一些随机的LaTeX\LaTeX公式，不定期更新。

f(z0)=12πi∮Cf(z)z−z0dzf\left(z_0\right)=\frac{1}{2\pi{}i}\oint\limits_C\frac{f(z)}{z-z_0}\mathrm{d}z

f(n)(z0)=n!2πi∮Cf(z)(z−z0)n+1dz(n=1,2,⋯ )f^{(n)}\left(z_0\right)=\frac{n!}{2\pi{}i}\oint\limits_C\frac{f(z)}{\left(z-z_0\right)^{n+1}}\mathrm{d}z\quad\left(n=1,2,\cdots\right)

∮Cdzz−z0=2πi\oint\limits_C\frac{\mathrm{d}z}{z-z_0}=2\pi{}i

∮Cdz(z−z0)n+1=0(n=1,2,⋯ )\oint\limits_C\frac{\mathrm{d}z}{\left(z-z_0\right)^{n+1}}=0\quad\left(n=1,2,\cdots\right)

Res[f(z),z0]=lim⁡z→z0(z−z0)f(z)\mathrm{Res}\left[f(z),z_0\right]=\lim_{z\to{}z_0}(z-z_0)f(z)

Res[f(z),z0]=1(m−1)!lim⁡z→z0dm−1dzm−1{(z−z0)mf(z)}\mathrm{Res}\left[f(z),z_0\right]=\frac{1}{\left(m-1\right)!}\lim_{z\to{}z_0}\frac{\mathrm{d}^{m-1}}{\mathrm{d}z^{m-1}}\left\{(z-z_0)^mf(z)\right\}

Res[f(z),z0]=P(z0)Q′(z0)\mathrm{Res}\left[f(z),z_0\right]=\frac{P(z_0)}{Q'(z_0)}

∫∞∞R(x)eaixdx=∫∞∞R(x)cos⁡axdx+i∫∞∞R(x)sin⁡axdx\int_\infty^\infty{}R(x)e^{aix}\mathrm{d}x=\int_\infty^\infty{}R(x)\cos{}ax\mathrm{d}x+i\int_\infty^\infty{}R(x)\sin{}ax\mathrm{d}x

TT∗=1−k−1k+1λ2≜τ(λ)\frac{T}{T^*}=1-\frac{k-1}{k+1}\lambda^2\triangleq\tau(\lambda)

pp∗=(TT∗)kk−1=(1−k−1k+1λ2)kk−1≜π(λ)\frac{p}{p^*}=\left(\frac{T}{T^*}\right)^{\frac{k}{k-1}}=\left(1-\frac{k-1}{k+1}\lambda^2\right)^{\frac{k}{k-1}}\triangleq\pi(\lambda)

ρρ∗=(TT∗)1k−1=(1−k−1k+1λ2)1k−1≜ε(λ)\frac{\rho}{\rho^*}=\left(\frac{T}{T^*}\right)^{\frac{1}{k-1}}=\left(1-\frac{k-1}{k+1}\lambda^2\right)^{\frac{1}{k-1}}\triangleq\varepsilon(\lambda)

TT∗=11+k−12Ma2≜τ(Ma)\frac{T}{T^*}=\frac{1}{1+\dfrac{k-1}{2}{Ma}^2}\triangleq\tau(Ma)

pp∗=(TT∗)kk−1=(1+k−12Ma2)−kk−1≜π(Ma)\frac{p}{p^*}=\left(\frac{T}{T^*}\right)^{\frac{k}{k-1}}=\left(1+\frac{k-1}{2}{Ma}^2\right)^{-\frac{k}{k-1}}\triangleq\pi(Ma)

ρρ∗=(TT∗)1k−1=(1+k−12Ma2)−1k−1≜ε(Ma)\frac{\rho}{\rho^*}=\left(\frac{T}{T^*}\right)^{\frac{1}{k-1}}=\left(1+\frac{k-1}{2}{Ma}^2\right)^{-\frac{1}{k-1}}\triangleq\varepsilon(Ma)

m˙=Kp∗AT∗q(λ)=KpAT∗y(λ)\dot{m}=K\frac{p^*A}{\sqrt{T^*}}q(\lambda)=K\frac{pA}{\sqrt{T^*}}y(\lambda)

m˙V+pA=k+12km˙ccrz(λ)=(2k+1)1k−1p∗Aq(λ)z(λ)=p∗Af(λ)=pπ(λ)Af(λ)=pAr(λ)\begin{align*} \quad\dot{m}V+pA&=\frac{k+1}{2k}\dot{m}c_{cr}z(\lambda)\\ &=\left(\frac{2}{k+1}\right)^{\frac{1}{k-1}}p^*Aq(\lambda)z(\lambda)\\ &=p^*Af(\lambda)\\ &=\frac{p}{\pi(\lambda)}Af(\lambda)\\ &=\frac{pA}{r(\lambda)} \end{align*}

dpp=kMa21−Ma2⋅dAA\frac{\mathrm{d}p}{p}=\frac{k{Ma}^2}{1-{Ma}^2}\cdot\frac{\mathrm{d}A}{A}

dρρ=Ma21−Ma2⋅dAA\frac{\mathrm{d}\rho}{\rho}=\frac{{Ma}^2}{1-{Ma}^2}\cdot\frac{\mathrm{d}A}{A}

dTT=(k−1)Ma21−Ma2⋅dAA\frac{\mathrm{d}T}{T}=\frac{\left(k-1\right){Ma}^2}{1-{Ma}^2}\cdot\frac{\mathrm{d}A}{A}

dVV=−(11−Ma2)dAA\frac{\mathrm{d}V}{V}=-\left(\frac{1}{1-{Ma}^2}\right)\frac{\mathrm{d}A}{A}

dMaMa=−1+k−12Ma2Ma2⋅dAA\frac{\mathrm{d}Ma}{Ma}=-\frac{1+\dfrac{k-1}{2}{Ma}^2}{{Ma}^2}\cdot\frac{\mathrm{d}A}{A}

σr0=σt0=3+μ8ρω2ra2+σa\sigma_{r0}=\sigma_{t0}=\frac{3+\mu}{8}\rho\omega^2r_a^2+\sigma_a

rd(hσr)dr+σr−σt+ρω2r2=0r\frac{\mathrm{d}(h\sigma_r)}{\mathrm{d}r}+\sigma_r-\sigma_t+\rho\omega^2r^2=0

y0=C1sin⁡kx+C2cos⁡kx+C3sinh⁡kx+C4cosh⁡kxy_0=C_1\sin{}kx+C_2\cos{}kx+C_3\sinh{}kx+C_4\cosh{}kx

ωi=(kil)2l2EIρA\omega_i=\frac{\left(k_il\right)^2}{l^2}\sqrt{\frac{EI}{\rho{}A} }

打公式的时候遇到连续的{或}要用空格分开。

{QMαy}i+1=(1000l100l22EIlEI10l36EIl22EIl1)i{QMαy}i\begin{Bmatrix} Q\\ M\\ \alpha\\ y \end{Bmatrix}_{i+1} = \begin{pmatrix} 1 & 0 & 0 & 0\\ l & 1 & 0 & 0\\ \dfrac{l^2}{2EI} & \dfrac{l}{EI} & 1 & 0\\ \dfrac{l^3}{6EI} & \dfrac{l^2}{2EI} & l & 1 \end{pmatrix}_i \begin{Bmatrix} Q\\ M\\ \alpha\\ y \end{Bmatrix}_i