## Saturday, January 10, 2015

### in which, this semester, i must hide my heretic ways.

so i just wrote my first lecture of 2015 .. and it reads like algebra, if only because it's supposed to read that way.

it's a first lecture in an undergraduate course on complex analysis. i've never taught it before.

something worries me; i wonder if i'm the right guy to teach it.

---
i'm not worried about screwing it up, not like last fall's numerical analysis ..

.. more on that, later ..

.. but now i have to come to terms with the subject. you see, i've deemed the subject unnecessary and consistently tried to avoid all aspects of complex analysis in my mathematical life.

yes, i know; it's the mathematical version of being a bigot or committing heresy!

perhaps it's just my first exposure to the subject [1]. perhaps it's the nature of the research problems i've chosen to study over the years. perhaps i generally take a pessimistic viewpoint in maths and a statement of the form ..

.. "every differentiable function at a point is infinitely differentiable at that same point and can be represented as a power series" ..

.. just sounds too good to be true, that there must have been some mistake. ye gods: what kind of dark, forbidden magic have we wrought?

---
now, of all things, i have to teach it.

with luck, my students won't inherit my own prejudices on the matter. i'll even attempt to sell them on its good points, if only for the sake of being a responsible mathematical role model.

on a related note, my first lecture can be summed up as such:

if you take the usual coordinate page, view it as a 2-dimensional linear subspace of R^4, rotate it appropriately, and project it down to its original domain, then you can make sense of square roots of negative numbers, provided that you treat those image vectors as 2x2 matrices.

---
notes:

[1] it's fair to say that my first course in it was uninspiring and largely i skipped all of the lectures except the exam periods. how i did "well" in that course is beyond me ..

.. but to be fair, i went to my favorite coffeehouse three times a week, armed with the textbook, ordered a large coffee, and would work through the problem sets from scratch. my barista friend would see me there regularly; she once asked me if i really liked that book or class, because i was reading it so much, to which i guffawed.

#### 1 comment:

Leonid said...

.. "every differentiable function at a point is infinitely differentiable at that same point and can be represented as a power series" ..

.. is indeed too good to be true. A counterexample is f(z)=|z|^2, which is complex differentiable at z=0, but...