## Possible Downtime and More Random Math

http://www.neildickson.com and http://www.codecortex.com may or may not be down on Sunday or Monday.  If they are, I blame Scotiabank for:

• needing 6-8 business days to make a simple transaction that can be made in less than half a second because of two little things I like to call: the internet and the 21st century
• sending two versions of a credit card within a week or two where activating the wrong one makes them both not work
• needing over a week to make and send a new credit card instead of just creating one on the spot in about 30 seconds as they easily could (just need a punch for the numbers and a magnetic strip writer)
• charging me an outrageous $54 annual fee for the privilege of carrying a piece of plastic around that didn’t work anyway • charing a record high 30% interest rate on the credit card even though I’ve never missed a payment • paying a record low 0.25% interest rate on my savings account, and even then only on amounts over$5,000; less than 1/3 of what RBC pays, and less than 1/4 of what HSBC pays
• working only 9:30am-4:00pm to avoid dealing with customers who have jobs

Anyway, hopefully the sites don’t go down, and here’s some more random math-ish stuff.  Suppose you have a function in terms of its Taylor series about zero (it’s Maclauren series):

$f(x) = \displaystyle\sum_{i=0}^{\infty} a_i x^i$

How can you find the square of the function?  In hind sight, it should have been obvious, but it’s:

$(f(x))^2 = \displaystyle\sum_{i=0}^{\infty} \left(\displaystyle\sum_{j=0}^{i} a_j a_{i-j}\right) x^i$

In fact, more generally, if we also have $g(x) = \sum_{i=0}^{\infty} b_i x^i$:

$f(x)g(x) = \displaystyle\sum_{i=0}^{\infty} \left(\displaystyle\sum_{j=0}^{i} a_j b_{i-j}\right) x^i$

You can also calculate $\displaystyle\frac{f(x)}{g(x)}$ fairly simply:

$\displaystyle\frac{f(x)}{g(x)} = \frac{\displaystyle\sum_{i=0}^{\infty} a_i x^i}{\displaystyle\sum_{i=0}^{\infty} b_i x^i} = \displaystyle\sum_{i=0}^{\infty} c_i x^i$
$\displaystyle\sum_{i=0}^{\infty} a_i x^i = \left(\displaystyle\sum_{i=0}^{\infty} b_i x^i\right)\left(\displaystyle\sum_{i=0}^{\infty} c_i x^i\right)$

$a_0 = b_0 c_0$
$a_1 = b_1 c_0 + b_0 c_1$
$a_2 = b_2 c_0 + b_1 c_1 + b_0 c_2$
$a_3 = b_3 c_0 + b_2 c_1 + b_1 c_2 + b_0 c_3$

$\cdots$

$\begin{bmatrix} b_0 & 0 & 0 & 0 & \cdots \\ b_1 & b_0 & 0 & 0 & \cdots \\ b_2 & b_1 & b_0 & 0 & \cdots \\ b_3 & b_2 & b_1 & b_0 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix} \begin{bmatrix} c_0 \\ c_1 \\ c_2 \\ c_3 \\ \vdots \end{bmatrix} = \begin{bmatrix} a_0 \\ a_1 \\ a_2 \\ a_3 \\ \vdots \end{bmatrix}$

Simply use forward substitution to solve for the values of $c_i$.

It’s much harder to work out what non-integer powers of the Taylor series are, though.  The pattern for $\sqrt{f(x)}$ seems too complex to figure out from 5 terms.  What’d be really cool is to get the series for $f(g(x))$, since then most anything else would be pretty easy to figure out.

~ by Neil Dickson on August 22, 2009.

### One Response to “Possible Downtime and More Random Math”

1. Scotiabank was good when I was at Carleton since they were the only bank on campus. But now that I’ve graduated, I’ve dropped them for greener pastures. I’m convinced their computer systems haven’t been updated since the 1970s! And I can never find anything on their online banking. You would not believe the crap I had to endure just to upgrade my “student” credit card to a full one.