now this course is all about complex numbers and I've got a list of recommended books here er well no in fact I seem to have left it behind never mind they're all out of print anyway now let me write up a definition where's the chalk gone ah here it is SNAP ah let me take another piece
now definition 1.1. is ah um of course I haven't said what this section's called yet oh it doesn't seem to have a name anyway it's all about convergence of power series you did something like it in real analysis didn't you don't you remember well he should have done it in his lectures and I don't have time to do that now
now definition 1.1. scribble scribble can you read that at the back no oh well sit further forward then can you read it at the front ah come to think of it I can't read it either perhaps if I turn on this light ah no not that one another one oh well the cord was a bit frayed I suppose well look that symbol is a capital sigma yes what's the problem yes well green seems to be the only colour they have left in the box probably because nobody in his right mind uses it so they leave it for me
well look perhaps if I explain it in words it's all in the textbooks anyway I can't help it if they're missing from the library people eat them or something well now I'll draw a diagram you don't have to copy this exactly because it's slightly wrong anyway this is diagram 2 good question I think I forgot to draw diagram 1 anyway as I say it doesn't help much phew let me take my jacket off a bit rip oh well I sewed that button on myself you can tell can't you
now let me digress a minute about the history of the subject here it was discovered by Cauchy or do I mean Gauss one of those people and he sent a copy of his paper to someone else who well anyway it's very important and has a lot of applications such as er such as well anyway you will see applications in your other courses I expect of course they don't use the same notation but then they don't have the same ideas of rigour as we do and now let's write down the first result lemma 1.2
lemma 1.2 oh I haven't actually defined radius of convergence yet have I still let me write it up and we can decide what it means later well I seem to have a few minutes left so I'd better start the proof let n be this, and r be this and v be that and n be that no on second thoughts I'm already using n now so I'll call it nu pardon no it's a nu a greek letter you must have seen it before all right call it v if you like but we're already using v still it won't cause confusion
now multiply this out and obviously what we get is er clearly um oh that can't be right what have I done wrong here can you see the mistake maybe I lost a minus sign somewhere search me oh dear it's time to finish isn't it well give me just 5 more minutes and I'll finish this off and oh maybe I should do this bit again more carefully ah that should have been a nu maybe no it should be a v or maybe an r oh well look I'll finish this next time I'm sure I've got most of the details right it's really very elementary after all I haven't done anything advanced yet...