Maths hacks

August 18, 2016 — January 31, 2023

faster pussycat
mathematics

1 Handy inequalities

Young’s inequality for products:

If a0 and b0 are nonnegative real numbers and if p>1 and q>1 are real numbers such that 1p+1q=1, then abapp+bqq. Equality holds if and only if ap=bq. a=b=2 arises a lot in statistics.

My favourite generalisation: If 0pi1 with ipi=1 then iaipiipiai Equality holds if and only if all the ais with non-zero pis 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. () Suppose f(x)=hf(x) and g(x)=kg(x) for constants h and k. … Then f(x)g(x)dx=1hk(f(x)g(x)f(x)g(x))+C.

3 Maclaurin integration

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consider the integral ex2dx 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+x33+x510+x742+x9216+ 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: f(x)dx=u=0(dudxu(xf(x))n=0(1x)n+u+1v=1u+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

Bruda. 2022. Maclaurin Integration: A Weapon Against Infamous Integrals.”
Pease. 1959. A Useful Integral Formula.” The American Mathematical Monthly.