e^{-\frac{A}{z} }$? 


Here the strength of the non-perturbative effect $(A)$ and the divergence of the asymptotic series are related. 


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*But also in what sense is the $Ce^{-\frac{A}{z}}$ ``non-analytic at $z=\infty$"? That all the possible Taylor series coefficients go to $0$ on taking the $z=\infty$ limit? Is this a signature of being a trans-series? 

 A: The equation in question, $(2.17)$ in the book, is
$$
\phi'(z)+A\phi(z)=\frac Az\tag{1}
$$
Note that $(1)$ can be written in a way to generate an asymptotic series solution:
$$
\phi(z)=\frac1z-\frac{\phi'(z)}{A}\tag{2}
$$
Iterating this as a recursion, we get an asymptotic approximation to a solution, $(2.18)$ in the book:
$$
\phi_0(z)=\frac1z+\frac1{Az^2}+\frac2{A^2z^3}+\frac6{A^3z^4}+\cdots\tag{3}
$$
Given any two solutions to $(1)$, their difference must satisfy the homogeneous equation
$$
\phi_h'(z)+A\phi_h(z)=0\tag{4}
$$
The solution to $(4)$ is
$$
\phi_h(z)=Ce^{-Az}\tag{5}
$$
Thus, as mentioned in $(2.19)$ of the book, the general solution would be
$$
\phi_0(z)+Ce^{-Az}\tag{6}
$$
I think one point that the author is trying to make is that the asymptotic series for all of these solutions is given in $(3)$ since $\phi_h(z)$ decays much faster than any power of $\frac1z$. Unfortunately, there seem to be some confusing typos in point 2. following $(2.19)$ related to an earlier comment, $(2.4)$ in the book.
