B Math B.1 Rant on units management B.3 Gaussian Integrals

B.2 Basic Taylor Series

(September 1, 2023)

You are expected to know what a Taylor series is, but we will develop facility with them as the course develops. You are expected to have the following memorized:

$\displaystyle\sin(x)=$	$\displaystyle x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}++\mathcal{O}(x^{7})$	(B.7)
$\displaystyle\cos(x)=$	$\displaystyle 1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\mathcal{O}(x^{6})$	(B.8)
$\displaystyle e^{x}=$	$\displaystyle 1+x+\frac{1}{2!}x^{2}+\mathcal{O}(x^{3})$	(B.9)
$\displaystyle\log(1+x)=$	$\displaystyle x-\frac{1}{2}x^{2}+\frac{1}{3}x^{3}+\mathcal{O}(x^{4})$	(B.10)
$\displaystyle(1+x)^{\alpha}=$	$\displaystyle 1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^{2}+\frac{\alpha(\alpha-% 1)(\alpha-2)}{3!}x^{3}+\mathcal{O}(x^{4})$	(B.11)
$\displaystyle\frac{1}{1+x}=$	$\displaystyle 1-x+x^{2}+\mathcal{O}(x^{3})$	(B.12)

These get me through life.

They can be integrated, etc, so for example the expression for $\log(1+x)$ follows from the geometric series $1/(1+x)$ by integration:

\log(1+x)=\int_{0}^{x}\frac{{\rm d}x^{\prime}}{1+x^{\prime}}=\int_{0}^{x}{\rm d% }x^{\prime}\,\left[1-x^{\prime}+(x^{\prime})^{2}+\mathcal{O}((x^{\prime})^{3})% \right]=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}+\mathcal{O}(x^{4})

(B.13)