06-math.Rmd

# Useful Math Facts {#math}

## Exponents and Logarithms
for positive constants $a$ and $b$: 
 
$a^0 = 1$ 

$a^{-n}=\frac{1}{a^n}$ for positive integer $n$

$a^{x+y}=a^x a^y$

$a^{x-y}=\frac{a^x}{a^y}$

$(a^x)^y=a^{xy}$

$(ab)^x=a^xb^x$


$\log_a(x) = y$ implies $a^{y} = x$

If you're working in statistics it's usually a safe bet that $\log(x)$ means $\log_e(x)$ not $\log_{10}(x)$.

$\log(1) = 0$

$\log(e) = 1$

$\log(xy)=\log(x) + \log(y)$

$\log(x/y)=\log(x)-\log(y)$

## Sequences and Series

### Monotonicity

A sequence of numbers $\{x_1,x_2,\ldots\}$ is *monotone increasing* if $x_n \le x_{n+1}$ for all $n$. If $x_n \ge x_{n+1}$ for all $n$, the sequence is *monotone decreasing*

### Geometric series

$\sum\limits_{k=0}^{n} a^k =\frac{1- a^{n+1}}{1-a}$

for $|a|<1$ $\sum\limits_{k=0}^{\infty} a^k =\frac{1}{1-a}$

### Taylor series

$e^x=\sum\limits_{n=0}^{\infty}\frac{x^n}{n!}$

### Binomial Theorem

$(x+y)^n = \sum\limits_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}$

## Calculus - Derivatives

$f(x)=c \;(\text{constant}) \qquad f'(x)=0$

$f(x) = x^{r},\, r \in \mathbb{R} \qquad f'(x) = rx^{r-1}$ (Power Rule)

$f(x) = e^x \qquad f'(x) = e^x$

$f(x)=\log(x) \qquad f'(x) =\frac{1}{x}$

$f(x)=g(x)\pm h(x) \qquad f'(x)=g'(x) \pm h'(x)$

$f(x)=g(x)h(x) \qquad f'(x)= g(x)h'(x)+g'(x)h(x)$  (Product Rule)

$f(x) =\frac{g(x)}{h(x)} \qquad f'(x)=\frac{g'(x)h(x)-g(x)h'(x)}{[h(x)]^{2}}\,$  (Quotient Rule)

$f(x)=g \circ h(x) = g(h(x)) \qquad f'(x) = g'(h(x))h'(x)$  (Chain Rule)

### Maxima and Minima

$f(x)$ has a *minimum* at $c$ if $f'(c)=0$ and $f''(c)>0$

$f(x)$ has a *maximum* at $c$ if $f'(c)=0$ and $f''(c)<0$

## Calculus - Integrals

$f(x) = c \qquad \int f(x)\,dx = cx$

$f(x) = x^r \qquad \int f(x)\,dx = \frac{x^{r+1}}{r+1},\,r \ne -1$

$f(x) = \frac{1}{x} \qquad \int f(x)\,dx = \log(x)$

$f(x) = e^x \qquad \int f(x)\,dx = e^x$

$f(x) = cg(x) \qquad \int f(x)\,dx = c \int g(x)\, dx$

$f(x) = g(x)\pm h(x) \qquad \int f(x)\,dx = \int g(x)\, dx \pm \int h(x)\, dx$

### Integration by Parts

$\int_{a}^{b} f(x)g'(x)\, dx = f(x) g(x)|_{a}^{b} - \int_{a}^{b} f'(x)g(x)\, dx$

Equivalently, let $u=f(x)$ and $dv= g'(x)\, dx$ then $\int u\, dv = uv -\int v du$

## Miscellaneous

### Gamma function

The Gamma function is defined as $\Gamma(\alpha) =\int\limits_{0}^{\infty}x^{\alpha-1}e^{-x}\, dx$ for $\alpha > 0$

$\Gamma(1)=1$

$\Gamma(\alpha)=(\alpha-1)\Gamma(\alpha-1)$

$\alpha\Gamma(\alpha)=\Gamma(\alpha+1)$

$\Gamma(n)=(n-1)!$ for integer $n>0$

### Beta Function

The Beta function is defined as $B(\alpha, \beta)=\int_{0}^{1}x^{\alpha-1}(1-x)^{\beta-1}\,dx$ for $\alpha, \beta > 0$

$B(\alpha, \beta) = B(\beta, \alpha)$
 
$B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}$