将技巧转化为方法：偶函数级数逼近的进阶应用Turning a trick into a technique
作者探讨如何将前文提到的数学技巧系统化发展为正式方法。通过从偶函数中减去倍数关系，可以创建高阶近似表达式，这种方法利用了偶函数仅包含偶次项的特性。研究展示了如何重复验证一个技巧的有效性，从而将其提升为可信赖的数学方法。
John
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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