公式

三角公式

三角函数

$$\begin{aligned} \sin x &= \frac{e^{ix} - e^{-ix}}{2i} \\ \cos x &= \frac{e^{ix} + e^{-ix}}{2} \end{aligned}$$

加减

$$\begin{aligned} \sin(x+y) &= \sin x \cos y + \cos x \sin y \\ \sin(x-y) &= \sin x \cos y - \cos x \sin y \\ \cos(x+y) &= \cos x \cos y - \sin x \sin y \\ \cos(x-y) &= \cos x \cos y + \sin x \sin y \end{aligned}$$

和差化积

$$\begin{aligned} \sin x + \sin y &= 2 \sin \frac{x+y}{2} \cos \frac{x-y}{2} \\ \sin x - \sin y &= 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2} \\ \cos x + \cos y &= 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2} \\ \cos x - \cos y &= -2 \sin \frac{x+y}{2} \sin \frac{x-y}{2} \end{aligned}$$

积化和差

$$\begin{aligned} \sin x \sin y &= \frac{1}{2} [\cos (x-y) - \cos (x+y)] \\ \cos x \cos y &= \frac{1}{2} [\cos (x+y) + \cos (x-y)] \\ \sin x \cos y &= \frac{1}{2} [\sin (x+y) + \sin (x-y)] \\ \cos x \sin y &= \frac{1}{2} [\sin (x+y) - \sin (x-y)] \end{aligned}$$

万能公式

$$\begin{aligned} \sin x &= \frac{2 \tan \frac{x}{2}}{1 + \tan^2 \frac{x}{2}} \\ \cos x &= \frac{1 - \tan^2 \frac{x}{2}}{1 + \tan^2 \frac{x}{2}} \\ \tan x &= \frac{2 \tan \frac{x}{2}}{1 - \tan^2 \frac{x}{2}} \end{aligned}$$

算符

微分算符 $\nabla$

$$\nabla = \frac{\partial}{\partial x}i + \frac{\partial}{\partial y}j + \frac{\partial}{\partial z}k$$

梯度

$$\nabla f = \frac{\partial f}{\partial x}i + \frac{\partial f}{\partial y}j + \frac{\partial f}{\partial z}k$$

散度

$$\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$

旋度

$$\nabla \times \vec{F} = \begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}$$

拉普拉斯算符 $\Delta$

$$\Delta = \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$$

Kronecker delta

$$\delta_{ij} = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}$$

Stirling 公式

$$n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$$

$$\ln n! = n \ln n - n + \frac{1}{2} \ln(2\pi n) + O\left(\frac{1}{n}\right)$$

Wallis 公式

$$\frac{\pi}{2} = \lim_{n\to \infty} \left[\frac{(2n)!!}{(2n-1)!!}\right]^2 \frac{1}{2n+1}$$

Euler 公式

$$e^{ix} = \cos x + i \sin x$$

量子力学

Schrodinger 方程

$$i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi$$

$$\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V$$

定态薛定谔方程

$$\hat{H} \psi = E \psi\qquad \Psi = \psi e^{-iEt/\hbar}$$

动量算符

$$\hat{p} = -i\hbar \nabla$$

含时期望值

$$\frac{d<Q>}{dt} = \frac{i}{\hbar} <[Q,H]>+<\frac{\partial\hat Q}{\partial t}>$$

能量期望值

$$<H> = <E>$$

海森堡不确定性关系

$$\sigma_x \sigma_p \geq \frac{\hbar}{2}$$

广义不确定性关系

$$\sigma_A \sigma_B \geq \left|\frac{1}{2i} <[A,B]> \right|$$

积分公式

三角函数

$$\int x \sin(ax) dx = \frac{1}{a^2} \sin(ax) - \frac{x}{a} \cos(ax)$$

$$\int x \cos(ax) dx = \frac{1}{a^2} \cos(ax) + \frac{x}{a} \sin(ax)$$

指数积分

$$\int_0^\infty x^n e^{-x/a} dx = n! a^{n+1}$$

Gauss 积分

$$\int_{-\infty}^\infty e^{-x^2/2\sigma^2} dx = \sqrt{2\pi\sigma^2}$$

$$\int_0^{\infty} x^{2n}e^{-x^2/a^2} dx = \sqrt{\pi} \frac{(2n)!}{n!} \left(\frac{a}{2}\right)^{2n+1}$$

$$\int_0^{\infty} x^{2n+1}e^{-x^2/a^2} dx = \frac{n!}{2} a^{2n+2}$$

分部积分

$$\int u dv = uv - \int v du$$

Taylor 展开

$$f(x) = \sum_{n=0}^{\infty} c_n (x-a)^n$$

$$c_n = \frac{f^{(n)}(a)}{n!}$$

Laurent 展开

$$f(z) = \sum_{n=-\infty}^{\infty} c_n (z-z_0)^n$$

$$c_n = \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z-z_0)^{n+1}} dz$$

Fourier 级数

$$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i\frac{2\pi n}{T}x}$$

$$c_n = \langle f(x), e^{i\frac{2\pi n}{T}x} \rangle = \frac{1}{T} \int_{0}^{T} f(x) e^{-i\frac{2\pi n}{T}x} dx$$

Fourier 变换

$$F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} dx$$

Fourier 逆变换

$$f(x) = \int_{-\infty}^{\infty} F(k) e^{ikx} dk$$

Laplace 变换

Laplace 变换

$F(s) = L[f(t)]$

$$F(s) = \int_0^\infty f(t) e^{-st} dt$$

Laplace 逆变换

$f(t) = L^{-1}[F(s)]$

$$f(t) = \frac{1}{2\pi i} \int_{\gamma-i\infty}^{\gamma+i\infty} F(s) e^{st} ds$$

Gauss 公式

$$\int_V \nabla \cdot \vec{F} dV = \oint_S \vec{F} \cdot d\vec{S}$$

Stokes 公式

$$\oint_C \vec{F} \cdot d\vec{l} = \int_S (\nabla \times \vec{F}) \cdot d\vec{S}$$

Green 公式

$$\oint_C \vec{F} \cdot d\vec{l} = \int_S (\nabla \times \vec{F}) \cdot d\vec{S}$$

Cauchy 公式

$$f(z_0)=\frac{1}{2\pi i} \oint_C \frac{f(z)}{z-z_0} dz$$

不等式

均值不等式

算术平均数

$$\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}$$

几何平均数

$$\sqrt[n]{a_1 a_2 \cdots a_n} \leq \frac{a_1 + a_2 + \cdots + a_n}{n}$$

调和平均数

$$\frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}} \leq \frac{a_1 + a_2 + \cdots + a_n}{n}$$

Cauchy-Schwarz 不等式

$$\langle f,g \rangle^2 \leq \langle f,f \rangle \langle g,g \rangle$$

Minkowski 不等式(三角不等式)

$$\|f + g\|_p \leq \|f\|_p + \|g\|_p$$

Holder 不等式

$p,q,r\in [1,\infty)$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$

$$\|fg\|_r \leq \|f\|_p \|g\|_q$$

Schur 不等式

For $x,y,z\geq 0\in \mathbb{R}$ and $t>0$:

$$x^t(x-y)(x-z)+y^t(y-x)(y-z)+z^t(z-x)(z-y)\geq 0$$

Gronwall 不等式

$\beta,u$ are continuous real functions on $[a,\infty)$ or $[a,b]$ or $[a,b)$, if $\beta(t)\geq 0$ and $u(t)$ satisfies:

$$\frac{d}{dt} u(t) \leq \beta(t) u(t)$$

then:

$$u(t) \leq u(a) e^{\int_a^t \beta(s) ds}$$

能量模不等式

$S(x)$ is cubic spline function (polynomial of degree 3), $\phi(x)\in C^2[a,b]$ and they satisfy the same interpolation conditions, then:

$$\int_a^b |\phi''(x)|^2 dx \geq \int _a^b | S''(x)|^2 dx$$

重要函数

Gamma 函数

$$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} dt$$

Beta 函数

$$B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt$$

Riemann Zeta 函数

$$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}=\frac{1}{\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x-1} dx \qquad(\mathrm{Re}(s)>1)$$

Bessel 贝塞尔函数

$$J_n(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m!(n+m)!} \left(\frac{x}{2}\right)^{n+2m}$$

Legendre 勒让德多项式

$$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2-1)^n$$

Hermite 厄米多项式

$$H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}$$

Laguerre 拉盖尔多项式

$$L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^n e^{-x})$$

Chebyshev 切比雪夫多项式

$$T_n(x) = \cos(n \arccos x)$$

Bernoulli 多项式

Is a series of $B_n(x)$ that satisfies:

$$\begin{cases} B_0(x) = 1, \\ B_n'(x) = n B_{n-1}(x) \quad \int_0^1 B_n(x) dx = 0,\quad x\in \mathbb{N} \end{cases}$$

Bernoulli 数:

$$b_n = n! B_n(0) ,\quad n= 2,3,\ldots$$