==== 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 ====
  * [[fibonacci sequence]]
  * [[Leibniz integral rule]]

==== 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.
    * See also: https://www.johndcook.com/blog/2016/12/31/sums-of-consecutive-powers/

==== pages with math outside of this wiki ====
use cases | to copy paste syntax
  * https://liorsinai.github.io/mathematics/2020/08/27/secant-mercator.html
  * https://en.wikipedia.org/wiki/Fibonacci_sequence
  * https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors