- Equations 30 and 31 should read as follows: $\setCounter{29}$
\begin{eqnarray}
\int_a^{x_0-\epsilon} \frac{f(x)}{(x-x_0)^{n+1}}\mbox{d}x &=& -\sum_{k=0}^{n-1} \frac{f^{(k)}(x_0)}{k!(n-k)} \left(\frac{1}{(-\epsilon)^{n-k}} – \frac{1}{(a-x_0)^{n-k}}\right) \nonumber \\
&& + \frac{f^{(n)}(x_0)}{n!} \left(\ln \epsilon – \ln(x_0-a)\right)\nonumber \\
&& + \sum_{k=n+1}^{\infty} \frac{f^{(k)}(x_0)}{k!(k-n)} \left((-\epsilon)^{k-n}-(a-x_0)^{k-n}\right)
\end{eqnarray}\begin{eqnarray}
\int_{x_0+\epsilon}^b \frac{f(x)}{(x-x_0)^{n+1}}\mbox{d}x &=& -\sum_{k=0}^{n-1} \frac{f^{(k)}(x_0)}{k!(n-k)} \left(\frac{1}{(b-x_0)^{n-k}} – \frac{1}{\epsilon^{n-k}}\right) \nonumber \\
&& + \frac{f^{(n)}(x_0)}{n!} \left(\ln (b-x_0) – \ln\epsilon\right)\nonumber \\
&& + \sum_{k=n+1}^{\infty} \frac{f^{(k)}(x_0)}{k!(k-n)} \left((b-x_0)^{k-n}-\epsilon^{k-n}\right)
\end{eqnarray}In the original paper, the correct index $(k-n)$ in the third terms of the right-hand sides of equations 30 and 31 was erroneously encoded as $(n-k)$. The post has been appropriately updated.
Thanks to Avram Guiterrez, Kenneth Jhon Remo and Aster Niel Abellano (PUP student-OJTs) for pointing out this typographical error.
- The signs in equations (25), (26) and (28) have been inadvertently interchanged. They should read $\setCounter{24}$
\begin{equation}\label{boundary1}
\Phi^{+}(x_0)= \#\!\!\!\int_a^b \frac{f(x)}{(x-x_0)^{n+1}} \mbox{d}x + i \pi \frac{f^{(n)}(x_0)}{n!} .
\end{equation}\begin{equation}\label{boundary2}
\Phi^{-}(x_0)= \#\!\!\!\int_a^b \frac{f(x)}{( x-x_0)^{n+1}} \mbox{d}x – i \pi \frac{f^{(n)}(x_0)}{n!} ,
\end{equation}
$\setCounter{27}$\begin{eqnarray}
\Phi^{\pm}(x_0)=\mathrm{Int}^{\mp}(x_0) .
\end{eqnarray}