This is a very difficult book yet I'm enjoying it.  I'm well educated as I believe are many of the people here, half in languages, the other half computer science.  And have read a great deal in physics and math since leaving college - or, if you prefer, university.  The latter is an aggregation of the former but the distinction is mostly cultural.<br><br>Does Penrose explain everything needed?  No, of course not.  Other commenters refer to "expensive" secondary sources - I've been able to remedy most of the holes I find with wikipedia pages and youtube videos.  He introduces the Cauchy-Riemann equations at a point without going into any depth.  I watched a youtube video and they seemed straightforward so when the book returned to the subject explaining it in detail, it was all familiar.<br><br>Why are holomorphic functions important?  Paragraph one alone of the relevant wikipedia page should tell you that.  Although based on stuff I've learned over the yrs it's probably very non-rigorous but I thought: to the extent that analysis is tied to the calculus, and to the extent that what Penrose is dealing with here is complex analysis, it makes a whole more sense to deal first with functions that are smooth and differentiable - as opposed to functions that are not amenable to the calculus.  Put that way, seems obvious to me.<br><br>Read the book, if necessary, very slowly.  Wander around a while - a couple of hrs or days if you prefer.  Look some stuff up on the web - then return to the book.<br><br>I find that my brain will process the book differently on different occasions.  Sometimes I pick it up, it's not clicking, and I put it down again.  Then I return, hrs or a day or so later, and suddenly it makes sense again.  Go easy on yourself - there's no race to finish the book.  Savor it and appreciate on one occasion, feeling dense and then on the subsequent occasion feeling a nice shaft of light hitting your mind.  There are connections all over the place in this book and they make for a lovely landscape.