Examples of the Stieltjes transformation

I give the definition and two examples of the Stieltjes transformation.
If \( f \) is a function with support \( I \subseteq \mathbb{R} \), then for \( z \in\mathbb{C} - I\), the Stieltjes transformation \( S(z) \) is
\begin{equation}\label{eq:20151219a}
S(z) = \int_{-\infty}^{+\infty} \frac{f(t)}{t-z} dt
\end{equation} The inverse of the Stieltjes transformation is \begin{equation}\label{eq:20151219b} f(z) = \lim_{ \epsilon \overset{ > }{\to} 0} \frac{ S(z+ i \epsilon) - S(z- i \epsilon)}{2 \pi i} \end{equation} I deduce \eqref{eq:20151219b} with a sloppy calculation: \begin{equation*}S(z+ i \epsilon) - S(z- i \epsilon) = 2 i \epsilon \int_{-\infty}^{+\infty} f(t) dt \frac{1}{(t-z)^2 + \epsilon^2} \end{equation*} Because \begin{equation*} \lim_{ \epsilon \overset{ > }{\to} 0} \frac{\epsilon}{x^2 + \epsilon^2} = \pi \delta(x) \end{equation*} I get \begin{equation*} \lim_{ \epsilon \stackrel{>}{\to} 0} \left( S(z+ i \epsilon) - S(z- i \epsilon)\right) = 2 i \int_{-\infty}^{+\infty} f(t) dt\ \pi \delta(t - z) = 2 i \pi f(z) \end{equation*}

It seems that the Stieltjes transformation is often used in probability theory. I therefore take the examples from probability theory. Example 1
The simplest probability distribution is the uniform distribution. Thus I take \( f(t) = 1 \) if \( t \in [ 0,a ] \) and \( f(t) = 0 \) otherwise. Then one can calculate that \begin{equation*} S(z) = \log \left( \dfrac{z-a}{z} \right) \stackrel{\mathrm{def}}{=} \mathrm{Log}(z-a) - \mathrm{Log} z \end{equation*} with \( \mathrm{Log} \) the logarithm with branch cut on \( \left] - \infty, 0 \right] \) and \( \mathrm{Log} 1 = 0 \). With this definition, \( \log \left( \dfrac{z-a}{z} \right) \) has a branch cut on the interval \( [0,a ] \). If one moves from the lower half plane over the branch cut to the upper half plane, the value of \( \log \left( \dfrac{z-a}{z} \right) \) jumps with the amount \( 2 \pi i \).
Here is a graph of the function \( \log \left( \dfrac{z-a}{z} \right) \)

\( \mathrm{Re} \left( \log \left( \dfrac{z-a}{z} \right) \right) \) for \( a = 1 \)

\( \mathrm{Im} \left( \log \left( \dfrac{z-a}{z} \right) \right) \) for \( a = 1 \)

Hence,

• If \( z \not\in [ 0, a] \), then \( S(z) \) is continuous, thus \( \displaystyle\lim_{ \epsilon \stackrel{>}{\to} 0} \left( S(z+ i \epsilon) - S(z- i \epsilon)\right) = 0 \)
• If \( z \in [ 0, a] \), then \( \displaystyle\lim_{ \epsilon \stackrel{>}{\to} 0} \left( S(z+ i \epsilon) - S(z- i \epsilon)\right) = 2 \pi i \)

which is indeed formula \eqref{eq:20151219b}

Example 2
The second example is motivated by Wigner's semicircle distribution from random matrix theory. Take  \( f(t) = \sqrt{ 1 - t^2} \) if \( t \in [ -1 , 1 ] \) and \( f(t) = 0 \) otherwise. By integrating the contour integral around the branch cut, one can calculate that \begin{equation*} S(z) = \int_{-1}^1 \frac{\sqrt{1 - t^2}}{t -z} dt = \pi \left( \sqrt{z - 1}\sqrt{z + 1} - z \right) \end{equation*} Here, in the right hand side, the complex square root \( \sqrt{z} \) is defined with branch cut on the negative real axis. It then follows that \( \sqrt{z - 1}\sqrt{z + 1} \) has a branch cut on \( [-1 , 1] \). Also, one can see that if \( x \in [-1,1] \) and \( z = x + i \epsilon \) with \( \epsilon > 0 \), then \( \sqrt{z^2 - 1} = i \sqrt{1 - x^2 } \).
Here is a graph of the function \( \sqrt{z - 1}\sqrt{z + 1} - z\)

\( \mathrm{Re} \left( \sqrt{z - 1}\sqrt{z + 1} - z \right) \)

\( \mathrm{Im} \left( \sqrt{z - 1}\sqrt{z + 1} - z \right) \)

Hence,

• If \( z \not\in [ -1, 1] \), then \( S(z) \) is continuous, thus \( \displaystyle\lim_{ \epsilon \stackrel{>}{\to} 0} \left( S(z+ i \epsilon) - S(z- i \epsilon)\right) = 0 \)
• If \( z \in [-1, 1] \), then \begin{equation*} \lim_{ \epsilon \stackrel{>}{\to} 0} \left( S(z+ i \epsilon) - S(z- i \epsilon)\right) = \pi \left( i \sqrt{1 - x^2 } - (- i) \sqrt{1 - x^2 }\right) = 2 \pi i \sqrt{1 - x^2 } \end{equation*}

which is again formula \eqref{eq:20151219b}

Remark: I decided to write this post because the Stieltjes transformation is used in random matrix theory. I wanted therefore to understand the Stieltjes transformation on some easy examples.