How does one obtain series with logarithmic terms? For some functions Wolfram|Alpha gives me series expansions which have some logarithmic terms. For example, for Tricomi confluent hypergeometric function $U(n,2,r)$ I get:
$$U(n,2,r)=\frac1{(n-1)r\Gamma(n-1)}+\frac{\psi^{(0)}(n)+\color{red}{\log(r)}+2\gamma-1}{\Gamma(n-1)}+\cdots.$$
In general, what is the procedure to obtain such terms? I guess the $r^{-1}$ term could be obtained by Laurent expansion, but for the above function its usual formula for coefficients won't work due to the branch point.
 A: The following is an excerpt from this answer.

The main theme here is Singularity Analysis which describes the asymptotic behavior of a large class of functions with moderate growth and decay and the strongly connected asymptotic behavior of the coefficients of their generating functions.
Functions with moderate growth and decay:
The functions under consideration have as central elements of an expansion at a singularity $\zeta$ 
  \begin{align*}
\left(1-\frac{z}{\zeta}\right)^{-\alpha}\left(\log\frac{1}{1-\frac{z}{\zeta}}\right)^\beta\qquad\qquad \alpha,\beta\in\mathbb{C}\tag{1}
\end{align*}
  These elements contribute asymptotically  terms of type
  \begin{align*}
\zeta^{-n}n^{\alpha-1}\left(\log n\right)^\beta
\end{align*}
We see depending on $\alpha$ and $\beta$ we have either powers or logarithmic contributions or both parts to consider.

Note: The basic facts mentioned here are from the classic Analytic Combinatorics (chap. VI) by P. Flajolet and R. Sedgewick which contains a detailed elaboration of singularity analysis.
A: I am not sure that this could be the answer you expect.
Consider $$y=U(n,2,r)\implies y'=-n\, U(n+1,3,r)$$ As you say, the development of $y'$ is a Laurent series but you can reduce to a Taylor series considering $r^2\,y'$ and get $$r^2\,y'=-\frac{1}{(n-1) \Gamma (n-1)}+\frac{r}{\Gamma (n-1)}-\frac{r^2 (n (-2 \psi
   ^{(0)}(n+1)-2 \log (r)-4 \gamma +3))}{4 \Gamma (n-1)}+O\left(r^3\right)$$ So $$y'=-\frac{1}{(n-1)  \Gamma (n-1)r^2}+\frac{1}{ \Gamma (n-1)r}-\frac{n (-2 \psi
   ^{(0)}(n+1)-2 \log (r)-4 \gamma +3)}{4 \Gamma (n-1)}+O\left(r\right)$$ and, integrating, $$y=\frac{1}{(n-1) r \Gamma (n-1)}+\frac{\log (r)}{\Gamma (n-1)}+\frac{n r (2 \psi
   ^{(0)}(n+1)+2 \log (r)+4 \gamma -5)}{4 \Gamma (n-1)}+O\left(r^2\right)$$ which can simplify to $$y=\frac{1}{r \Gamma (n)}+\frac{\log (r)}{\Gamma (n-1)}+\frac{n r \left(2 H_n+2 \log
   (r)+2 \gamma -5\right)}{4 \Gamma (n-1)}+O\left(r^2\right)$$
