傅里叶 fourier 级数nx公式

三角函数nx积分

\[ \int_{0}^{\pi} \sin(nx) dx = \frac{1 - \cos(\pi n)}{n} = \frac{(-1)^n + 1}{n} \] \[ \int_{0}^{\pi} \cos(nx) dx = \frac{\sin(\pi n)}{n} = 0 \]

\[ \int_{0}^{2\pi} \sin(nx) dx = 0 \]

\[ \int_{0}^{2\pi} \cos(nx) dx = 0 \]

含x的三角nx积分

\[ \int_0^\pi x \sin(nx) dx = \frac{-\pi \cos(\pi n)}{n} = \frac{(-1)^{n-1} \pi}{n^2} \]

\[ \int_0^\pi x \cos(nx) dx = \frac{\cos(\pi n) - 1}{n^2} = \frac{(-1)^n - 1}{n^2} \]

\[ \int_0^{2\pi} x \sin(nx) dx = -\frac{2\pi}{n} \]

\[ \int_0^{2\pi} x \cos(nx) dx = 0 \]

含 x^2 三角nx积分

\[ \int_{0}^{\pi} x^{2} \sin(nx) dx = \frac{(2 - \pi^{2}n^{2}) \cos(\pi n) - 2}{n^{3}} \]

\[ \int_{0}^{\pi} x^{2} \cos(nx) dx = \frac{2\pi \cos(\pi n)}{n^{2}} = \frac{2\pi (-1)^{n}}{n^{2}} \]

\[ \int_{0}^{2\pi} x^{2} \sin(nx) dx = \frac{-4\pi^{2}}{n} \]

\[ \int_{0}^{2\pi} x^{2} \cos(nx) dx = \frac{4\pi}{n^{2}} \]