数学备忘录：基础数学

三角函数

三角函数的积化和差

\[ \begin{gather} \sin\alpha\cos\beta=\frac{1}{2}\left[\sin(\alpha+\beta)+\sin(\alpha-\beta)\right] \\ \cos\alpha\sin\beta=\frac{1}{2}\left[\sin(\alpha+\beta)-\sin(\alpha-\beta)\right] \\ \cos\alpha\cos\beta=\frac{1}{2}\left[\cos(\alpha+\beta)+\cos(\alpha-\beta)\right] \\ \sin\alpha\sin\beta=-\frac{1}{2}\left[\cos(\alpha+\beta)-\cos(\alpha-\beta)\right] \end{gather} \]

三角函数的和差化积

\[ \begin{gather} \sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \\ \sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} \\ \cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \\ \cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} \end{gather} \]

常见函数的幂级数展开式（略去收敛域）

\[ \begin{gather} \mathrm{e}^x=\sum_{n=0}^\infty\frac{x^n}{n!}=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots \\ \sin x=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}x^{2n+1}=x-\frac{1}{3!}x^3+\frac{1}{5!}x^5-\cdots \\ \cos x=\sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}x^{2n}=1-\frac{1}{2!}x^2+\frac{1}{4!}x^4-\cdots \\ \tan x=x+\frac{1}{3}x^3+\frac{2}{15}x^5+\frac{17}{315}x^7+\frac{62}{2835}x^9+\cdots \\ \ln(1+x)=\sum_{n=0}^\infty\frac{(-1)^n}{n+1}x^{n+1}=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots \\ \ln(1-x)=-\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}=-\left(x+\frac{x^2}{2}+\frac{x^3}{3}+\cdots\right) \\ (1+x)^\alpha=\sum_{n=0}^\infty \begin{pmatrix} \alpha \\ n \end{pmatrix} x^{n} \\ (1\pm x)^{1/2}=1\pm\frac12x-\frac{1\cdot1}{2\cdot4}x^2\pm\frac{1\cdot1\cdot3}{2\cdot4\cdot6}x^3-\frac{1\cdot1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}x^4\pm\cdots \\ (1\pm x)^{3/2}=1\pm\frac32x+\frac{3\cdot1}{2\cdot4}x^2\mp\frac{3\cdot1\cdot1}{2\cdot4\cdot6}x^3+\frac{3\cdot1\cdot1\cdot3}{2\cdot4\cdot6\cdot8}x^4\mp\cdots \\ (1\pm x)^{5/2}=1\pm\frac52x+\frac{5\cdot3}{2\cdot4}x^2\pm\frac{5\cdot3\cdot1}{2\cdot4\cdot6}x^3-\frac{5\cdot3\cdot1\cdot1}{2\cdot4\cdot6\cdot8}x^4\mp\cdots \\ (1\pm x)^{-1/2}=1\mp\frac12x+\frac{1\cdot3}{2\cdot4}x^2\mp\frac{1\cdot3\cdot5}{2\cdot4\cdot6}x^3+\frac{1\cdot3\cdot5\cdot7}{2\cdot4\cdot6\cdot8}x^4\mp\cdots \\ (1\pm x)^{-3/2}=1\mp\frac32x+\frac{3\cdot5}{2\cdot4}x^2\mp\frac{3\cdot5\cdot7}{2\cdot4\cdot6}x^3+\frac{3\cdot5\cdot7\cdot9}{2\cdot4\cdot6\cdot8}x^4\mp\cdots \\ (1\pm x)^{-5/2}=1\mp\frac52x+\frac{5\cdot7}{2\cdot4}x^2\mp\frac{5\cdot7\cdot9}{2\cdot4\cdot6}x^3+\frac{5\cdot7\cdot9\cdot11}{2\cdot4\cdot6\cdot8}x^4\mp\cdots \\ (1\pm x)^{-1}=1\mp x+x^2\mp x^3+x^4\pm\cdots \\ (1\pm x)^{-2}=1\mp 2x+3x^2\mp 4x^3+5x^4\mp\cdots \\ \arcsin x=x+\frac{1}{2}\frac{x^3}{3}+\frac{1\cdot3}{2\cdot4}\frac{x^5}{5}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\frac{x^7}{7}+\cdots \\ \arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots \\ \arccos x=\frac{\pi}{2}-\arcsin x, \quad \mathrm{arccot} x=\frac{\pi}{2}-\arctan x \\ \mathrm{arccsc} x=\arcsin\frac{1}{x}, \quad \mathrm{arcsec} x=\arccos\frac{1}{x} \end{gather} \]

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多重积

• \(\mathbf a\cdot(\mathbf b\times\mathbf c)=\mathbf b\cdot(\mathbf c\times\mathbf a)=\mathbf c\cdot(\mathbf a\times\mathbf b)\)
• \(\mathbf a\times(\mathbf b\times\mathbf c)=\mathbf b(\mathbf a\cdot\mathbf c)-(\mathbf a\cdot\mathbf b)\mathbf c\)
• \((\mathbf a\times\mathbf b)\cdot(\mathbf c\times\mathbf d)=(\mathbf a\cdot\mathbf c)(\mathbf b\cdot\mathbf d)-(\mathbf a\cdot\mathbf d)(\mathbf b\cdot\mathbf c)\)

矢量微分算符 \(\nabla\)

• \(\nabla\cdot(u\mathbf a)=(\nabla u)\cdot\mathbf a+u(\nabla\cdot\mathbf a)\)
• \(\nabla\times(u\mathbf a)=(\nabla u)\times\mathbf a+u(\nabla\times\mathbf a)\)
• \(\nabla\cdot(\mathbf a\times\mathbf b)=\mathbf b\cdot(\nabla\times\mathbf a)-\mathbf a\cdot(\nabla\times\mathbf b)\)
• \(\nabla\times(\mathbf a\times\mathbf b)=(\mathbf b\cdot\nabla)\mathbf a-(\mathbf a\cdot\nabla)\mathbf b+\mathbf a(\nabla\cdot\mathbf b)-\mathbf b(\nabla\cdot\mathbf a)\)
• \(\nabla(\mathbf a\cdot\mathbf b)=(\mathbf b\cdot\nabla)\mathbf a+(\mathbf a\cdot\nabla)\mathbf b+\mathbf b\times(\nabla\times\mathbf a)\)
• 梯度的旋度：\(\nabla\times\nabla u=\nabla\times(\nabla u)=0\)
• 旋度的散度：\(\nabla\cdot(\nabla\times\mathbf a)=0\)
• \(\nabla\times(\nabla\times\mathbf a)=\nabla(\nabla\cdot\mathbf a)-\nabla^2\mathbf a\)

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Laplace 算符 \(\nabla^2\)

• 极坐标系中的 Laplace 算符

\[ \nabla^2=\frac1r\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2}{\partial\phi^2} =\frac{1}{r^2}\left[r\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}\right)+\frac{\partial^2}{\partial\phi^2}\right] \]

• 柱坐标系中的 Laplace 算符

\[ \begin{align} \nabla^2 & = \frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial}{\partial\rho}\right)+\frac{1}{\rho^2}\frac{\partial^2}{\partial\phi^2}+\frac{\partial^2}{\partial z^2} \\ & = \frac{1}{\rho^2}\left[\rho\frac{\partial}{\partial\rho}\left(\rho\frac{\partial}{\partial\rho}\right)+\frac{\partial^2}{\partial\phi^2}\right]+\frac{\partial^2}{\partial z^2} \end{align} \]

• 球坐标系中的 Laplace 算符

\[ \begin{align} \nabla^2 & = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\phi^2} \\ & = \frac{1}{r^2\sin^2\theta}\left[\sin^2\theta\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)+\sin\theta\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{\partial^2}{\partial\phi^2}\right] \end{align} \]