TITLE: Does Wikipedia misstate Glasser's master theorem
QUESTION [8 upvotes]: Citing from Wikipedia and one of its references, MathWorld, following hold:
Glasser's master Theorem For $f$ integrable,  $\Phi(x) = |a|x - \sum_{i=1}^N \frac{|\alpha_i|}{x-\beta_i}$ and $a$, $\alpha_i$, $\beta_i$ arbitrary real constants the identity
\begin{equation}
    \mathrm{PV}\int_{-\infty}^\infty f(\Phi(x)) dx =
    \mathrm{PV} \int_{-\infty}^\infty f(x) dx
    \label{Glasser}
    \tag{1}
\end{equation}
holds.
Now consider
\begin{align*}
    \Phi_1(x) &= |a|x - \sum_{i=1}^N \frac{|\alpha_i|}{x-\beta_i} \\
    \Phi_2(x) &= x - \sum_{i=1}^N \frac{|a\alpha_i|}{x-|a|\beta_i}
\end{align*}
Then, by Glasser's theorem \ref{Glasser}
$$\mathrm{PV}\int_{-\infty}^\infty f(\Phi_1(x)) dx =
\mathrm{PV} \int_{-\infty}^\infty f(x) dx =
\mathrm{PV}\int_{-\infty}^\infty f(\Phi_2(x)).$$
However, under the change of variables $y = |a| x$
\begin{equation}
    \mathrm{PV}\int_{-\infty}^\infty f(\Phi_1(x)) dx =
    \frac{1}{|a|}\mathrm{PV} \int_{-\infty}^\infty f(\Phi_2(y)) dy.
    \label{my Idea}
    \tag{2}
\end{equation}
Thus, I assume Glasser's theorem only holds for $|a| = 1$; a quick numerical check seems to support Eq. \ref{my Idea}. Is Wikipedia and MathWorld wrong about this?

REPLY [4 votes]: Thank you for noticing this.  Your argument is correct. (An alternative way to see the error is simply to set every $α_i$ to 0.)  I fixed the Wikipedia page (it now uses $x-a$ in place of $|a|x$) and submitted a correction to MathWorld.