Maybe everyone is just bored of talking to me. And adriana is too lazy to go through the maths e-mails so she is putting off replying to them, probably.

I'm not sure. She washed up all our dishes after we ate at her house, but another time I saw her just drop some food under the sofa (it was a bit of sweetcorn from her pizza) and she didn't bother to pick it up. That was pretty gross. A sign of laziness, maybe?

She was sloppy, in that case. But I can't think of other encounters where she was.

I don't know about that, but she does sometimes go to school in track bottoms or sometimes even pyjamas.

She told me. Although most of the time she sleeps in the track-bottoms then goes to school wearing the same ones. She said she doesn't care...

I guess so...

She was nice to talk to though, even though she is strange. Made many of my otherwise boring days interesting.

Easier said than done, I even think about F sometimes.

I just need to move on to another girl who interests me, although there aren't any at the moment.

Maths reminds me of F sometimes. I can do that though.

I am trying to get to grips with contour integrals but I'm not too sure how they work, my knowledge is pretty limited. Do you know a lot about it?

Is the residue of some function f(z) always the co-efficient of 1/(z - a) in its Laurent series? A lot of the time I just end up finding the residue to be zero...

Okay, see you later.

And is it also true that if the residue is 0 then the contour evaluates to 0?

I thought that we had to evaluate 2*pi*i*(sum of the residues)... if the sum of the residues is 0, then...

Okay...

I guess I could try doing something simple like

We know this evaluates to π, so we could maybe try using a contour to try and get the answer too.

Considering instead

where C is a simple closed curve -- a semi-circle in the upper half of the complex plane, from negative infinity to positive infinity, so it encloses the entire real integral, including one of the poles (z = i).

However, I asked Wolfram for the series expansion of 1/(1 + z^2), and it's giving me a series expansion at z = i, and another at z = -i. Since my contour encloses the pole at z = i, I'm interested in the residue of that series.

But I don't know how they found that Laurent series... can you tell me how they found that series expansion that includes the negative power of (z - i)?