In many literature it is known that an operator monotone function can be represented in a form $$f(x)=a+bx+\int_{0}^{\infty}\left(\frac{t}{1+t^{2}}-\frac{1}{t+x} \right)d\mu(t).$$ I also know that the typical examples of operator monotone function is $-\frac{1}{x}$ and $\log x.$ For the first example I guess that $a=b=0,$ but I do not know how to find $\mu.$
Any help would be appreciated.
Thank you so much.