泰勒展开($|x-x_{0}|<1$)
$$
\begin{align}
f(x)=
\mathop{\sum_{n=0}^{\infty}}\frac{f^{(n)}\left(x_{0}\right)}{n!}\left(x-x_{0}\right)^{n}
\,.
\end{align}
$$常见的有:
$$
\begin{align}
\frac{1}{1-x}&=\mathop{\sum_{n=0}^{\infty}}\left(x\right)^{n}
\,,\\
e^{x}&=\mathop{\sum_{n=0}^{\infty}}\frac{x^{n}}{n!}
\,,
\end{align}
$$和
$$
\begin{align}
\sin (x)
&=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots
=\sum_{n=0}^{\infty}\left(-1\right)^{n}\frac{x^{2n+1}}{\left(2n+1\right)!}
\,,\\
\cos (x)
&=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots
=\sum_{n=0}^{\infty}(-1)^{x}\frac{x^{2n}}{(2n)!}
\,.
\end{align}
$$矢量
两个矢量的叉乘为
$$
\begin{align}
\vec{c}&=\vec{a}\times\vec{b}
\,,\\
c_{k}&=\epsilon_{ijk}a_{i}b_{j}
\,,
\end{align}
$$其中,$\vec{a}$ 和 $\vec{b}$ 是极矢量(polar vector),而 $\vec{c}$ 称为轴矢量(axial vector)或赝矢量(pseudovector)。极矢量的性质符号(正负号)在空间反演对称下发生改变,而赝矢量的性质符号保持不变。因此,标量三重积,即一个极矢量和一个赝矢量的标量积,
$$
\begin{align}
\vec{a}\cdot\left(\vec{b}\times\vec{c}\right)
=\vec{b}\cdot\left(\vec{c}\times\vec{a}\right)
=\vec{c}\cdot\left(\vec{a}\times\vec{b}\right)
=\sum_{i,\ j,\ k}\varepsilon_{ijk}a_{i}b_{j}c_{k}
\,,
\end{align}
$$是一个赝标量,因为其性质符号发生了改变。另一类矢量为
$$
\vec{a}\times\left(\vec{b}\times\vec{c}\right)=\left(\vec{a}\cdot\vec{c}\right)\vec{b}-\left(\vec{a}\cdot\vec{b}\right)\vec{c}
\,,
$$它满足雅可比行列式
$$
\vec{a}\times(\vec{b}\times\vec{c})
+\vec{b}\times(\vec{c}\times\vec{a})
+\vec{c}\times(\vec{a}\times\vec{b})
=0
\,.
$$调和级数(Harmonic series)
$$
\sum_{m=1}^{\infty}\frac{1}{m}=1+\frac{1}{2}+\frac{1}{3}+\cdots
\,.
$$列维-奇维塔(Levi-Civita)符号
$$
\epsilon_{ijk}\epsilon_{imn}=\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km}
\,.
$$矩阵乘法
$$
c_{ij}=a_{ik}b_{kj}
\,.
$$行列式
$$
\det A =\epsilon_{i_{1}i_{2}\dots i_{n}}a_{1i_{1}}a_{2i_{2}}\cdots a_{ni_{n}}
\,.
$$泡利矩阵
$$
\begin{align}
\sigma_{1}
&=
\begin{pmatrix}
0 & 1
\\
1 & 0
\end{pmatrix}
\,,\\
\sigma_{2}
&=
\begin{pmatrix}0
& -i
\\
i & 0
\end{pmatrix}
\,,\\
\sigma_{3}
&=
\begin{pmatrix}
1 & 0
\\
0 & -1
\end{pmatrix}
\,.
\end{align}
$$因此,
$$
\begin{align}
\left(\sigma^{i}\right)^{2} &= 1
\,,\\
\left(\sigma^{i}\right)^{T} &= \left(\sigma^{i}\right)^{*}
\,,\\
\sigma_{i} \sigma_{j} &= \delta_{ij} + i \epsilon_{ijk} \sigma_{k}
\,,\\
\mathrm{Tr}\left(\sigma^{i}\right) &= 0
\,,\\
\mathrm{Tr}\left(\sigma^{i}\sigma^{j}\right) &= 2\delta^{ij}
\,,\\
\mathrm{Tr}\left(\sigma^{i}\sigma^{j}\sigma^{k}\right) &= 2 i \epsilon^{ijk}
\,,\\
\exp(i\theta\sigma_{k}) &= \cos\theta + i \sigma_{k}\sin\theta
\,.
\end{align}
$$对于矩阵 $A$ 有
$$
\begin{align}
\exp(A)
&=\sum_{j=0}^{\infty}\frac{1}{j!}A^{j}
\,,\\
\sin(A)
&=\sum_{j=0}^{\infty}\frac{(-1)^{j}}{(2j+1)!}A^{2j+1}
\,,\\
\cos(A)
&=\sum_{j=0}^{\infty}\frac{(-1)^{j}}{(2j)!}A^{2j}
\,,\\
e^{A}
&=\underset{N\to\infty}{\lim}\left(1+\frac{A}{N}\right)^{N}
\,,\\
\det e^{A}
&=e^{\mathrm{Tr}A}
\,.
\end{align}
$$贝克-坎贝尔-豪斯多夫(Baker–Campbell–Hausdorff)公式
$$
\begin{align}
e^{A}Be^{-A}
&=B+\left[A,B\right]+\frac{1}{2!}\left[A,\left[A,B\right]\right]+\frac{1}{3!}\left[A,\left[A,\left[A,B\right]\right]\right]+\cdots
\,,\\
e^{A+B}
&=e^{A}e^{B}e^{-\frac{1}{2}\left[A,B\right]+\cdots}
\,.
\end{align}
$$球坐标中的拉普拉斯
$$
\nabla^{2}V=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial V}{\partial r}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial V}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\left(\frac{\partial^{2}V}{\partial\phi^{2}}\right)
\,.
$$修正贝塞尔函数
$$
\begin{align}
\int_{0}^{\infty}dx\ \frac{xe^{-\beta\sqrt{\gamma^{2}+x^{2}}}}{\sqrt{\gamma^{2}+x^{2}}}\sin(bx)
&=\frac{\gamma b}{\sqrt{\beta^{2}+b^{2}}}K_{1}\left(\gamma\sqrt{\beta^{2}+b^{2}}\right)
\,,\\
\int_{0}^{\infty}dx\ xe^{-\beta\sqrt{\gamma^{2}+x^{2}}}\sin(bx)
&=\frac{b\beta\gamma^{2}}{\beta^{2}+b^{2}}K_{2}\left(\gamma\sqrt{\beta^{2}+b^{2}}\right)
\,.
\end{align}
$$高斯积分
$$
\int_{-\infty}^{\infty}e^{-ax^{2}}=\sqrt{\frac{\pi}{a}}
\,.
$$高斯函数
$$
f(x)=a\cdot\exp\left(-\frac{(x-b)^{2}}{2c^{2}}\right)
\,.
$$具有期望值 $\mu=b$ 和标准差 $\sigma^{2}=c^{2}$ 的正态分布随机变量的概率密度函数为
$$
f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\frac{(x-\mu)^{2}}{\sigma^{2}}\right)
\,.
$$三角函数和差恒等式
$$
\begin{align}
\tan(\alpha+\beta)
&=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}
\,,\\
\tan(\alpha-\beta)
&=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}
\,.
\end{align}
$$欧拉数
$$
\begin{align}
e
&=\underset{n\to\infty}{\lim}\left(1+\frac{1}{n}\right)^{n}=2.71828\dots
\,,\\
e
&=\underset{x\to0}{\lim}\left(1+x\right)^{\frac{1}{x}}
\,,\\
e
&=\sum_{n=0}^{\infty}\frac{1}{n!}
\,,\\
e^{x}
&=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}
\,.
\end{align}
$$$\pi$
$\pi$ 是圆周与直径的比值,也可以通过积分定义为(Karl Weierstrass)
$$
\pi=\int_{-1}^{1}\frac{dx}{\sqrt{1-x^{2}}}
\,.
$$正弦定理
$$
\frac{a}{\sin\angle A} = \frac{b}{\sin\angle B} = \frac{c}{\sin\angle C} = 2 R
$$其中,$R$ 为三角形的外接圆半径。
余弦定理
$$
\begin{align}
\vec{c}
&=\vec{a}-\vec{b}
\\
c^{2}
&=a^{2}+b^{2}-2ab\cos\angle(a,b)
\,.
\end{align}
$$导数极值
若一个函数 $f(x)$ 满足 $f^{\prime}(x_{0})=f^{\prime}(x_{0})=f^{\prime\prime}(x_{0})=\cdots=f^{(n)}(x_{0})=0$ 和 $f^{(n+1)}(x_{0})\neq0\ (n\in\mathbb{Z}, n\ge3)$,则其在 $x=x_{0}$ 处为极值;并且
- 若 $n$ 为奇数且 $f^{(n+1)}(x_{0})<0$,则 $f(x_{0})$ 为极大值;
- 若 $n$ 为奇数且 $f^{(n+1)}(x_{0})>0$,则 $f(x_{0})$ 为极大值;
- 若 $n$ 为偶数,则 $x_{0}$ 为拐点。
黎曼 $\zeta$ 函数
$$
\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}
\,,
$$其中,$s$ 为复数。它的积分表述为
$$
\begin{align}
\zeta(s)
&=\frac{1}{\Gamma(s)}\int_0^\infty dx \frac{^{s-1}}{e^x - 1}
\\
&=\frac{1}{(1-2^{1-s})\Gamma(s)}\int_0^\infty dx \frac{x^{s-1}
}{e^x + 1}
\,,
\end{align}
$$其中,$\Gamma(s)$ 是欧拉 $\Gamma$ 函数。常遇到的 $\zeta$ 函数值为
$$
\begin{align}
\zeta(2)=\frac{\pi^2}{6}
\,,\\
\zeta(4)=\frac{\pi^4}{90}
\,.
\end{align}
$$欧拉 $\Gamma$ 函数
$$
\Gamma(n)=(n-1)!
$$其中,$n\in\mathbb{Z}^{+}$。它的积分表述为:
$$
\Gamma(z)=\int_0^\infty x^{z-1}e^{-t}dt
\,,\quad
\mathrm{Re}(z)>0
\,.
$$狄拉克 $\delta$ 函数
$$
\int_{-\infty}^{\infty}\delta(x)dx=1
\,,
$$其中,
$$
\delta(x)=\begin{cases}
+\infty & x=0\\
0 & x\ne0
\end{cases}
\,.
$$高斯误差函数(Gauss error function)
$$
\text{Erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt
\,.
$$标量场 $\varphi(\vec{r})$ 的梯度为矢量场:
$$
\begin{align}
\nabla\varphi = \vec{e}_{1} \frac{\partial \varphi}{\partial x_{1}} + \vec{e}_{2} \frac{\partial \varphi}{\partial x_{2}} + \vec{e}_{3} \frac{\partial \varphi}{\partial x_{3}}
\,.
\end{align}
$$矢量场 $\vec{a}(\vec{r})$ 的散度为标量场
$$
\text{div}\ \vec{a}(\vec{r})
\equiv
\nabla \cdot \vec{a}(\vec{r})
\equiv
\sum_{j=1}^{3}\frac{\partial a_{j}}{\partial x_{j}}
\,.
$$拉普拉斯算符为梯度场的散度
$$
\text{div grad}\ \varphi
= \sum_{j=1}^{3}\frac{\partial^{2}\varphi}{\partial x_{j}^{2}}
\equiv\Delta\varphi
\,.
$$矢量场的旋度为
$$
\begin{align}
\text{curl}\ \vec{a}
&\equiv \nabla\times \vec{a}
= \sum_{i,\ j,\ k}\varepsilon_{ijk}\frac{\partial a_{j}}{\partial x_{i}} \vec{e}_{k}
\\
&=\vec{e}_{1} \left(\frac{\partial a_{3}}{\partial x_{2}}-\frac{\partial a_{2}}{\partial x_{3}}\right) + \vec{e}_{2} \left(\frac{\partial a_{1}}{\partial x_{3}}-\frac{\partial a_{3}}{\partial x_{1}}\right) + \vec{e}_{3} \left(\frac{\partial a_{2}}{\partial x_{1}}-\frac{\partial a_{1}}{\partial x_{2}}\right)
\,.
\end{align}
$$二项分布(Binomial Distribution)
$$
\begin{align}
P\{X=k\}=
\binom{n}{k} p^{k} (1-p)^{n-k}\,,\quad k=0,1,2,\cdots
\,,
\end{align}
$$其中,$\binom{n}{k}=\frac{n!}{k!(n-k)!}$。
泊松分布
$$
\begin{align}
P\{X=k\}=\frac{\lambda^k e^{-\lambda}}{k!}\,,\quad k=0,1,2,\cdots
\,,
\end{align}
$$其中,$\lambda>0$。相应的泊松极限定理为
$$
\begin{align}
\lim_{n\to\infty}
\binom{n}{k} p_{n}^{k} (1-p_{n})^{n-k}=\frac{\lambda^k e^{-\lambda}}{k!}
\,.
\end{align}
$$欧拉-马歇罗尼常数(Euler-Mascheroni constant):调和级数与自然对数差值的极限
$$
\begin{align}
\gamma
=\lim_{n\to\infty}\left[ \sum_{k=1}^{n}\frac{1}{k}-\ln n \right]
=\int_{1}^{\infty}\left(\frac{1}{\lfloor x\rfloor}-\frac{1}{x}\right)\ dx
=0.5772\cdots
\,,
\end{align}
$$其中,$\lfloor~\rfloor$ 为下取整函数。