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.

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