Turning a trick into a technique

Someone said a technique is a trick that works twice. I wanted to see if I could get anything interesting by turning the trick in the previous post into a technique. The trick created a high-order approximation by subtracting a multiple one even function from another. Even functions only have even-order terms, and by using the right multiple you can cancel out the second-order term as well. For an example, I’d like to approximate the Bessel function J0(x) by the better known cosine function. Both are even functions. J0(x) = 1 − x2/4 + x4/64 + … cos(x) = 1 − x2/2 + x4/24 + … and so 2 J0(x) − cos(x) = 1 − x4/96 + … which means J0(x) ≈ (1 + cos(x))/2 is an excellent approximation for small x. Let’s try this for a couple examples. J0(0.2) = 0.990025 and (1 + cos(0.2))/2 = 0.990033. J0(0.05) = 0.99937510 and (1 + cos(0.05))/2 = 0.99937513.The post Turning a trick into a technique first appeared on John D. Cook.