### Table of Contents

### correlation

t-statistic for correlation, $r$ \[ t = r \sqrt{\frac{n-2}{1-r^2}} \]

The reverse equation is \[ r = \frac{t}{\sqrt{t^2 + n-2}} \]

Derivation of the reverse equation: \begin{align*} & t = r \sqrt{\frac{n-2}{1-r^2}} & \\ & \frac{t^2}{r^2} = \frac{n-2}{1-r^2} & \\ & \frac{t^2}{n-2} = \frac{r^2}{1-r^2} & \\ & \frac{t^2}{t^2 + n-2} = r^2 & \\ & r = \sqrt{\frac{t^2}{t^2 + n-2}} & \end{align*}

See also:

- https://stackoverflow.com/questions/2632628/left-align-block-of-equations - shows a trick to left align an equation array

### sec(x) + tan(x)

\begin{align*} \sec x + \tan x & = \tan \left( \frac{x}{2} + \frac{\pi}{4} \right) \\ & = \sqrt{\frac{1 + \sin x}{1 - \sin x}} \end{align*}

I came across this simplification while reading https://liorsinai.github.io/mathematics/2020/08/27/secant-mercator.html which talks about the integral of the secant which in turn has applications in Mercator map.

### pages in this wiki

### links I came across

- What is the formula for sum of the p-th powers of the first n positive integers?
- Use Faulhaber's formula in described in https://en.wikipedia.org/wiki/Faulhaber%27s_formula
- The formula involves binomial coefficients and Bernoulli numbers.