This question is related to the folowing links. https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula https://en.wikipedia.org/wiki/Stieltjes_transformation
I am trying to understand Stieltjes transformation and it appears remarkably similar to Cauchy integral formula. There may be a connection between the transformation and the formula. I am unable to put both of them in a general context, is there a one?
The Cauchy integral theorem is true for any analytical functions and it is general. Further, the Cauchy theorem is necessary to show Stieltjes transform. The Stieltjes transform is also general, however it has a certain interesting properties (under certain condition) that there is an one-to-one correspondence between the Stieltjes transformation of the function and its transform. There are 4 conditions which need to be true for the one-to-one correspondence as mentioned above. A good link is https://www.youtube.com/watch?v=9OO-gqj3N0c.