Analytic Function Theory by Einar Hille

[quark]

Every year, I give DS a little welcome to the next school year gift. This year, I was lazy and bought him an Amazon gift card. So DS did some browsing and sent me a link to use his gift card to purchase this book. He insists that a math mentor he met at his recent summer program swears by it. I was a little leery and scrolled down to read reviews, slightly worried that only one person had reviewed it. Imagine my surprise and delight when I saw that the 5-star reviewer was our very own mathwonk!

 

I love this board. :001_wub:

 

If anyone else has any advice for DS on how to approach the book or can be willing to provide some hints should he need them while self studying (I don't see solutions?), I'd be even more grateful!

[Mike in SA]

Wow.  If his mentor feels he has the background for it, then he should be able to simply read and gain value.  I haven't worked this particular text (it looks pretty solid and well written), but can give general advice...

 

As always, the examples provided should be solved independently BEFORE attempting to read the example.  Much like with AOPS, trying to work every problem (with or without a solution) provides value in itself.

 

Other than that, I would be most concerned with having appropriate background first.  There are a number of good texts which could provide additional reference to shore up weak spots when they rear up - just plan for periodic sidetracking for weeks at a time.  On the plus side, that'll be a taste of real math research - it stops being spoon-fed after a while, and more than one reference becomes essential.  I certainly wouldn't set the pace by dividing page count by school days.

 

I'd be happy to recommend an additional source when desired.  We (DW and I) have discovered a few interesting routes to prep for some of the topics in that text.  I'm sure the other mathematicians on the board would be happy, as well.

[quark]

Thanks Mike! Gosh, I gave up setting a pace and dividing page count years ago. I think the last time I tried that with anything, he was about 7. :D

 

So glad to know he has a resource for questions! What background would you recommend (bolded)? He has completed linear alg, diff eq and the calculus1-3 series and has been working regularly with the AoPS Polymath research materials. No formal analysis yet. I'll pass your suggestions on so he can start looking up topics!

 

Wow.  If his mentor feels he has the background for it, then he should be able to simply read and gain value.  I haven't worked this particular text (it looks pretty solid and well written), but can give general advice...

 

As always, the examples provided should be solved independently BEFORE attempting to read the example.  Much like with AOPS, trying to work every problem (with or without a solution) provides value in itself.

 

Other than that, I would be most concerned with having appropriate background first.  There are a number of good texts which could provide additional reference to shore up weak spots when they rear up - just plan for periodic sidetracking for weeks at a time.  On the plus side, that'll be a taste of real math research - it stops being spoon-fed after a while, and more than one reference becomes essential.  I certainly wouldn't set the pace by dividing page count by school days.

 

I'd be happy to recommend an additional source when desired.  We (DW and I) have discovered a few interesting routes to prep for some of the topics in that text.  I'm sure the other mathematicians on the board would be happy, as well.

 

[Mike in SA]

My guess is that he's probably ready.

 

AoPS Precalc gives a good intro to complex functions; a Calculus course along the lines of Anton will be rigorous enough to prep for a course in analysis.  I personally don't feel that the AP calculus courses provide adequate prep, and for students taking that route, a course in intermediate analysis is often beneficial prior to formal complex analysis.  This text, however, seems to start early enough in the development of complex analysis that makes me believe intermediate analysis wouldn't add much.

 

Now, another branch that comes to mind is algebra/group theory.  Some background there would be helpful, but not required - it's more about the nature of thinking in a pure math course.  Knowing something about isomorphism, homomorphism, and polymorphism (for example) in the abstract sense of the terms will bring more relevance to topics when reached.  I wouldn't worry too much right now - the terms will come up over and over again as he progresses.

 

 

[quark]

Thanks so much Mike. No his calc was not AP and while I feel the CC could have been better he does a lot of extra stuff on his own. He has abstract algebra and group theory fundamentals too. His former math mentor included the basics with some proving thrown in over 2 years and then kid went deeper with AoPS Group Theory Seminar last summer. He plans to take abstract algebra formally for credit at the uni in spring. Glad he has some background because he seems eager to start. I don't know how far he will go into the book due to an already packed fall but he looks eager. He will very likely follow up with more formal classes in about a year too.

 

Thanks again!