Markov-type functions I'd like to have some informations about Markov-type functions (or Cauchy-type):
\[ f(z)=\int_{\Gamma} \frac{\mathrm{d}\gamma(\xi)}{\xi-z}.\]
$\gamma$ is a positive measure with compact support $\subset \bar{R} $.
I'm wondering if there is a reference that explains the study of such functions and gives some examples ($\exp(z)$,$z^{-1/2}$,$\frac{log(1+z)}{z}$,...)
I'm interested in rational approximation of Markov functions to bound an error estimation in the domain of matrix functions.       
 A: Rational approximations of Markov functions are expressed in terms of orthogonal polynomials with respect to $\gamma$ and related functions, and as Alexandre Eremenko pointed out you'll find more details by looking at Padé approximation; the error terms are somehow explicit. An excellent book for the topic is "Rational approximations and orthogonality" of Nikishin and Sorokin.   
Btw, these functions are moment generating functions, since
$$
\int \frac{d\gamma(x)}{z-x}=\sum_{k=0}^\infty z^{-k-1}\int x^k d\gamma(x),
$$
and encodes all the information concerning $\gamma$.
A: Theorem 3 in Chapter VI of Akhiezer and Glazman “Theory of Linear Operators in Hilbert Space,” Ungar, New York, 1963, gives a characterization, a holomorphic function $f$ that maps the upper half plane to the lower half plane can be represented as $\int_{\mathbb{R}} \frac{1}{z-x} \gamma({\rm d}x)$ with a finite measure $\gamma$ if and only if
$$
\limsup_{y\to\infty}y|f(iy)|<\infty.
$$
The $\limsup$ then is equal to $\gamma(\mathbb{R})$.
