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Unicode字符⨍ (U+2A0D)表示Finite Part Integral
Hadamard regularization是一种通过删除一些发散项，只保留有限部分来正则化发散积分的方法
例如$⨍_{-1}^1\frac{dt}{t^2}=-2$
定义
若柯西主值$\displaystyle {\mathcal {C}}\int _{a}^{b}{\frac {f(t)}{t-x}}\,dt\quad ({\text{for }}a<x<b)$存在
则$\displaystyle {\frac {d}{dx}}\left({\mathcal {C}}\int _{a}^{b}{\frac {f(t)}{t-x}}\,dt\right)={\mathcal {H}}\int _{a}^{b}{\frac {f(t)}{(t-x)^{2}}}\,dt\quad ({\text{for }}a<x<b).$
在上面的例子中,$$\mathcal {C}\int_{-1}^1\frac{dt}{t-x}=\lim_{\epsilon\to0}\int_{-1}^{-\epsilon}\frac{dt}{t-x}+\int_{\epsilon}^1\frac{dt}{t-x}=\log{1-x\over1+x}$$
- Integrate[1/(t-x),{t,-1,1},PrincipalValue->True]
根据定义$$\mathcal H\int_{-1}^1\frac{dt}{(t-x)^2}=\frac{d}{dx}\frac{1-x}{x+1}=-\frac{2}{(x+1)^2}$$
令$x=0$
$$\mathcal H\int_{-1}^1\frac{dt}{t^2}=-2$$
上面的 Hadamard 有限部分积分（对于 $a < x < b$）也可以由以下等效定义给出：
$ {\displaystyle {\mathcal {H}}\int _{a}^{b}{\frac {f(t)}{(t-x)^{2}}}\,dt=\lim _{\varepsilon \to 0^{+}}\left\{\int _{a}^{x-\varepsilon }{\frac {f(t)}{(t-x)^{2}}}\,dt+\int _{x+\varepsilon }^{b}{\frac {f(t)}{(t-x)^{2}}}\,dt-{\frac {f(x+\varepsilon )+f(x-\varepsilon )}{\varepsilon }}\right\},} $ $ {\displaystyle {\mathcal {H}}\int _{a}^{b}{\frac {f(t)}{(t-x)^{2}}}\,dt=\lim _{\varepsilon \to 0^{+}}\left\{\int _{a}^{b}{\frac {(t-x)^{2}f(t)}{((t-x)^{2}+\varepsilon ^{2})^{2}}}\,dt-{\frac {\pi f(x)}{2\varepsilon }}-{\frac {f(x)}{2}}\left({\frac {1}{b-x}}-{\frac {1}{a-x}}\right)\right\}.} $ |
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