Maths hacks

1 Handy inequalities

Young’s inequality for products:

If $a\ge 0$ and $b\ge 0$ are nonnegative real numbers and if $p>1$ and $q>1$ are real numbers such that $\frac{1}{p}+\frac{1}{q}=1$, then $$ab\le \frac{a^p}{p}+\frac{b^q}{q}.$$ Equality holds if and only if $a^p=b^q$. $a=b=2$ arises a lot in statistics.

My favourite generalisation: If $0\le p_i\le 1$ with $\sum _i{p}_i=1$ then $$\prod _i{a}_i^{p_i}\le \sum _i{p}_i{a}_i$$ Equality holds if and only if all the $a_i$s with non-zero $p_i$s are equal.

2 exp, sinh, cos etc

Avoid having to integrate by parts twice

Suppose $f(x)$ and $g(x)$ are functions that are each proportional to their second derivative. These include exponential, circular, and hyperbolic functions. Then the integral of $f(x)g(x)$ can be computed in closed form with a moderate amount of work. There’s a formula that can compute all these related integrals in one fell swoop. (Pease 1959) Suppose $$f^{\prime \prime }(x)=hf(x)$$ and $$g^{\prime \prime }(x)=kg(x)$$ for constants $h$ and $k$. … Then $$\int f(x)g(x)dx=\frac{1}{h-k}(f^{\prime }(x)g(x)-f(x)g^{\prime }(x))+C\text{.}$$

3 Maclaurin integration

(Bruda 2022)

consider the integral $$\int e^{x^2}dx$$ The most common approach to evaluating this integral is to expand it as a power series and integrate term-by-term, which yields $$C+x+\frac{x^3}{3}+\frac{x^5}{10}+\frac{x^7}{42}+\frac{x^9}{216}+\cdots$$ as the antiderivative, with $C$ as the constant of integration. Maclaurin Integration is an alternative solution by series (although not a power series, since it involves the function itself in the solution) that eliminates the nuisance of calculation completely.

…the formula is simply: $$\int f(x)dx=-\sum _{u=0}^{\mathrm{\infty }}(\frac{d^u}{d{x}^u}(xf(x))\sum _{n=0}^{\mathrm{\infty }}\frac{(1-x{)}^{n+u+1}}{\prod _{v=1}^{u+1}(n+v)})+C,$$ where $C$ is the constant of integration.…

This formula is valid only if: 1. $f(x)$ is defined on the domain $(0,2)$, 2. $f(x)$ is continuous on $(0,2)$, and 3. $xf(x)$ has derivatives of all orders on $(0,2)$.

4 References