Darboux's method is one way of estimating the coefficients of generating functions involving roots.
It is easier than Singularity Analysis, but it applies to a smaller set of functions.
We will make use of the "Big O" notation.
as 
which means there exists positive real numbers
such that if
:

Alternatively, we can say that

for positive integer
, meaning there exists a positive real number
and positive integer
such that:
for 
Theorem due to Wilf[1].
If we have a function
where
where
has a radius of convergence greater than
and an expansion near 1 of
, then:
^{\beta }f(z)=\sum _{j=0}^{m}f_{j}{\frac {n^{-\beta -j-1}}{\Gamma (-\beta -j)}}+O(n^{-m-\beta -2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59ebdc6be74262b8d77bdc7959a202c9dfe0a11e)
The theorem is a bit abstract, so I will show an example of how you might use it before going into the proof.
Taking an example from Wilf[2]:

is a complete function, so its radius of convergence is greater than 1.
Near 1 it can be expanded using the Taylor series:

Therefore, for
:
![{\displaystyle [z^{n}]{\frac {e^{-z/2-z^{2}/4}}{\sqrt {1-z}}}=e^{-3/4}{\frac {n^{-{\frac {1}{2}}}}{\Gamma ({\frac {1}{2}})}}+O(n^{-{\frac {3}{2}}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67929786694dc34d2b2587d973c308113bfea710)
Or, if we want more precision we can set
:
![{\displaystyle [z^{n}]{\frac {e^{-z/2-z^{2}/4}}{\sqrt {1-z}}}=e^{-3/4}{\frac {n^{-{\frac {1}{2}}}}{\Gamma ({\frac {1}{2}})}}+e^{-3/4}{\frac {n^{-{\frac {3}{2}}}}{\Gamma (-{\frac {1}{2}})}}+O(n^{-{\frac {5}{2}}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/556f802bc6d6e34857e42ecd73d73e6ad493b35b)
and so on.
Proof due to Wilf[3].
We have:

and:

By factoring out
from the last sum:

Therefore:

We have to prove that:
![{\displaystyle [z^{n}]\sum _{j=0}^{m}f_{j}(1-z)^{\beta +j}=\sum _{j=0}^{m}f_{j}{\frac {n^{-\beta -j-1}}{\Gamma (-\beta -j)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1f9171542889b980efe6622a4e7095b75538dd)
^{\beta +m+1}g(z)=O(n^{-m-\beta -2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65970044c651e53385c2128cfa00b035293d6359)
By applying #Lemma 1:
![{\displaystyle [z^{n}]\sum _{j=0}^{m}f_{j}(1-z)^{\beta +j}=\sum _{j=0}^{m}f_{j}[z^{n}](1-z)^{\beta +j}\sim \sum _{j=0}^{m}f_{j}{\frac {n^{-\beta -j-1}}{\Gamma (-\beta -j)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5406986cc8fb43fa3cda8d450e1d1385501b2c7c)
(by #Lemma 1)
(because, by assumption in the theorem, the radius of convergence
is greater than
and Cauchy's inequality tells us that
and
)
(for
constants and assuming that
).

because
because
.
Putting it all together:
^{\beta +m+1}g(z)=O(\theta ^{\frac {n}{2}})+O(n^{-m-\beta -2})=O(n^{-m-\beta -2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93149f1ff8a68187862626453c13bc53b85ca102)
because
[4] because
[5].
^{\beta }\sim {\frac {n^{-\beta -1}}{\Gamma (-\beta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47f81026f6b6c905f991a59232ef98385b494d74)
Proof:
[6]
where
is the rising factorial.
We can apply a similar theorem to functions with multiple singularities. From Wilf[7] and Szegő[8].
If
is analytic in
, has a finite number of singularities
on the unit circle
and in the neighbourhood of each singularity has the expansion

Then we have the asymptotic series
![{\displaystyle [z^{n}]f(z)=\sum _{v\geq 0}\sum _{k=1}^{r}c_{v}^{(k)}{\binom {\alpha _{k}+v\beta _{k}}{n}}(-e^{i\phi k})^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/860c6df152d2f0833d88ec0c874a1876a36335c1)
- Szegő, Gabor (1975). Orthogonal Polynomials (4th ed.). American Mathematical Society.
- Wilf, Herbert S. (2006). Generatingfunctionology (PDF) (3rd ed.). A K Peters, Ltd.