数学备忘录：基础数学

三角函数

三角函数的积化和差

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

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}+\dots \\ \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-\dots \\ \cos x=\sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}x^{2n}=1-\frac{1}{2!}x^2+\frac{1}{4!}x^4-\dots \\ \tan x=x+\frac{1}{3}x^3+\frac{2}{15}x^5+\frac{17}{315}x^7+\frac{62}{2835}x^9+\dots \\ \ln(1+x)=\sum_{n=0}^\infty\frac{(-1)^n}{n+1}x^{n+1}=x-\frac{x^2}{2}+\frac{x^3}{3}-\dots \\ \ln(1-x)=-\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}=-\left(x+\frac{x^2}{2}+\frac{x^3}{3}+\dots\right) \\ (1+x)^\alpha=\sum_{n=0}^\infty \begin{pmatrix} \alpha \\ n \end{pmatrix} x^{n} \\ (1\pm x)^{1/2}=1\pm\frac{1}{2}x-\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\dots \\ (1\pm x)^{3/2}=1\pm\frac{3}{2}x+\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\dots \\ (1\pm x)^{5/2}=1\pm\frac{5}{2}x+\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\dots \\ (1\pm x)^{-1/2}=1\mp\frac{1}{2}x+\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\dots \\ (1\pm x)^{-3/2}=1\mp\frac{3}{2}x+\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\dots \\ (1\pm x)^{-5/2}=1\mp\frac{5}{2}x+\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\dots \\ (1\pm x)^{-1}=1\mp x+x^2\mp x^3+x^4\pm\dots \\ (1\pm x)^{-2}=1\mp 2x+3x^2\mp 4x^3+5x^4\mp\dots \\ \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}+\dots \\ \arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\dots \\ \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

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

• 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

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