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更新线性代数
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870138612 committed Nov 4, 2024
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- 变限积分重要结论
- $f(x)$为可积的奇函数$\Rightarrow\begin{cases}\int_0^xf(t)dt、\int_0^xf(t)dt+C皆为偶函数\\ \int_a^xf(t)dt为偶函数(a \not = 0)\end{cases}$.
- $f(x)$为可积的偶函数$\Rightarrow\begin{cases}\int_0^xf(t)dt为奇函数\\ \int_a^xf(t)(a\not = 0)\begin{cases} 若 \int_a^xf(t)dt = \int_0^xf(t)dt,为奇函数 \\ 若\int_a^xf(t)dt \not = \int_0^xf(t)dt,为非奇非偶函数\end{cases}\end{cases}$.
- $f(x)$是可积且以T为周期的周期函数,则$\int_0^xf(t)dt$是以$T$为周期的周期函数$\Leftrightarrow \int_0^Tf(x)dx=0$.
- $f(x)$是可积且以$T$为周期的周期函数,则$\int_0^xf(t)dt$是以$T$为周期的周期函数$\Leftrightarrow \int_0^Tf(x)dx=0$.
- 连续的奇函数的一切原函数都是偶函数,连续的偶函数的原函数中只有一个原函数为奇函数.


- 反常积分计算时,注意识别奇点(端点,内部)
- 例如$\int_0^2f(x)dx$的奇点为x=1,则应该拆分为$\int_0^1f(x)dx+\int_1^2f(x)dx$.
- 例如$\int_0^2f(x)dx$的奇点为$x=1$,则应该拆分为$\int_0^1f(x)dx+\int_1^2f(x)dx$.
- 奇点处的值一般使用极限求得,分别对应奇点处的左极限和右极限.
- $\Gamma$函数

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