Maxima-区分分段函数 [英] Maxima - differentiating a piecewise function

问题描述

假设您有一个由间隔定义的函数，例如

Suppose you have a function defined by intervals, such as

f(x):=block(if x<0 then x^2 else x^3);

当我们将其与

diff(f(x),x);

我们得到

d/dx (if x<0 then x^2 else x^3)

我想得到

(if x<0 then 2*x else 3*x^2)

有没有办法获得这样的结果?

Is there a way to obtain such result?

推荐答案

这是对"if"表达式使用简化规则的另一种方法.此处未解决的部分是检测不连续性并为这些位置生成增量函数.如果要忽略它们，则可以定义FOO返回0.这部分在这里尚未解决.如果有兴趣，我可以尝试一下.

Here's a different approach using a simplification rule for "if" expressions. The unsolved part here is to detect discontinuities and generate delta functions for those locations. If you want to ignore those, you can define FOO to return 0. Note that I didn't attempt to implement the function discontinuities; that part is unsolved here. I can give it a try if there is interest.

(%i1) display2d : false $
(%i2) matchdeclare ([aa, bb, cc], all, xx, symbolp) $
(%i3) 'diff (if aa then bb else cc, xx) $
(%i4) tellsimpafter (''%, apply ("if", [aa, diff (bb, xx), true, diff (cc, xx)]) + FOO (aa, bb, cc, xx)) $
(%i5) FOO (a, b, c, x) := 'lsum ((ev (c, x = d) - ev (b, x = d)) * delta (d, x), d, discontinuities (a, x)) $
(%i6) diff (if x > 0 then x^2 else x^3, x);
(%o6) (if x > 0 then 2*x else 3*x^2)+'lsum((d^3-d^2)*delta(d,x),d,
                                           discontinuities(x > 0,x))

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