Trigonometric Functions
$\sin{z}={\dfrac{\e^{iz}-\e^{-iz}}{2i}}, \quad \cos{z}={\dfrac{\e^{iz}+\e^{-iz}}{2}}, \quad \tan{z}=-i\cdot{\dfrac {\e^{iz}-\e^{-iz}}{\e^{iz}+\e^{-iz}}};$
| $\theta$ | $\sin(\theta)$ | $\cos(\theta)$ | $\tan(\theta)$ | $\csc(\theta)$ | $\sec(\theta)$ | $\cot(\theta)$ |
| $\dfrac{\pi}{6} = 30^\circ$ | $\dfrac{1}{2}$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{\sqrt{3}}{3}$ | $2$ | $\dfrac{2\sqrt{3}}{3}$ | $\sqrt{3}$ |
| $\dfrac{\pi}{3} = 60^\circ$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{1}{2}$ | $\sqrt{3}$ | $\dfrac{2\sqrt{3}}{3}$ | $2$ | $\dfrac{\sqrt{3}}{3}$ |
| $\dfrac{\pi}{4} = 45^\circ$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $1$ | $\sqrt{2}$ | $\sqrt{2}$ | $1$ |
| $\dfrac{\pi}{12} = 15^\circ$ | $\dfrac{\sqrt{6}-\sqrt{2}}{4}$ | $\dfrac{\sqrt{6}+\sqrt{2}}{4}$ | $2-\sqrt{3}$ | $\sqrt{6}+\sqrt{2}$ | $\sqrt{6}-\sqrt{2}$ | $2+\sqrt{3}$ |
| $\dfrac{\pi}{8} = 22.5^\circ$ | $\dfrac{\sqrt{2-\sqrt{2}}}{2}$ | $\dfrac{\sqrt{2+\sqrt{2}}}{2}$ | $\sqrt{2}-1$ | $\sqrt{4+2\sqrt{2}}$ | $\sqrt{4-2\sqrt{2}}$ | $\sqrt{2}+1$ |
| $\dfrac{\pi}{10} = 18^\circ$ | $\dfrac{\sqrt{5}-1}{4}$ | $\dfrac{\sqrt{10+2\sqrt{5}}}{4}$ | $\sqrt{\dfrac{5-2\sqrt{5}}{5}}$ | $\sqrt{5}+1$ | $\sqrt{\dfrac{10-2\sqrt{5}}{5}}$ | $\sqrt{5+2\sqrt{5}}$ |
| $\dfrac{\pi}{5} = 36^\circ$ | $\dfrac{\sqrt{10-2\sqrt{5}}}{4}$ | $\dfrac{\sqrt{5}+1}{4}$ | $\sqrt{5-2\sqrt{5}}$ | $\sqrt{\dfrac{10+2\sqrt{5}}{5}}$ | $\sqrt{5}-1$ | $\sqrt{\dfrac{5+2\sqrt{5}}{5}}$ |
$\sin(-x)=-\sin{x}, \quad\cos(-x)=\cos{x}, \quad\tan(-x)=-\tan{x};$
$\sin(x\pm\pi)=-\sin{x}, \quad\cos(x\pm\pi)=-\cos{x}, \quad\tan(x\pm\pi)=\tan{x};$ $\sin(\pi-x)=\sin{x}, \quad\cos(\pi-x)=-\cos{x}, \quad\tan(\pi-x)=-\tan{x};$
$\sin\left(x\pm\dfrac{\pi}{2}\right)=\pm\cos{x}, \quad \cos\left(x\pm\dfrac{\pi}{2}\right)=\mp\sin{x}, \quad \tan\left(x\pm\dfrac{\pi}{2}\right)=-\cot{x};$ $\sin\left(\dfrac{\pi}{2}-x\right)=\cos{x}, \quad \cos\left(\dfrac{\pi}{2}-x\right)=\sin{x}, \quad \tan\left(\dfrac{\pi}{2}-x\right)=\cot{x};$
$\sin\left(x\pm\dfrac{\pi}{4}\right)=\dfrac{\sqrt{2}}{2}(\sin{x}\pm\cos{x}), \quad \cos\left(x\pm\dfrac{\pi}{4}\right)=\dfrac{\sqrt{2}}{2}(\cos{x}\mp\sin{x}), \quad \tan\left(x\pm\dfrac{\pi}{4}\right)=\dfrac{\sin{x}\pm\cos{x}}{\cos{x}\mp\sin{x}}=\dfrac{\tan{x}\pm1}{1\mp\tan{x}};$
$\sin\left(\dfrac{\pi}{4}-x\right)=\dfrac{\sqrt{2}}{2}\left(\cos x-\sin x\right), \quad \cos\left(\dfrac{\pi}{4}-x\right)=\dfrac{\sqrt{2}}{2}\left(\cos x+\sin x\right), \quad \tan\left(\dfrac{\pi}{4}-x\right)=\dfrac{1-\tan x}{1+\tan x};$
$\sin^2{x}+\cos^2{x}=1, \quad \sec^2{x}-\tan^2{x}=1, \quad \csc^2{x}-\cot^2{x}=1;$
$\sin\left(x\pm{y}\right)=\sin{x}\cos{y}\pm\cos{x}\sin{y}, \quad \cos\left(x\pm{y}\right)=\cos{x}\cos{y}\mp\sin{x}\sin{y}, \quad \tan\left(x\pm{y}\right)=\dfrac{\tan{x}\pm\tan{y}}{1\mp\tan{x}\tan{y}}, \quad \cot\left(x\pm y\right)=\dfrac{\cot x\cot y\mp 1}{\cot y\pm\cot x};$
$\sin{2x}=2\sin{x}\cos{x}, \quad \cos{2x}=\begin{cases}\cos^2{x}-\sin^2{x}\\2\cos^2{x}-1\\1-2\sin^2{x}\end{cases}, \quad \tan{2x}=\dfrac{2\tan{x}}{1-\tan^2{x}}, \quad \cot2x=\dfrac{\cot^{2}x-1}{2\cot x};$
$\sin{3x}=\begin{cases}3\sin{x}-4\sin^3{x}\\3\sin{x}\cos^2x-\sin^3x\\4\sin{x}\cos^2{x}-\sin{x}\end{cases}, \quad \cos{3x}=\begin{cases}4\cos^3{x}-3\cos{x}\\\cos^3x-3\cos{x}\sin^2x\\\cos{x}-4\cos{x}\sin^2{x}\end{cases}, \quad \tan{3x}=\dfrac{3\tan{x}-\tan^3{x}}{1-3\tan^2{x}};$
$ \sin(nx)=\displaystyle\sum_{k=0\atop 2k+1\le n}{\binom{n}{2k+1}(-1)^k\sin^{2k+1}x\cos^{n-(2k+1)}x}, \cos(nx)=\displaystyle\sum_{k=0\atop 2k\le n}{\binom{n}{2k}(-1)^k\sin^{2k}x\cos^{n-2k}x}, \tan(nx)=\dfrac{\sum_{k=0\atop 2k+1\le n}{\binom{n}{2k+1}(-1)^k\tan^{2k+1}x}}{\sum_{k=0\atop 2k\le n}{\binom{n}{2k}(-1)^k\tan^{2k}x}}; $
$\cos(n\theta)+i\sin(n\theta)=(\cos\theta+i\sin\theta)^n$, $\begin{bmatrix}\cos(n\theta)\\\sin(n\theta)\end{bmatrix}=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}^n\begin{bmatrix}1\\0\end{bmatrix}$;
$\sin^2\dfrac{x}{2}=\dfrac{1-\cos{x}}{2}, \quad \cos^2\dfrac{x}{2}=\dfrac{1+\cos{x}}{2}, \quad \tan^2\dfrac{x}{2}=\dfrac{1-\cos{x}}{1+\cos{x}}; \qquad \sin^2x=\dfrac{1-\cos2x}{2}, \quad \cos^2x=\dfrac{1+\cos2x}{2}; \quad \tan^2x=\dfrac{1-\cos2x}{1+\cos2x};$
$\sin{x}\cdot\cos{y}=\dfrac{\sin(x+y)+\sin(x-y)}{2}; \quad \sin{x}\cdot\sin{y}=\dfrac{\cos(x-y)-\cos(x+y)}{2}, \quad \cos{x}\cdot\cos{y}=\dfrac{\cos(x+y)+\cos(x-y)}{2};$
$\sin{x}\pm\sin{y}=2\sin\left(\dfrac{x\pm{y}}{2}\right)\cos\left(\dfrac{x\mp{y}}{2}\right); \quad \cos{x}+\cos{y}=2\cos\left(\dfrac{x+y}{2}\right)\cos\left(\dfrac{x-y}{2}\right), \quad \cos{x}-\cos{y}=-2\sin\left(\dfrac{x+y}{2}\right)\sin\left(\dfrac{x-y}{2}\right);$
$\sin{x}+\cos{y}=2\sin\left(\dfrac{x+y+\frac{\pi}{2}}{2}\right)\sin\left(\dfrac{x-y+\frac{\pi}{2}}{2}\right), \quad \sin{x}-\cos{y}=-2\cos\left(\dfrac{x+y+\frac{\pi}{2}}{2}\right)\cos\left(\dfrac{x-y+\frac{\pi}{2}}{2}\right); $
Inverse Trigonometric Functions
$\sin^{-1}(z)=-i\cdot\ln\left(iz+{\sqrt{1-z^{2}}}\right), \quad \cos^{-1}(z)=-i\cdot\ln\left(z\pm{\sqrt{z^{2}-1}}\right), \quad \tan^{-1}(z)={\dfrac{1}{2i}}\ln\left({\dfrac{1+iz}{1-iz}}\right), \quad \mathrm{atan2}(y,x)={\dfrac{1}{2i}}\ln\left({\dfrac{x+yi}{x-yi}}\right);$
$\sin^{-1}(-x)=-\sin^{-1}(x),\quad \tan^{-1}(-x)=-\tan^{-1}(x),\quad \csc^{-1}(-x)=-\csc^{-1}(x),\quad \operatorname{atan2}(y,x)=-\operatorname{atan2}(-y,x) \ne-\operatorname{atan2}(y,-x)=\operatorname{atan2}(-y,-x);$
$\csc^{-1}{x}=\sin^{-1}\left(\dfrac{1}{x}\right), \quad \sec^{-1}{x}=\cos^{-1}\left(\dfrac{1}{x}\right); \quad \tan^{-1}{x}=\cot^{-1}\left(\dfrac{1}{x}\right)\ (x\gt0); \qquad \sin^{-1}{x}+\cos^{-1}{x}=\dfrac{\pi}{2}, \quad \tan^{-1}{x}+\cot^{-1}{x}=\dfrac{\pi}{2}, \quad \sec^{-1}{x}+\csc^{-1}{x}=\dfrac{\pi}{2};$
$\operatorname{atan2}(y,x)=\begin{cases} \tan^{-1}(\frac{y}{x})&x>0\\ \tan^{-1}(\frac{y}{x})+\pi &x\lt0,\ y\ge0\\ \tan^{-1}(\frac{y}{x})-\pi &x\lt0,\ y\lt0\\ \frac{\pi}{2}&x=0,\ y\gt0\\ -\frac{\pi}{2}&x=0,\ y\lt0 \end{cases} = \begin{cases} \cot^{-1}(\frac{x}{y})&y>0\\\cot^{-1}(\frac{x}{y})-\pi &y\lt0\\ 0&y=0,\ x\gt0\\ \pi&y=0,\ x\lt0\end{cases} =2\tan^{-1}\left(\dfrac{y}{\sqrt{x^2+y^2}+x}\right) =2\tan^{-1}\left(\dfrac{\sqrt{x^2+y^2}-x}{y}\right);$
$a\sin(kx)+b\cos(kx) =\sqrt{a^{2}+b^{2}}\cos\left(kx-\operatorname{atan2}(a,b)\right) =\sqrt{a^{2}+b^{2}}\sin\left(kx+\operatorname{atan2}(b,a)\right);$
$\sin^{-1}(x)=\cos^{-1}\left(\sqrt{1-x^2}\right)=\tan^{-1}\left(\dfrac{x}{\sqrt{1-x^2}}\right),\quad \cos^{-1}(x)=\sin^{-1}\left(\sqrt{1-x^2}\right)=\tan^{-1}\left(\dfrac{\sqrt{1-x^2}}{x}\right);\quad (0\lt x\lt1)$
$\sin(\cos^{-1}{x})=\sqrt{1-x^2}, \quad \sin(\tan^{-1}{x})=\dfrac{x}{\sqrt{1+x^2}}, \quad \cos(\sin^{-1}{x})=\sqrt{1-x^2}, \quad \cos(\tan^{-1}{x})=\dfrac{1}{\sqrt{1+x^2}}, \quad \tan(\cos^{-1}{x})=\dfrac{\sqrt{1-x^2}}{x}, \quad \tan(\sin^{-1}{x})=\dfrac{x}{\sqrt{1-x^2}};$
$\cos(\mathrm{atan2}(y,x))=\dfrac{x}{\sqrt{x^2+y^2}}, \quad \sin(\mathrm{atan2}(y,x))=\dfrac{y}{\sqrt{x^2+y^2}}; \qquad \tan^{-1}(x)\pm\tan^{-1}(y)=\mathrm{atan2}(x\pm y, 1\mp xy);$
$\tan(\operatorname{atan2}(y,x))=\dfrac{y}{x},\quad \cot(\operatorname{atan2}(y,x))=\dfrac{x}{y};\qquad \operatorname{atan2}(\sin{x},\cos{x})=x;\ \ (-\pi\lt x\lt\pi)$
$\sin^{-1}(\sin(x))=\begin{cases}x,&|x|\lt \frac{\pi}{2}\\\sgn(x)\pi-x,&\frac{\pi}{2}\lt|x|\lt \frac{3\pi}{2}\end{cases}, \quad \cos^{-1}(\sin(x))=\begin{cases}\frac{\pi}{2}-x,&|x|\lt \frac{\pi}{2}\\x-\sgn(x)\pi+\frac{\pi}{2},&\frac{\pi}{2}\lt|x|\lt\frac{3\pi}{2}\end{cases}, \\ \sin^{-1}(\cos(x))=\begin{cases}\frac{\pi}{2}-|x|,&|x|\lt\pi\\|x|-\frac{3\pi}{2},&\pi\lt|x|\lt 2\pi\end{cases}, \quad \cos^{-1}(\cos(x))=\begin{cases}|x|,&|x|\lt\pi\\2\pi-|x|,&\pi\lt|x|\lt 2\pi\end{cases};$
$\tan^{-1}(\tan(x))=x-\pi\operatorname{round}\left(\dfrac{x}{\pi}\right), \quad \tan^{-1}(\cot(x))=-x+\pi\left(\dfrac{1}{2}+\operatorname{floor}\left(\dfrac{x}{\pi}\right)\right), \\ \cot^{-1}(\cot(x))=x-\pi\operatorname{floor}\left(\dfrac{x}{\pi}\right), \quad \cot^{-1}(\tan(x))=-x+\pi\left(\dfrac{1}{2}+\operatorname{round}\left(\dfrac{x}{\pi}\right)\right);$
$\sin(2\sin^{-1}x)=2x\sqrt{1-x^2},\quad
\sin(2\cos^{-1}x)=2x\sqrt{1-x^2};\quad
\cos(2\sin^{-1}x)=1-2x^2,\quad
\cos(2\cos^{-1}x)=2x^2-1;\\
\sin(2\tan^{-1}x)=\dfrac{2x}{x^2+1},\quad
\cos(2\tan^{-1}x)=\dfrac{1-x^2}{x^2+1},\quad
\tan(2\tan^{-1}x)=\dfrac{2x}{1-x^2};\\$
$\sin(2\,\mathrm{atan2}(y,x))=\dfrac{2xy}{x^2+y^2}, \quad
\cos(2\,\mathrm{atan2}(y,x))=\dfrac{x^2-y^2}{x^2+y^2}, \quad
\tan(2\,\mathrm{atan2}(y,x))=\dfrac{2xy}{x^2-y^2};$
$\sin(3\sin^{-1}x)=3x-4x^3,\quad \cos(3\cos^{-1}x)=4x^3-3x;\quad \sin(3\cos^{-1}x)=(4x^2-1)\sqrt{1-x^2},\quad \cos(3\sin^{-1}x)=(1-4x^2)\sqrt{1-x^2};\\ \sin(3\tan^{-1}x)=\dfrac{3x-x^3}{(x^2+1)^{3/2}},\quad \cos(3\tan^{-1}x)=\dfrac{1-3x^2}{(x^2+1)^{3/2}},\quad \tan(3\tan^{-1}x)=\dfrac{x^3-3x}{3x^2-1};\\$
$\sin\left(\dfrac{1}{2}\sin^{-1}(x)\right)=\dfrac{1}{2}\left(\sqrt{1+x}-\sqrt{1-x}\right)=\dfrac{1}{\sqrt{2}}\cdot\dfrac{x}{\sqrt{\sqrt{1-x^2}+1}},\quad
\cos\left(\dfrac{1}{2}\sin^{-1}(x)\right)=\dfrac{1}{2}\left(\sqrt{1+x}+\sqrt{1-x}\right)=\dfrac{1}{\sqrt{2}}\cdot\sqrt{\sqrt{1-x^2}+1},
\\
\tan\left(\dfrac{1}{2}\sin^{-1}(x)\right)=\dfrac{x}{\sqrt{1-x^2}+1};\quad
\sin\left(\dfrac{1}{2}\cos^{-1}(x)\right)=\dfrac{\sqrt{1-x}}{\sqrt{2}},\quad
\cos\left(\dfrac{1}{2}\cos^{-1}(x)\right)=\dfrac{\sqrt{1+x}}{\sqrt{2}},\quad
\tan\left(\dfrac{1}{2}\cos^{-1}(x)\right)=\dfrac{\sqrt{1-x}}{\sqrt{1+x}};\\
\sin\left(\dfrac{1}{2}\tan^{-1}(x)\right)=\dfrac{1}{\sqrt{2}}\cdot\dfrac{x}{\sqrt{(1+x^2)+\sqrt{1+x^2}}},\quad
\cos\left(\dfrac{1}{2}\tan^{-1}(x)\right)=\dfrac{1}{\sqrt{2}}\cdot\sqrt{\dfrac{1}{\sqrt{1+x^2}}+1},\quad
\tan\left(\dfrac{1}{2}\tan^{-1}(x)\right)=\dfrac{\sqrt{x^2+1}-1}{x}=\dfrac{x}{\sqrt{x^2+1}+1};\\
\sin\left(\dfrac{1}{2}\mathrm{atan2}(y,x)\right)=\dfrac{1}{\sqrt{2}}\cdot\dfrac{y}{\sqrt{(x^2+y^2)+x\sqrt{x^2+y^2}}},
\quad
\cos\left(\dfrac{1}{2}\mathrm{atan2}(y,x)\right)=\dfrac{1}{\sqrt{2}}\cdot\sqrt{\dfrac{x}{\sqrt{x^2+y^2}}+1};$
Trigonometric Equations
Solutions of $a\sin(x)=b$: $x=\arcsin\left(\dfrac{b}{a}\right)+2\pi n$ and $x=\pi-\arcsin\left(\dfrac{b}{a}\right)+2\pi n$;
Solutions of $a\cos(x)=b$: $x=\pm\arccos\left(\dfrac{b}{a}\right)+2\pi n$;
Solutions of $a\tan(x)=b$: $x=\arctan\left(\dfrac{b}{a}\right)+\pi n$; Solutions of $a\cos(x)+b\sin(x)=0$: $x=\arctan\left(-\dfrac{a}{b}\right)+\pi n$;
Solutions of $a\cos(x)+b\sin(x)+c=0$: $x=2\arctan\left(\dfrac{b\pm\sqrt{a^2+b^2-c^2}}{a-c}\right)+2\pi n$;
Power, Exponential, and Logarithm
For real numbers:
$a^x=\e^{x\ln{a}}$;
$a^0=1$,
$a^{-x}=\dfrac{1}{a^x}$,
$a^\frac{x}{y}=\sqrt[y]{a^x}$;
$a^x a^y=a^{x+y}$,
$\dfrac{a^x}{a^y}=a^{x-y}$,
$(a^x)^y=a^{xy}$,
$(ab)^x=a^x b^x$,
$\left(\dfrac{a}{b}\right)^x=\dfrac{a^x}{b^x}$;
$\sqrt{x}=x^{1/2}$; $\sqrt{x}\sqrt{y}=\sqrt{xy}$, $\sqrt{\dfrac{1}{x}}=\dfrac{1}{\sqrt{x}}$, $\sqrt{x^2}=|x|$;
$a^{\log_a{b}}=b$,
$\log_a{b}=\dfrac{\log_c{b}}{\log_c{a}}=\dfrac{\ln{b}}{\ln{a}}$;
$\log_a{1}=0$,
$\log_a{a}=1$;
$\log_a{b}+\log_a{c}=\log_a{bc}$,
$\log_a{b}-\log_a{c}=\log_a\left(\dfrac{b}{c}\right)$,
$-\log_b{x}=\log_b\left(\dfrac{1}{x}\right)$;
$\log_a{b^k}=k\cdot\log_a{b}$,
$\log_{a^k}{b}=\dfrac{1}{k}\log_a{b}$,
$\log_a{b}\cdot\log_b{c}=\log_a{c}$;
$a+\log_b{x}=\log_b(b^ax)$;
$\log_b{\sqrt{x}}=\dfrac{1}{2}\log_b{x}$;
$\exp(a+bi)=\e^a(\cos{b}+i\sin{b}), \quad \ln(a+bi)=\ln\sqrt{a^2+b^2}+\mathrm{atan2}(b,a)\,i;$ $\sqrt{a+bi}=\sqrt{\dfrac{\sqrt{a^{2}+b^{2}}+a}{2}}+i\cdot\sgn(b)\sqrt{\dfrac{\sqrt{a^{2}+b^{2}}-a}{2}}$;
For general complex numbers with $a,b\ne0$:
$a^x=\e^{x\ln{a}}$, $\dfrac{1}{a}=\e^{-\ln{a}}$; $a^0=1$, $a^{-x}=\dfrac{1}{a^x}$; $a^x a^y=a^{x+y}$, $\dfrac{a^x}{a^y}=a^{x-y}$;
$\sqrt{a^2}=\begin{cases}a,&\Re(a)>0;\ \Re(a)=0,\Im(a)>0\\-a,&\Re(a)\lt0;\ \Re(a)=0,\Im(a)\lt0\end{cases}$; $\left(\sqrt{a}\right)^2=a$;
$\e^{\ln(x)}=x$; $\ln(\e^x)=x \quad (\Im(x)\lt\pi)$; $a^{\ln(x)/\ln(a)}=x \quad (x\ne0, a\ne0,1)$;
Properties of complex conjugate
$\overline{z\pm w}=\overline{z}\pm\overline{w},\quad \overline{zw}=\overline{z}\cdot\overline{w},\quad \overline{\left(\dfrac{z}{w}\right)}=\dfrac{\overline{z}}{\overline{w}};\qquad z\overline{z}=|z|^2,\quad z^{-1}=\dfrac{\overline{z}}{|z|^2},\quad \overline{z^n}=\left(\overline{z}\right)^n\ (n\in\Z),\quad \exp(\overline{z})=\overline{\exp(z)},\quad \ln(\overline{z})=\overline{\ln(z)};\quad$
Hyperbolic Functions
$\sinh{x}={\dfrac{\e^{x}-\e^{-x}}{2}}={\dfrac{\e^{2x}-1}{2\e^{x}}}={\dfrac{1-\e^{-2x}}{2\e^{-x}}}, \quad \cosh{x}={\dfrac{\e^{x}+\e^{-x}}{2}}={\dfrac{\e^{2x}+1}{2\e^{x}}}={\dfrac{1+\e^{-2x}}{2\e^{-x}}};$
$\tanh{x}={\dfrac{\sinh{x}}{\cosh{x}}}={\dfrac{\e^{x}-\e^{-x}}{\e^{x}+\e^{-x}}}={\dfrac{\e^{2x}-1}{\e^{2x}+1}}={1-\dfrac{2}{\e^{2x}+1}}, \quad \coth{x}={\dfrac{\cosh{x}}{\sinh{x}}}={\dfrac{\e^{x}+\e^{-x}}{\e^{x}-\e^{-x}}}={\dfrac{\e^{2x}+1}{\e^{2x}-1}}={1+\dfrac{2}{\e^{2x}-1}};$
$\sinh^{-1}(x)=\ln\left(x+{\sqrt{x^{2}+1}}\right), \quad \cosh^{-1}(x)=\ln\left(x\pm{\sqrt{x^{2}-1}}\right), \quad \tanh^{-1}(x)={\dfrac{1}{2}}\ln\left({\dfrac{1+x}{1-x}}\right), \quad \coth^{-1}(x)={\dfrac{1}{2}}\ln\left({\dfrac{x+1}{x-1}}\right);$
$\sin(ix)=i\cdot\sinh{x}, \quad \cos(ix)=\cosh{x}, \quad \tan(ix)=i\cdot\tanh{x}; \quad \sinh(ix)=i\cdot\sin{x}, \quad \cosh(ix)=\cos{x}, \quad \tanh(ix)=i\cdot\tan{x};$
$\sin^{-1}(iz)=i\cdot\sinh^{-1}(z), \quad \tan^{-1}(iz)=i\cdot\tanh^{-1}(z); \quad \sinh^{-1}(iz)=i\cdot\sin^{-1}(z), \quad \tanh^{-1}(iz)=i\cdot\tan^{-1}(z);$
$\cosh{x}+\sinh{x}=\e^{x}, \quad \cosh{x}-\sinh{x}=\e^{-x}; \quad \cosh^2{x}-\sinh^2{x}=1, \quad \mathrm{sech}^2{x}+\tanh^2{x}=1, \quad \coth^2{x}-\mathrm{csch}^2{x}=1;$
$\sinh\left(a\pm{b}\right)=\sinh{a}\cosh{b}\pm\cosh{a}\sinh{b}, \quad \cosh\left(a\pm{b}\right)=\cosh{a}\cosh{b}\pm\sinh{a}\sinh{b}, \quad \tanh\left(a\pm{b}\right)=\dfrac{\tanh{a}\pm\tanh{b}}{1\pm\tanh{a}\tanh{b}};$
$\sinh{2x}=2\sinh x\cosh x, \quad \cosh{2x}=\sinh^{2}{x}+\cosh^{2}{x}=2\sinh^{2}x+1=2\cosh^{2}x-1, \quad \tanh{2x}=\dfrac{2\tanh x}{1+\tanh^2x};$
$\sinh{3x}=\begin{cases}4\sinh^3{x}+3\sinh{x}\\4\sinh{x}\cosh^2{x}-\sinh{x}\\3\sinh{x}\cosh^2{x}+\sinh^3{x}\end{cases}, \quad \cosh{3x}=\begin{cases}4\cosh^3{x}-3\cosh{x}\\4\sinh^2{x}\cosh{x}+\cosh{x}\\3\sinh^2{x}\cosh{x}+\cosh^3{x}\end{cases}, \quad \tanh{3x}=\dfrac{3\tanh{x}+\tanh^3{x}}{1+3\tanh^2{x}};$
$\sinh{x}\cdot\sinh{y}=\dfrac{1}{2}\left(\cosh(x+y)-\cosh(x-y)\right), \quad \cosh{x}\cdot\cosh{y}=\dfrac{1}{2}\left(\cosh(x+y)+\cosh(x-y)\right), \quad \sinh{x}\cdot\cosh{y}=\dfrac{1}{2}\left(\sinh(x+y)+\sinh(x-y)\right);$
$\sinh^2(x)=\dfrac{1}{2}(\cosh(2x)-1), \quad \cosh^2(x)=\dfrac{1}{2}(\cosh(2x)+1), \quad \sinh(x)\cdot\cosh(x)=\dfrac{1}{2}\sinh(2x);$
$\sinh{x}\pm\sinh{y}=2\sinh\left(\dfrac{x\pm{y}}{2}\right)\cosh\left(\dfrac{x\mp{y}}{2}\right); \quad \cosh{x}+\cosh{y}=2\cosh\left(\dfrac{x+y}{2}\right)\cosh\left(\dfrac{x-y}{2}\right), \quad \cosh{x}-\cosh{y}=2\sinh\left(\dfrac{x+y}{2}\right)\sinh\left(\dfrac{x-y}{2}\right);$
$\sinh(\cosh^{-1}x)=\sqrt{x^2-1}, \quad \cosh(\sinh^{-1}x)=\sqrt{x^2+1}, \quad \tanh(\sinh^{-1}x)=\dfrac{x}{\sqrt{x^2+1}}, \\ \quad \tanh(\cosh^{-1}x)=\dfrac{\sqrt{x^2-1}}{x}, \quad \sinh(\tanh^{-1}x)=\dfrac{x}{\sqrt{1-x^2}}, \quad \cosh(\tanh^{-1}x)=\dfrac{1}{\sqrt{1-x^2}}; \qquad(x\in\R);$
Limit
$\lim[f(x)\pm{g(x)}]=\lim{f(x)}\pm\lim{g(x)}; \quad \lim[f(x)\cdot{g(x)}]=\lim{f(x)}\cdot\lim{g(x)}; \quad \lim\dfrac{f(x)}{g(x)}=\dfrac{\lim{f(x)}}{\lim{g(x)}}; \\ \lim[c\cdot{f(x)}]=c\cdot\lim{f(x)}; \quad \lim[f(x)]^{n}=[\lim{f(x)}]^{n},\> n\in{N};$
If $ \>{y_n}\le{x_n}\le{z_n}\> $ and $\>\lim{y_n}=\lim{z_n}=a\>$, then $\>\lim{x_n}=a\>$;
If $ \>\lim\dfrac{f(x)}{g(x)}=1\> $, then $\>\lim[{f(x)\cdot{h(x)}}]=\lim[{g(x)\cdot{h(x)}}]\>$, $\>\lim\dfrac{h(x)}{f(x)}=\lim\dfrac{h(x)}{g(x)} \> $;
L'Hôpital's Rule: If $\>\lim{f(x)}=\lim{g(x)}=0\>$ or $\>\lim{f(x)}=\lim{g(x)}=\infty\>$, then $\>\lim\dfrac{f(x)}{g(x)}=\lim\dfrac{f'(x)}{g'(x)}\>$;
$\text{L'Hôpital's Rule for other indeterminates: } \quad \lim{0\cdot\infty}\Rightarrow\lim{\dfrac{0}{1/\infty}}, \quad \lim{\infty_1-\infty_2}\Rightarrow\lim\dfrac{1/\infty_1-1/\infty_2}{1/(\infty_1\infty_2)}, \quad \lim{\infty^0}\Rightarrow\exp\left(\lim\dfrac{0}{1/\ln\infty}\right), \\\qquad \lim{1^\infty}\Rightarrow\exp\left(\lim\dfrac{\ln1}{1/\infty}\right), \quad \lim{ (0^+)^{0} }\Rightarrow \exp\left(\lim\dfrac{0}{1/\ln0^+}\right), \quad \lim{\log_{0}{0}}\Rightarrow \lim\dfrac{\ln0}{\ln0}, \quad \lim{\log_{1}{1}}\Rightarrow \lim\dfrac{\ln1}{\ln1};$
Derivative
$\ddx[a\cdot{f(x)}+b\cdot{g(x)}]=a\cdot{f'(x)}+b\cdot{g'(x)}; \quad$ $\ddx[f(x)\cdot{g(x)}]=f'(x)\cdot{g(x)}+f(x)\cdot{g'(x)};$
$\ddx[f_1(x)\cdot{f_2(x)}\cdot{f_3(x)}]=f'_1(x)\cdot{f_2(x)}\cdot{f_3(x)}+f_1(x)\cdot{f'_2(x)}\cdot{f_3(x)}+f_1(x)\cdot{f_2(x)}\cdot{f'_3(x)}; \quad$ “Each item take turns at derivation”
$\ddx\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{f'(x)g(x)-f(x)g'(x)}{g^2(x)}, \quad \ddx\left(\dfrac{1}{u(x)}\right)=\dfrac{u'(x)}{u^2(x)};$
$\ddx(f\circ{g})(x)=(f'\circ{g})(x)\cdot{g'(x)}; \quad$ $\ddx{f^{-1}(x)}=\dfrac{1}{(f'\circ{f^{-1}})(x)}; \quad$ $\dfrac{\d^n}{\d^n{x}}[f(x)\cdot{g(x)}]=\displaystyle\sum\limits_{k=0}\limits^{n}\dbinom{n}{k}\cdot{f^{(n-k)}(x)}\cdot{g^{(k)}(x)}$;
$\ddx{C}=0; \quad$ $\ddx{|x|}=\dfrac{x}{|x|};$
$\ddx{x^\mu}=\mu{x^{\mu-1}}; \quad$ $\ddx{\e^x}=\e^x, \quad \ddx{a^x}=a^x\ln{a}; \quad$ $ \ddx{\dfrac{1}{x}}=-\dfrac{1}{x^2}, \quad \ddx{\sqrt{x}}=\dfrac{1}{2\sqrt{x}}, \quad \ddx{\sqrt[3]{x}}=\dfrac{1}{3}x^{-\frac{2}{3}};$
$\ddx\ln|x|=\dfrac{1}{x}, \quad \ddx{\log_a|x|}=\dfrac{1}{x\ln{a}}$;
$\ddx\sin{x}=\cos{x}, \quad \ddx{\cos{x}}=-\sin{x}, \quad \ddx{\tan{x}}=\sec^2{x}, \quad \ddx\csc{x}=-\csc{x}\cot{x}, \quad \ddx{\sec{x}}=\sec{x}\tan{x}, \quad \ddx\cot{x}=-\csc^2{x};$
$\ddx\sin^{-1}{x}=\dfrac{1}{\sqrt{1-x^2}}, \quad \ddx\cos^{-1}{x}=-\dfrac{1}{\sqrt{1-x^2}}, \quad \ddx\tan^{-1}{x}=\dfrac{1}{1+x^2}, \quad \ddx\cot^{-1}{x}=-\dfrac{1}{1+x^2};$
$\ddx\sinh{x}=\cosh{x}, \quad \ddx\cosh{x}=\sinh{x}, \quad \ddx\tanh{x}=\sech^{2}{x}, \quad \ddx\csch{x}=-\coth{x}\cdot\csch{x}, \quad \ddx\sech{x}=-\sech{x}\cdot\tanh{x}, \quad \ddx\coth{x}=-\csch^2{x};$
$\ddx\sinh^{-1}{x}=\dfrac{1}{\sqrt{x^2+1}}, \quad \ddx\cosh^{-1}{x}=\dfrac{1}{\sqrt{x^2-1}}, \quad \ddx\tanh^{-1}{x}=\ddx\coth^{-1}{x}=\dfrac{1}{1-x^2};$
$\ddx\mathrm{erf}(x)=\dfrac{2}{\sqrt{\pi}}\e^{-x^2};$
Taylor Series: $ f(x)=\displaystyle\sum\limits_{k=0}\limits^{n}{\dfrac{(x-x_0)^k\cdot{f^{(k)}}(x_0)}{k!}}+R_n$; Maclaurin Series: $ f(x)=\displaystyle\sum\limits_{k=0}\limits^{n}{\dfrac{x^k\cdot{f^{(k)}}(0)}{k!}}+R_n$;
Arc length and curvature
$(x(t),y(t))$: $\d{s}=\sqrt{x'^2+y'^2}\d{t}$; $(x(t),y(t),z(t))$: $\d{s}=\sqrt{x'^2+y'^2+z'^2}\d{t}$; $y=f(x)$: $\d{s}=\sqrt{y'^2+1}\d{x}$; $r=r(\theta)$: $\d{s}=\sqrt{r^2+r'^2}\d{\theta}$;
Curvature: $\kappa=\lim\limits_{\Delta s\to 0}{\dfrac{\Delta\theta}{\Delta{s}}}=\left|\dfrac{\d\theta}{\d{s}}\right|$;
$\vec{r}=\mathbf{r}(t)$: $\kappa(t)=\dfrac{|\mathbf{r}'(t)\times\mathbf{r}''(t)|}{|\mathbf{r}'(t)|^{3}}$; $y=f(x)$: $\kappa(x)={\dfrac{y''}{\left(1+{y'}^{2}\right)^{\frac{3}{2}}}}$; $r=r(\theta)$: $\kappa(\theta)={\dfrac{r^{2}+2{r'}^{2}-r\,r''}{\left(r^{2}+{r'}^{2}\right)^{\frac{3}{2}}}}$;
Unit tangent: $T=\dfrac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}$; Normal unit vector: $N=\dfrac{T'(t)}{|T'(t)|}$; Binormal unit vector: $B=T\times N$;
Frenet–Serret formula: $\bmat{T'\\N'\\B'\\}=\bmat{0&\kappa&0\\-\kappa&0&\tau\\0&-\tau&0}\bmat{T\\N\\B}$, where $\tau=-\dfrac{\d{B}}{\d{s}}\cdot N$ is the torsion;
Kinematics
Velocity and acceleration in Cartesian coordinate: $\vec{p}=x\vec{i}+y\vec{j}+z\vec{k}$, $\vec{v}=\dfrac{\d{x}}{\d{t}}\vec{i}+\dfrac{\d{y}}{\d{t}}\vec{j}+\dfrac{\d{z}}{\d{t}}\vec{k}$, $\vec{a}=\dfrac{\d^2{x}}{\d{t}^2}\vec{i}+\dfrac{\d^2{y}}{\d{t}^2}\vec{j}+\dfrac{\d^2{z}}{\d{t}^2}\vec{k}$;
Velocity and acceleration in polar coordinate $\begin{cases}r=r(t)\\\theta=\theta(t)\end{cases}$: $\vec{p}=r\vec{e_r}$, $\vec{v}=\dot{r}\vec{e_r}+r\dot{\theta}\vec{e_\theta}$, $\vec{a}=(\ddot{r}-r\dot{\theta}^2)\vec{e_r}+(2\dot{r}\dot{\theta}+r\ddot{\theta})\vec{e_\theta}$;
Velocity and acceleration in eigen coordinate:
$\vec{v}=v\vec{e_t}$,
$\vec{a}=\dfrac{\d{v}}{\d{t}}\vec{e_t}+\dfrac{v^2}{R}\vec{e_n}$,
$R$ is the curvature radius;
($\vec{a}$ can also be calculated by projecting acceleration vector in $\vec{e_t}$ and
$\vec{e_n}$ directions)
Derivatives of unit vectors in polar coordinate: $\dfrac{\d}{\d{t}}{\vec{e_r}}=\dot{\theta}\vec{e_\theta}$, $\dfrac{\d}{\d{t}}{\vec{e_\theta}}=-\dot{\theta}\vec{e_r}$;
Integral
$\Int{a\cdot{f(x)}\pm b\cdot{g(x)}}\d{x}=a\Int{f(x)}\pm b\Int{g(x)}\d{x};$
$\Int{(f\circ{g})(x)\cdot{g}'(x)}\d{x}=\Int{f(u)}\d{u}{\huge\vert}_{u=g(x)}, \quad \Int{(f\circ{g})(x)}\d{x}=\Int{\dfrac{1}{g'(x)}(f\circ{g})(x)}\d[g(x)];$
$\intd{f(x)}=\intd[u]{(f\circ{g})(u)\cdot{g'(u)}}{\huge\vert}_{u=g^{-1}(x)};$
$\Int{uv'}=uv-\Int{u'v}; \quad$ （Log / Inverse trig − Algebraic − Trigonometric − Exp）
$\intd{k}=kx+C; \quad \intd{|x|}=\dfrac{x|x|}{2}+C; \quad \intd{\dfrac{1}{x}}=\ln|x|+C, \quad \intd{x^\mu}=\dfrac{x^{\mu+1}}{\mu+1}+C \>(\mu\ne1); \quad \intd{\e^x}=\e^x+C, \quad \intd{a^x}=\dfrac{a^x}{\ln{a}}+C; \quad \intd{\sqrt{x}}=\dfrac{2}{3}x^{3/2}+C; \quad$
$\intd{\sin{x}}=-\cos{x}+C, \quad \intd{\cos{x}}=\sin{x}+C, \quad \intd{\tan{x}}=-\ln|\cos{x}|+C; \\ \intd{\cot{x}}=\ln{|\sin{x}|}+C, \quad \intd{\sec{x}}=\tanh^{-1}(\sin{x})+C, \quad \intd{\csc{x}}=\ln\left|\tan\dfrac{x}{2}\right|+C=-\tanh^{-1}(\cos{x})+C;$
$\intd{\sinh{x}}=\cosh{x}+C, \quad \intd{\cosh{x}}=\sinh{x}+C, \quad \intd{\tanh{x}}=-\ln(\cosh{x})+C; \\ \intd{\coth{x}}=\ln{|\sinh{x}|}+C, \quad \intd{\sech{x}}=\tan^{-1}(\sinh{x})+C, \quad \intd{\csch{x}}=-\coth^{-1}(\cosh{x})+C=\ln(|\e^{-x}-1|)-\ln(|\e^{-x}+1|)+C;$
$\intd{\ln{x}}=x(\ln(x)-1)+C; \quad$ $\intd{\sin^{-1}(x)}=x\sin^{-1}\left(x\right)+\sqrt{1-x^2}+C, \quad \intd{\cos^{-1}(x)}=x\cos^{-1}\left(x\right)-\sqrt{1-x^2}+C;$
$\intd{\tan^{-1}(x)}=x\tan^{-1}\left(x\right)-\dfrac{\ln\left(x^2+1\right)}{2}+C, \quad \intd{\cot^{-1}(x)}=x\cot^{-1}\left(x\right)+\dfrac{\ln\left(x^2+1\right)}{2}+C;$ $\intd{\e^{-x^2}}=\dfrac{\sqrt{\pi}}{2}\erf(x)+C;$
$\intd{\dfrac{1}{a^2+x^2}}=\dfrac{1}{a}\tan^{-1}\left(\dfrac{x}{a}\right)+C, \quad \intd{\dfrac{1}{a^2-x^2}}=\dfrac{1}{2a}\ln\left|\dfrac{x+a}{x-a}\right|+C=\dfrac{1}{a}\tanh^{-1}\left(\dfrac{x}{a}\right)+C=\dfrac{1}{a}\coth^{-1}\left(\dfrac{x}{a}\right)+C;$
$\intd{\dfrac{1}{\sqrt{a^2+x^2}}}=\sinh^{-1}\left(\dfrac{x}{|a|}\right)+C, \quad \intd{\dfrac{1}{\sqrt{a^2-x^2}}}=\sin^{-1}\left(\dfrac{x}{|a|}\right)+C, \quad \intd{\dfrac{1}{\sqrt{x^2\pm{a^2}}}}=\ln\left|\sqrt{x^2\pm{a^2}}+x\right|+C;$
$\intd{\sqrt{a^2-x^2}}=\dfrac{1}{2}\left(a^2\sin^{-1}\left(\dfrac{x}{|a|}\right)+x\sqrt{a^2-x^2}\right)+C; \quad \intd{\sqrt{a^2+x^2}}=\dfrac{1}{2}\left({a^2\sinh^{-1}\left(\dfrac{x}{|a|}\right)+x\sqrt{a^2+x^2}}\right)+C; \qquad$
$\intd{\sqrt{x^2-a^2}}=\dfrac{1}{2}\left(x\sqrt{x^2-a^2}-a^2\cosh^{-1}\left(\dfrac{x}{|a|}\right)\right)+C; \quad (x>0)$
$\intd{x\e^x}=x\e^x-\e^x+C; \quad \intd{x\sin(x)}=\sin(x)-x\cos(x)+C, \quad \intd{x\cos(x)}=\cos(x)+x\sin(x)+C;$
$\intabd{f(x)}{a}{a}=0$; $\intabd{f(x)}{a}{b}=-\intabd{f(x)}{b}{a}$; $\intabd{f(x)}{a}{c}=\intabd{f(x)}{a}{b}+\intabd{f(x)}{b}{c}$;
$\intabd{[f(x)\pm{g(x)}]}{a}{b}=\intabd{f(x)}{a}{b}\pm\intabd{g(x)}{a}{b}$, $\intabd{k\cdot{f(x)}}{a}{b}=k\cdot\intabd{f(x)}{a}{b}$, $\intabd{k_{1}f_{1}(x)\pm{k_2}f_{2}(x)}{a}{b}=k_{1}\intabd{f_1(x)}{a}{b}\pm{k_2}\intabd{f_2(x)}{a}{b}$;
$\ddx{\left({\intabd[t]{f(t)}{\varphi(x)}{\psi(x)}}\right)}=\psi'(x)f[\psi(x)]-\varphi'(x)f[\varphi(x)]$;
$\intabd{f(x)}{a}{b}=\intabd[t]{f[\varphi(t)]\varphi'(t)}{\varphi^{-1}(a)}{\varphi^{-1}(b)}$, $\intabd{f[\varphi(x)]\varphi'(x)}{a}{b}=\intabd{f(x)}{\varphi(a)}{\varphi(b)}=\intabd[\varphi(x)]{f[\varphi(x)]}{a}{b}$; $\intabd{f(\sin{x})}{0}{\pi\over2}=\intabd{f(\cos{x})}{0}{\pi\over2}$;
Newton-Leibniz Formula: $\intabd{f(x)}{a}{b}=F(b)-F(a)$;
Green Formula: $\Oint_LP\d{x}+Q\d{y}=\Iint_D\left(\dfrac{\partial{Q}}{\partial{x}}-\dfrac{\partial{P}}{\partial{y}}\right)\d{x}\d{y}$; ($A=\dfrac{1}{2}\Oint_Lx\d{y}-y\d{x}$);
Gauss Formula: $\Oiint_{\Sigma^+}P\,\d{y}\d{z}+Q\,\d{x}\d{z}+R\,\d{x}\d{y}=\Iiint_V\left(\dfrac{\partial{P}}{\partial{x}}+\dfrac{\partial{Q}}{\partial{y}}+\dfrac{\partial{R}}{\partial{z}}\right)\d{x}\d{y}\d{z}$; ($V=\dfrac{1}{3}\;\Oiint_{\Sigma}x\,\d{y}\d{z}+y\,\d{z}\d{x}+z\,\d{x}\d{y}$);
Stokes Formula: $\Oint_LP\d{x}+Q\d{y}+R\d{z}=\Iint_\Sigma\left(\dfrac{\partial{R}}{\partial{y}}-\dfrac{\partial{Q}}{\partial{z}}\right)\d{y}\d{z}+\left(\dfrac{\partial{P}}{\partial{z}}-\dfrac{\partial{R}}{\partial{x}}\right)\d{z}\d{x}+\left(\dfrac{\partial{Q}}{\partial{x}}-\dfrac{\partial{P}}{\partial{y}}\right)\d{x}\d{y}$;
Special Integrals
Elliptic integral of the first kind: $F(\varphi ,k)=F\left(\varphi \,|\,k^{2}\right)=F(\sin \varphi; k)=\displaystyle\int _{0}^{\varphi }{\dfrac {\d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}$; $\displaystyle K(k)=\int _{0}^{\tfrac {\pi }{2}}{\frac {\d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}=\int _{0}^{1}{\frac {\d t}{\sqrt {\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}}$;
Elliptic integral of the second kind: $\displaystyle E(\varphi ,k)=E\left(\varphi \,|\,k^{2}\right)=E(\sin \varphi ;k)=\int _{0}^{\varphi }{\sqrt {1-k^{2}\sin ^{2}\theta }}\,\d\theta$; $\displaystyle E(k)=\int _{0}^{\tfrac {\pi }{2}}{\sqrt {1-k^{2}\sin ^{2}\theta }}\,\d\theta =\int _{0}^{1}{\frac {\sqrt {1-k^{2}t^{2}}}{\sqrt {1-t^{2}}}}\,\d t$;
Euler's Integrals:
$\Beta(x,y)=\intabd[t]{t^{x-1}(1-t)^{y-1}}{0}{1}$,
$\Gamma(x)=\intabd[t]{t^{x-1}\e^{-t}}{0}{\infty}$;
$\Beta(x,y)=\Beta(y,x)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$;
$\Gamma(x+1)=x\Gamma(x)$;
$\Beta(x+1,y+1)=\dfrac{xy}{(x+y)(x+y+1)}\Beta(x,y)$,
$\Beta(x+1,y)=\dfrac{x}{x+y}\Beta(x,y)$;
$\Gamma(x)\Gamma(1-x)=\dfrac{\pi}{\sin(\pi x)}$,
$\Gamma(2x)=\dfrac{2^{2x-1}}{\sqrt{\pi}}\Gamma\left(x\right)\Gamma\left(x+\dfrac{1}{2}\right)$;
$\intabd{\cos^m(x)\sin^n(x)}{0}{\frac{\pi}{2}}=\dfrac{1}{2}\Beta\left(\dfrac{m+1}{2},\dfrac{n+1}{2}\right)=\dfrac{\Gamma\left(\frac{m+1}{2}\right)\Gamma\left(\frac{n+1}{2}\right)}{2\cdot\Gamma\left(\frac{m+n}{2}+1\right)}$, $\intabd{\sin^n(x)}{0}{\frac{\pi}{2}}=\intabd{\cos^n(x)}{0}{\frac{\pi}{2}}=\begin{cases}\dfrac{(n-1)!!}{n!!}\dfrac{\pi}{2},&n\;\text{even}\\\dfrac{(n-1)!!}{n!!},&n\;\text{odd}\end{cases}$, $\intabd{\tan^\alpha(x)}{0}{\frac{\pi}{2}}=\dfrac{\pi}{2\cos\left(\dfrac{\alpha\;\pi}{2}\right)}$;
Multivariable Derivative
$\dfrac{\partial\begin{bmatrix}f_1&f_2&\cdots\end{bmatrix}^T}{\partial\begin{bmatrix}x_1&x_2&\cdots\end{bmatrix}^T}=\begin{bmatrix}\frac{\partial{f_1}}{\partial{x_1}}&\frac{\partial{f_1}}{\partial{x_2}}&\cdots\\\frac{\partial{f_2}}{\partial{x_1}}&\frac{\partial{f_2}}{\partial{x_2}}&\cdots\\\vdots&\vdots&\ddots\end{bmatrix}$; $\dfrac{\partial(u,v)}{\partial(x,y)}=\begin{vmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{vmatrix}$, $\dfrac{\partial(u,v,w)}{\partial(x,y,z)}=\begin{vmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}&\frac{\partial u}{\partial z}\\\frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}&\frac{\partial v}{\partial z}\\\frac{\partial w}{\partial x}&\frac{\partial w}{\partial y}&\frac{\partial w}{\partial z}\end{vmatrix}$;
$\dfrac{\partial(u,v)}{\partial(s,t)}=\dfrac{\partial(u,v)}{\partial(x,y)}\cdot\dfrac{\partial(x,y)}{\partial(s,t)}$, $\dfrac{\partial(u,v)}{\partial(x,y)}\cdot\dfrac{\partial(x,y)}{\partial(u,v)}=1$, $\dfrac{\partial(u,v)}{\partial(x,y)}=-\dfrac{\partial(v,u)}{\partial(x,y)}$, $\dfrac{\partial(u,u)}{\partial(x,y)}=0$;
Area and volume elements ⭧
$\d{A}=\d{x}\d{y}=\dfrac{\partial(x,y)}{\partial(u,v)}\d{u}\d{v}$, $\d{V}=\d{x}\d{y}\d{z}=\dfrac{\partial(x,y,z)}{\partial(u,v,w)}\d{u}\d{v}\d{w}$;
Polar/cylindrical $\begin{aligned}x&=\rho\cos(\theta)\\y&=\rho\sin(\theta)\end{aligned}$: $\dfrac{\partial(x,y)}{\partial(\rho,\theta)}=\rho$; Spherical $\begin{aligned}x&=\rho\sin(\varphi)\cos(\theta)\\y&=\rho\sin(\varphi)\sin(\theta)\\z&=\rho\cos(\varphi)\end{aligned}$: $\dfrac{\partial(x,y,z)}{\partial(\rho,\theta,\varphi)}=\rho^2\sin(\varphi)$;
Parametric surface $\mathbf{r}=\mathbf{r}(u,v)$: $\d{A}=|\mathbf{r}_u'\times\mathbf{r}_v'|\d{u}\d{v}=\sqrt{\mathbf{r}_u'^2\mathbf{r}_v'^2-(\mathbf{r}_u'\cdot\mathbf{r}_v')^2}\ \d{u}\d{v}$;
Derivative of implicit functions
$F(x,y(x))=0$: $\displaystyle{\dfrac{\d{y}}{\d{x}}=-\frac{\frac{\partial{F}}{\partial{x}}}{\frac{\partial{F}}{\partial{y}}}}=-F_x'/F_y'$, $\displaystyle{\frac{\d^2 y}{\d x^2}=-\frac{\frac{\partial^2 F}{\partial x^2}\left(\frac{\partial F}{\partial y}\right)^2-2\frac{\partial^{2}F}{\partial x\partial y}\frac{\partial F}{\partial x}\frac{\partial F}{\partial y}+\frac{\partial^2 F}{\partial y^2}\left(\frac{\partial F}{\partial x}\right)^2}{\left(\frac{\partial F}{\partial y}\right)^3}}$;
$F(x,y,z(x,y))=0$: $\dfrac{\d z}{\d x}=-F_x'/F_z'$, $\dfrac{\d z}{\d y}=-F_y'/F_z'$;
Vector Calculus
Gradient of a scalar field: $\mathbf{grad}\ u=\nabla{u}=\dfrac{\partial{u}}{\partial{x}}\mathbf{i}+\dfrac{\partial{u}}{\partial{y}}\mathbf{j}+\dfrac{\partial{u}}{\partial{z}}\mathbf{k}$; Directional derivative: $\dfrac{\d}{\d{t}}u(\mathbf{x_0}+\mathbf{l}\cdot t)=\mathbf{grad}\ u(\mathbf{x_0})\cdot\mathbf{l}$;
$\nabla(C_1u_1+C_2u_2) = C_1\nabla{u_1}+C_2\nabla{u_2}$, $\nabla{u_1u_2}=u_1\nabla{u_2}+u_2\nabla{u_1}$, $\nabla{f(u)}=f'(u)\nabla{u}$;
Divergence of a vector field: $\mathbf{div}\ F=\nabla\cdot F=\dfrac{\partial F_x}{\partial x}+\dfrac{\partial F_y}{\partial y}+\dfrac{\partial F_z}{\partial z}$;
$\mathrm{div}(C_1\mathbf{v}_1+C_2\mathbf{v}_2) = C_1\mathrm{div}(\mathbf{v}_1)+C_2\mathrm{div}(\mathbf{v}_2)$, $\mathrm{div}(u\mathbf{v})=u\cdot\mathrm{div}(\mathbf{v})+\mathbf{v}\cdot\mathrm{grad}(u)$;
Curl of a vector field: $\mathbf{curl}\ F=\nabla\times{F}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\F_x&F_y&F_z\end{vmatrix} =\left(\dfrac{\partial F_{z}}{\partial y}-\dfrac{\partial F_{y}}{\partial z}\right)\mathbf{i} +\left(\dfrac{\partial F_{x}}{\partial z}-\dfrac{\partial F_{z}}{\partial x}\right)\mathbf{j} +\left(\dfrac{\partial F_{y}}{\partial x}-\dfrac{\partial F_{x}}{\partial y}\right)\mathbf{k}$;
Differential equations
Solve $\dfrac{\d{y}}{\d{x}}=f(x)$: $y=\Int{f(x)\d{x}}+C$;
Solve $\dfrac{\d{y}}{\d{x}}=f(x)g(y)$: $\dfrac{1}{g(y)}\d{y}=f(x)\d{x} \Longrightarrow\Int\dfrac{1}{g(y)}\d{y}=\Int f(x)\d{x}+C$;
Solve $\dfrac{\d{y}}{\d{x}}=\varphi\left(\dfrac{y}{x}\right)$: $z=\dfrac{y}{x} \Longrightarrow x\dfrac{\d{z}}{\d{x}}+z=\varphi(z)\Longrightarrow\dfrac{\d{z}}{\d{x}}=\dfrac{\varphi(z)-z}{x}$; $\varphi(z)-z\ne0: \Int\dfrac{\d{z}}{\varphi(z)-z}=\ln\left|\dfrac{x}{C}\right|; \quad \varphi(z)-z\equiv0: z=\dfrac{y}{x}=C$;
Solve $\dfrac{\d{y}}{\d{x}}+p(x)y=q(x)$: $y(x)=\e^{-\int p(x)\d{x}}\left[\Int q(x)\e^{\int p(x)\d{x}}\d{x}+C\right]$;
Solve $\dfrac{\d{y}}{\d{x}}+p(x)y=q(x)y^n$: $z=y^{1-n} \Longrightarrow \dfrac{\d{z}}{\d{x}}+(1-n)p(x)z=(1-n)q(x)$; (treat as above)
Solve $y^{(n)}=f(x)$: $\dfrac{\d^{n-1}y}{\d x^{n-1}}=\Int f(x)\d{x}+C_1$; (repeat)
Solve $y''=f(x,y')$: $p(x)=\dfrac{\d{y}}{\d{x}} \Longrightarrow \dfrac{\d{p}}{\d{x}}=f(x,p)$; $p(x)=\varphi(x,C_1) \Longrightarrow y=\Int \varphi(x,C_1)\d{x} +C_2$;
Solve $y''=f(y,y')$: $p(y)=\dfrac{\d{y}}{\d{x}} \Longrightarrow \dfrac{\d^2y}{\d x^2}=p\dfrac{\d{p}}{\d{y}}=f(y,p)$; $p(y)=\varphi(y,C_1) \Longrightarrow \Int\dfrac{\d{y}}{\varphi(y,C_1)}=x+C_2$;
Solve $y^{(n)}=f\left(x,y^{(n-1)}\right)$: $u(x)=y^{(n-2)} \Longrightarrow u''=f(x,u')$, solve using above and repeat;
Solve $y^{(n)}=f\left(y^{(n-1)},y^{(n-2)}\right)$: $u(x)=y^{(n-2)} \Longrightarrow u''=f(u',u)$, solve using above and repeat;
Solve $y''+py'+qy=0$: Let $r$ be the roots of $r^2+pr+q=0$; $y=\begin{cases} C_1\e^{r_1x}+C_2\e^{r_2x}, &r\in[r_1,r_2],\ r_1\ne r_2\\ \e^{rx}(C_1+C_2x), &r=r_1=r_2\\ \e^{ax}\left(C_1\cos(bx)+C_2\sin(bx)\right), &r=a\pm bi,\ b\ne0 \end{cases}$
Solve $y''+py'+qy=\e^{\lambda x}P(x)$: Find $Q(x)$ such that $Q''(x)+(2\lambda+p)Q'(x)+(\lambda^2+p\lambda+q)Q(x)=P(x)$, $\Longrightarrow y=\e^{\lambda x}Q(x)+(\text{solution of }y''+py'+qy=0)$;
$\lambda^2+p\lambda+q$ and $2\lambda+p$ are both zero when $\lambda$ is a multiple root of $\lambda^2+p\lambda+q=0$;
$P(x)=P_m(x)$ is a degree-$m$ polynomial: $Q(x)=\begin{cases} Q_m(x), &\lambda^2+p\lambda+q\ne0\\ xQ_m(x), &\lambda^2+p\lambda+q=0,\ 2\lambda+p\ne0\\ x^2Q_m(x), &\lambda^2+p\lambda+q=2\lambda+p=0\\ \end{cases}$, where $Q_m(x)$ is a polynomial of degree $m$;
$\e^{\lambda x}P(x)=\e^{ax}\left[P_m(x)\cos(bx)+P_n(x)\sin(bx)\right]$: $\e^{\lambda x}Q(x)=x^k\e^{ax}\left[R_1(x)\cos(bx)+R_2(x)\sin(bx)\right]$, $k=\begin{cases}0, &(a+bi)^2+p(a+bi)+q\ne0\\1, &(a+bi)^2+p(a+bi)+q=0\\\end{cases}$, $R_i(x)$ has degree $\max(m,n)$;
Solve $y''+p(x)y'+q(x)y=0$: Find any non-zero solution $y_1(x)$, let $y_2(x)=y_1(x)\Int\dfrac{\e^{-\int p(x)\d{x}}}{y_1(x)^2}\d{x}$, $\Longrightarrow y=C_1y_1(x)+C_2y_2(x)$;
Solve $y''+p(x)y'+q(x)y=f(x)$: Find any solution $y^*(x)$, $\Longrightarrow y(x)=y^*(x)+(\text{solution of }y''+p(x)y'+q(x)y=0)$;
If $y=y_1(x)$ satisfies $y''+p(x)y'+q(x)y=f_1(x)$ and $y=y_2(x)$ satisfies
$y''+p(x)y'+q(x)y=f_2(x)$,
then $y=C_1y_1(x)+C_2y_2(x)$ satisfies $y''+p(x)y'+q(x)y=C_1f_1(x)+C_2f_2(x)$;
Solve $x^2y''+pxy'+qy=f(x)$: $x=\e^t \Longrightarrow \dfrac{\d^2y}{\d t^2}+(p-1)\dfrac{\d y}{\d t}+qy=f(\e^t)$;
Matrix
$(AB)C=A(BC)$, $k(AB)=(kA)B=A(kB)$, $A(B+C)=AB+AC$, $(B+C)A=BA+CA$; $AI=IA=A$; $A^mA^n=A^{m+n}$, $\left(A^m\right)^n=A^{mn}$;
$(A^T)^T=A$, $(A+B)^T=A^T+B^T$, $(kA)^T=kA^T$, $(AB)^T=B^TA^T$, $(A_1A_2\dots A_k)^T=A_k^TA_{k-1}^T\dots A_1^T$; $A^TA=0\Rightarrow A=O$;
$A$ invertible: $\Longrightarrow$ $AB=O \Rightarrow A=O$, $AX=AY \Rightarrow X=Y$, $AX=O \Rightarrow X=O$, $AX=B \Rightarrow X=A^{-1}B$;
$A,B$ invertible: $\Longrightarrow$ $(A^{-1})^{-1}=A$, $(\lambda{A})^{-1}=\lambda^{-1}A^{-1}$, $(AB)^{-1}=B^{-1}A^{-1}$, $(A^T)^{-1}=(A^{-1})^T$;
$\det{AB}=\det{A}\cdot\det{B}$, $\det{A^T}=\det{A}$, $\det{A^{-1}}=\dfrac{1}{\det{A}}$;
Swap two rows/cols: $\det{A'}=-\det{A}$; Multiply a row/col by k: $\det{A'}=k\det{A}$; Add the multiple of a row/col to another row/col: $\det{A'}=\det{A}$;
$\det\pmat{a_1 & & * \\ & \ddots & \\ O & & a_n} = a_1a_2\dots{a_n}$, $\det\pmat{A_1 & & * \\ & \ddots & \\ O & & A_n} = |A_1||A_2|\dots|A_n|$, $\det{I}=1$;
$\vmat{\cdots&\cdots&\cdots\\[4pt]x_{1a}+x_{1b}&\cdots&x_{na}+x_{nb}\\[4pt]\cdots&\cdots&\cdots} =\vmat{\cdots&\cdots&\cdots\\[4pt]x_{1a}&\cdots&x_{na}\\[4pt]\cdots&\cdots&\cdots} +\vmat{\cdots&\cdots&\cdots\\[4pt]x_{1b}&\cdots&x_{nb}\\[4pt]\cdots&\cdots&\cdots}$; Vandermonde Determinant: $\vmat{1&1&1&\cdots&1\\x_1&x_2&x_3&\cdots&x_n\\ x_1^2&x_2^2&x_3^2&\cdots&x_n^2\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_1^{n-1}&x_2^{n-1}&x_3^{n-1}&\cdots&x_n^{n-1}}= \displaystyle{\prod\limits_{i\lt j}(x_j-x_i)}$;
$A^{-1}=\dfrac{A^*}{|A|}$; $AA^*=A^*A=\det(A)\cdot{I}$; $A^{-1}=\bmat{a&b\\c&d}^{-1}=\dfrac{1}{|A|}\bmat{d&-b\\-c&a}$;
Cramer's Rule: Solution of $\displaystyle{AX=\bmat{a_{11}&a_{12}&\cdots&a_{1n}\\[3pt]a_{21}&a_{22}&\cdots&a_{2n}\\[3pt] \vdots&\vdots&\ddots&\vdots\\[3pt]a_{n1}&a_{n2}&\cdots&a_{nn}} \bmat{x_1\\[3pt]x_2\\[3pt]\vdots\\[3pt]x_n}= \bmat{y_1\\[3pt]y_2\\[3pt]\vdots\\[3pt]y_n}=Y}$ is $x_i=\dfrac{|A_i|}{|A|}$, which $A_i$ is the matrix formed by replacing the ith column of $A$ by $Y$;
$R(A)=0 \iff A=O$, $R(A_{n\times{n}})=n \iff A \> \text{intertible}$; $R(kA)=R(A)(k\ne0)$, $R(A^T)=R(A)$, $R(A^*)=\begin{cases}n, &R(A)=n \\ 1, &R(A)=n-1 \\ 0, &R(A)\lt n-1\end{cases}$; $\Rk\bmat{A&O\\O&B}=R(A)+R(B)$;
Elementary transforms don’t change rank of a matrix; $A_{m\times{n}} \cong \bmat{I_{R(A)}&O\\O&O}_{m\times{n}}$;
2×2 matrix
Let $A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}$. $\det{A}=a_{11}a_{22}-a_{12}a_{21}$, $A^{-1}=\dfrac{1}{\det{A}}\begin{bmatrix}a_{22}&-a_{12}\\-a_{21}&a_{11}\end{bmatrix}$; $R(A)=1 \Longleftrightarrow A\ \text{rows and columns in proportion}$, $R(A)=0 \Longleftrightarrow A=\mathbf{O}$.
Characteristic polynomial: $\lambda^2-(a_{11}+a_{22})\lambda+(a_{11}a_{22}-a_{12}a_{21})$; $\lambda\mathbf{I}-A=\begin{bmatrix}\lambda-a_{11}&-a_{12}\\-a_{21}&\lambda-a_{22}\end{bmatrix}$
Eigenvalues: $\lambda=\dfrac{1}{2}\left(\left(a_{11}+a_{22}\right) \pm\sqrt{\left(a_{11}-a_{22}\right)^{2}+4a_{12}a_{21}}\right)$;
Vector
$\vec{a}+\vec{b}=\vec{b}+\vec{a}$, $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$, $k(l\cdot\vec{a})=(kl)\cdot\vec{a}$, $k(\vec{a}+\vec{b})=k\vec{a}+k\vec{b}$;
$|\vec{a}|=\sqrt{a_1^2+a_2^2+a_3^2}$, $|k\cdot\vec{a}|=|k|\cdot|\vec{a}|$; $\displaystyle{\left|\Prj_u{\vec{a}}\right|=|\vec{a}|\cdot\cos\left\lt \vec{a},u\right>}$, $\Prj_u(\vec{a}_1+\vec{a}_2)=\Prj_u\vec{a}_1+\Prj_u\vec{a}_2$; $|\vec{a}\pm\vec{b}|\le|\vec{a}|+|\vec{b}|$; $|\vec{a}\cdot\vec{b}|\le|\vec{a}|\cdot|\vec{b}|$;
$\cos\theta_x=\dfrac{a_1}{|\vec{a}|}$, $\cos\theta_y=\dfrac{a_2}{|\vec{a}|}$, $\cos\theta_z=\dfrac{a_3}{|\vec{a}|}$; $\vec{e}_\vec{a}=\dfrac{\vec{a}}{|\vec{a}|}=(\cos\theta_x,\cos\theta_y,\cos\theta_z)$; $\cos^2\theta_x+\cos^2\theta_y+\cos^2\theta_z=1$;
$\vec{a}\cdot\vec{b}=|\vec{a}|\cdot|\vec{b}|\cdot\cos\theta=a_1b_1+a_2b_2+a_3b_3$, $\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$, $(\lambda\cdot\vec{a})\cdot\vec{b}=\vec{a}\cdot(\lambda\cdot\vec{b})=\lambda\cdot(\vec{a}\cdot\vec{b})$, $(\vec{a}\pm\vec{b})\cdot\vec{c}=\vec{a}\cdot\vec{b}\pm\vec{a}\cdot\vec{c}$; $(\vec{a}\pm\vec{b})^2=\vec{a}^2\pm2\vec{a}\cdot\vec{b}+\vec{b}^2$, $\vec{a}^2=|\vec{a}|^2$;
$\vec{a}\perp\vec{b} \iff \vec{a}\cdot\vec{b}=0 \iff a_1b_1+a_2b_2+a_3b_3=0$; $\left|\Prj_\vec{a}\vec{b}\right|=\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|}=\vec{b}\cdot\vec{e}_\vec{a}$; $\displaystyle{\left\lt\vec{a},\vec{b}\right>}=\cos^{-1} \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|\cdot|\vec{b}|}=\cos^{-1} \dfrac{a_1b_1+a_2b_2+a_3b_3}{\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^3}}$, $\vec{a}\cdot\vec{b} \quad \begin{cases}>0,&\left\lt\vec{a},\vec{b}\right> \lt\dfrac{\pi}{2}\\\lt0,&\left\lt\vec{a},\vec{b}\right>>\dfrac{\pi}{2}\end{cases}$;
$\vec{a}\times\vec{b}=\vec{c} \> \equd{ \vec{c}\perp\vec{a}, \>\vec{c}\perp\vec{b}, \> \text{right-hand rule}} {|\vec{c}|=|\vec{a}|\cdot|\vec{b}|\cdot\sin\theta}= \vmat{\vec{i}&\vec{j}&\vec{k}\\a_1&a_2&a_3\\b_1&b_1&b_3}$; $\vec{a}\parallel\vec{b} \iff \vec{a}\times\vec{b}=0$, $\vec{a}\perp\vec{b} \iff |\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|$;
$\vec{a}\times\vec{b}=-\vec{b}\times\vec{a}$, $\vec{a}\times\vec{a}=0$; $(\lambda\vec{a})\times\vec{b}=\lambda(\vec{a}\times\vec{b})$; $(\vec{a}\pm\vec{b})\times\vec{c}=\vec{a}\times\vec{c}\pm\vec{b}\times\vec{c}$, $\vec{c}\times(\vec{a}\pm\vec{b})=\vec{c}\times\vec{a}\pm\vec{c}\times\vec{b}$;
$[\vec{a},\vec{b},\vec{c}]=(\vec{a}\times\vec{b})\cdot\vec{c}=\vmat{a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3}$; $[\vec{a},\vec{b},\vec{c}]=[\vec{b},\vec{c},\vec{a}]=[\vec{c},\vec{a},\vec{b}]$; $[\vec{a},\vec{b},(\lambda\vec{c}+\mu\vec{d})]=\lambda[\vec{a},\vec{b},\vec{c}]+\mu[\vec{a},\vec{b},\vec{d}]$;
Analytic Geometry
Basic Equations
$S_\text{cylinder}=2\pi{r}^2+2\pi{rl}$, $S_\text{cone}=\pi{r}^2+\pi{rl}$, $S_\text{sphere}=4\pi{r^2}$; $V_\text{cylinder}=Sh$, $V_\text{cone}=\dfrac{1}{3}Sh$, $V_\text{sphere}=\dfrac{4}{3}\pi{R^3}$;
$(\rho;\theta)\Longrightarrow(\rho\cdot\cos\theta,\>\rho\cdot\sin\theta)$, $(x,y)\Longrightarrow\left(\sqrt{x^2+y^2};\>\mathrm{atan2}(y,x)\right)$;
In any triangle:
$\dfrac{a}{\sin{A}}=\dfrac{b}{\sin{B}}=\dfrac{c}{\sin{C}}=2R$;
$c^2=a^2+b^2-2ab\cos{C}$, $\cos{A}=\dfrac{b^2+c^2-a^2}{2bc}$;
where $R=\dfrac{abc}{4\cdot Area}$ is the circumradius of the triangle;
$Area=\sqrt{p(p-a)(p-b)(p-c)}=rs$, where $s=\dfrac{1}{2}(a+b+c)$, $r$ denotes
the inradius;
Area of quadrilateral: $\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\theta}$, where $s=\dfrac{1}{4}(a+b+c+d)$, $\cos\theta=\cos[\frac{1}{2}(\angle{A}+\angle{C})]=\cos[\frac{1}{2}(\angle{B}+\angle{D})]$;
Distance from point $(x_0,y_0)$ to line $Ax+By+C=0$: $d=\dfrac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$;
Distance from point $(x_0,y_0,z_0)$ to plane $Ax+By+Cz+D=0$: $d=\dfrac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}$;
Distance between $y=mx+b$ and $y=mx+c$: $\dfrac{|b-c|}{\sqrt{m^2+1}}$;
Equations of Planar Straight Line
| 0) | General |
$Ax+By+C=0$ |
$m=-\dfrac{A}{B}$, $a=-\dfrac{C}{A}$, $b=-\dfrac{C}{B}$, $\vec{n}=(A,B)$ |
| 1) | Slope $m$ and y-intercept $b$ | $y=mx+b$ | $a=-\dfrac{b}{m}$ |
| 2) | Point $(x_0,y_0)$ and slope $m$ | $y-y_0=m(x-x_0)$ | $a=-\dfrac{y_0}{m}+x_0$, $b=y_0-mx_0$ |
| 3) | Points $(x_1,y_1)$ and $(x_2,y_2)$ | $\dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}$ | $m=\dfrac{y_2-y_1}{x_2-x_1}$, $a=\dfrac{x_1y_2-x_2y_1}{y_2-y_1}$, $b=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}$ |
| 4) | x and y intercepts $a$, $b$ | $\dfrac{x}{a}+\dfrac{y}{b}=1$ | $m=-\dfrac{b}{a}$ |
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m is the slope and a, b are x and y intercepts. $l_1 \parallel l_2 \iff m_1=m_2 $, $l_1=l_2 \iff m_1=m_2 \land b_1=b_2 $; $l_1 \perp l_2 \iff m_1m_2=-1 $; |
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Equations of Circle
| 1) | Center $C(a,b)$ and radius $r$ |
$(x-a)^2+(y-b)^2=r^2$ |
$\Rightarrow \> x^2+y^2+(-2a)x+(-2b)y+(a^2+b^2-r^2)=0$ |
| 2) | General | $x^2+y^2+Dx+Ey+F=0$ $\scriptsize{(D^2+E^2-4F>0)}$ |
$C=\left(-\dfrac{D}{2},-\dfrac{E}{2}\right)$, $r=\dfrac{1}{2}\sqrt{D^2+E^2-4F}$ |
Equation 2) is convenient in determining the equation of a circle through 3 given points. (Solve systems of linear equations. )
Quadratic Curves
Ellipse: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \quad (a>b>0)$; $x\in[-a,a]$, $y\in[-b,b]$, $c=\sqrt{a^2-b^2}$, $F=(\pm{c},0)$, $e=\dfrac{c}{a}\lt 1$;
Hyperbola: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1 \quad (a,b>0)$; $|x|\ge{a}$, $c=\sqrt{a^2+b^2}$, $F=(\pm{c},0)$, $e=\dfrac{c}{a}>1$; $\text{asymptote: }\dfrac{x}{a}\pm\dfrac{y}{b}=0$;
Parabola: $y^2=2px \quad (p>0)$; $\text{Focus: }\left(\dfrac{p}{2},0\right)$, $\text{Directrix: }x=-\dfrac{p}{2}$, $e=1$; $y=ax^2$: $F=\dfrac{1}{4a}$, $\text{directrix: } y=-\dfrac{1}{4a}$;
Equations of Plane
| 0) | General |
$Ax+By+Cz+D=0$ |
$\vec{n}=(A,B,C)$ |
| 1) | Normal $\vec{n}=(A,B,C)$ and point $P=(x_0,y_0,z_0)$ |
$A(x-x_0)+B(y-y_0)+C(z-z_0)=0$ $\small{\Longrightarrow Ax+By+Cz=Ax_0+By_0+Cz_0}$ |
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| 2) | Three points $P_i(x_i,y_i,z_i)$ $\scriptsize{(i=\{1,2,3\})}$ |
$\vmat{x-x_1&y-y_1&z-z_1\\x_2-x_1&y_2-y_1&z_2-z_1\\x_3-x_1&y_3-y_1&z_3-z_1}=0$ | |
| 3) | x, y and z intercepts $a$, $b$, $c$ | $\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$ | |
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$A=0$: parallel to x-axis; $A=B=0$: parallel to $xOy$ plane; $D=0$: through Origin; $\pi_1\perp\pi_2 \iff A_1A_2+B_1B_2+C_1C_2=0$, $\pi_1\parallel\pi_2 \iff \dfrac{A_1}{A_2}=\dfrac{B_1}{B_2}=\dfrac{C_1}{C_2}$; $\displaystyle\left\lt \pi_1,\pi_2\right> =\cos^{-1}\dfrac{|A_1A_2+B_1B_2+C_1C_2|}{\sqrt{A_1^2+B_1^2+C_1^2}\cdot\sqrt{A_2^2+B_2^2+C_2^2}}$; |
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Equations of Spatial Straight Line
| 1) | Point $P(x_0,y_0,z_0)$ and direction vector $\vec{s}=(a,b,c)$ |
$\dfrac{x-x_0}{a}=\dfrac{y-y_0}{b}=\dfrac{z-z_0}{c}$ | If $a$,$b$,or $c$ occurs as 0, then the numerator is also 0 |
| 2) | through $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$ |
$\dfrac{x-x_1}{x_2-x_1}=\dfrac{y-y_1}{y_2-y_1}=\dfrac{z-z_1}{z_2-z_1}$ | |
| 3) | Parameter | $\begin{cases}x=x_0+a\cdot{t}\\y=y_0+b\cdot{t}\\z=z_0+c\cdot{t}\end{cases}$ |
$P_0(x_0,y_0,z_0)$ and direction vector $\vec{s}=(a,b,c)$, same as #1; Usually for solving equations |
| 4) | Intersection of two planes | $\begin{cases}A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\end{cases}$ | $\vec{s}=\vmat{\vec{i}&\vec{j}&\vec{k}\\A_1&B_1&C_1\\A_2&B_2&C_2}$; |
Distance from point $M$ to line $P=P_0+\vec{s}\cdot{t}$: $d=\dfrac{|\vec{s}\times\vec{MP_0}|}{|\vec{s}|}$;
$l_1\parallel{l_2}\iff\vec{s}_1\parallel\vec{s}_2$, $l_1=l_2\iff\vec{s}_1\parallel\vec{s}_2\parallel\vec{P_1P_2}$; $l_1 \text{ intersect } l_2 \iff \vec{s}_1\nparallel\vec{s}_2\>\land\>[\vec{s_1},\>\vec{s_2},\>\vec{P_1P_2}]=0$; $\left\lt l_1,l_2\right>=\cos^{-1}\dfrac{|\vec{s}_1\cdot\vec{s}_2|}{|\vec{s}_1|\cdot|\vec{s}_2|}$, $l_1\perp{l_2}\iff\vec{s}_1\cdot\vec{s}_2=0$;
$l\parallel\pi\iff\vec{s}\cdot\vec{n}=0$, $l\subset\pi\iff\vec{s}\cdot\vec{n}=0\>\land\>Ax_0+By_0+Cz_0+D=0$; $l\text{ intersect }\pi\iff\vec{s}\cdot\vec{n}\ne0$, $\left\lt l,\pi\right>=\sin^{-1}\dfrac{|\vec{s}\cdot\vec{n}|}{|\vec{s}|\cdot|\vec{n}|}$;
Sheaf of Planes:
Let the equation of straight line $l$ be
$\equd{A_1x+B_1y+C_1z+D_1=0\qquad(1)}{A_2x+B_2y+C_2z+D_2=0\qquad(2)}$,
the equation of all planes through $l$ is given by: $\left(A_1x+B_1y+C_1z+D_1\right)+\lambda\left(A_2x+B_2y+C_2z+D_2\right)=0$, except #2 $(\lambda\rightarrow\infty)$;
Permutation and Combination
$n$ different types, $k$ choices:
Number of Orders: $n!$;
Permutations without Repetition: $P_k^n=\dfrac{n!}{(n-k)!}$; (Also denoted as $P^k_n$, $A^k_n$, $\sideset{^n}{_k}P$, $\sideset{_n}{_k}P$)
Permutations with Repetition: $U^n_k=n^k$;
Combinations without Repetition: $C^n_k=\dfrac{n!}{k!(n-k)!}$; (Also denoted as $C(n,k)$, $\dbinom{n}{k}$, $C^k_n$, $\sideset{^n}{_k}C$, $\sideset{_n}{_k}C$)
Combinations with Repetition: $H^n_k=\dfrac{(n+k-1)!}{k!(n-1)!}$; (Also denoted as $F^n_k$) $H^n_k=C^{n+k-1}_{k}=C^{n+k-1}_{n-1}$;
Number of Orders in Circular Permutation: $(n-1)!$;
$\dbinom{n}{k}=\dbinom{n}{n-k}=\dfrac{n}{k}\dbinom{n-1}{k-1}; \quad \dbinom{n}{k-1}+\dbinom{n}{k}=\dbinom{n+1}{k}, \quad \dbinom{n}{k}=\dbinom{n-1}{k-1}+\dbinom{n-1}{k}; \quad$
Statistics and Probability
Mean: $\mu=\displaystyle\frac{1}{N}\sum_{i=1}^{N}{x_i}$; Variance: $\sigma^2 =\displaystyle\frac{1}{N}\sum_{i=0}^{N}{(x_i-\mu)^2} =\frac{1}{N}\left(\sum_{i=1}^{N}{x_i^2}-\frac{1}{N}\left(\sum_{i=1}^{N}{x_i}\right)^2\right)$; Standard deviation: $\sigma=\sqrt{\sigma^2}$;
Variance of a sample: $s^2 =\displaystyle\frac{1}{n-1}\sum_{i=0}^{n}{(x_i-\mu)^2} =\frac{1}{n-1}\left(\sum_{i=1}^{n}{x_i^2}-\frac{1}{n}\left(\sum_{i=1}^{n}{x_i}\right)^2\right)$; Standard deviation of a sample: $s=\sqrt{s^2}$;
Random variable, expected value and variance
Discrete random variable: $\E(X)=\displaystyle\sum_i{x_i\P(x_i)}, \quad \var(X)=\displaystyle\sum_i{(x_i-\mu_X)\P(x_i)};$
Continuous random variable: $\E(X)=\displaystyle\int_{-\infty}^{\infty}{x\cdot p_X(x)\d{x}},\quad \var(X)=\displaystyle\int_{-\infty}^{\infty}{(x-\mu_X)^2\cdot p_X(x)\d{x}};$
$\E(aX+b)=a\E(X)+b,\quad \var(aX+b)=a^2\var(X),\quad \sigma_{aX+b}=|a|\cdot\sigma_X;$ $\var(X)=\E(X^2)-\E(X)^2;$ $\E(X\pm Y)=\E(X)\pm\E(Y);$
For independent random variables $X$, $Y$: $\E(XY)=\E(X)\E(Y);\quad \var(X+Y)=\var(X)+\var(Y),\ \ \sigma_{X+Y}=\sqrt{\sigma_X^2+\sigma_Y^2};$
* Probability distributions
Normal distribution: $PDF(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\exp\left(-\dfrac{\left(x-\mu\right)^{2}}{2\sigma^{2}}\right),\quad CDF(x)=\dfrac{1}{2}\left(1+\erf\left(\dfrac{x-\mu}{\sigma\sqrt{2}}\right)\right),\quad Quantile(p)=\mu+\sigma\sqrt{2}\operatorname{erf}^{-1}(2p-1); \\ \Pr(|X-\mu|\lt\sigma_X)\approx68.27\%,\quad \Pr(|X-\mu|\lt2\sigma_X)\approx95.45\%,\quad \Pr(|X-\mu|\lt3\sigma_X)\approx99.73\%;$
Exponential distribution: $x\ge0,\quad PDF(x)=\lambda\e^{-\lambda x},\quad CDF(x)=1-\e^{-\lambda x},\quad Quantile(p)=-\dfrac{\ln(1-p)}{\lambda};\qquad \mu=\dfrac{1}{\lambda},\quad \sigma^2=\dfrac{1}{\lambda^2};$
Binomial distribution: $\Pr(X=k)=\dbinom{n}{k}p^{k}(1-p)^{n-k};\qquad \mu=np,\quad \sigma^2=np(1-p);$
Geometric distribution with $k\ge1$:
$\Pr(X=k)=(1-p)^{k-1}p,\quad \Pr(X\le k)=1-(1-p)^{k};\qquad
\mu=\dfrac{1}{p},\quad \sigma^2=\dfrac{1-p}{p^2};$
Geometric distribution with $k\ge0$:
$\Pr(X=k)=(1-p)^{k}p,\quad \Pr(X\le k)=1-(1-p)^{k+1};\qquad
\mu=\dfrac{1-p}{p},\quad \sigma^2=\dfrac{1-p}{p^2};$
Poisson distribution: $PDF(k)=\dfrac{\lambda^{k}\e^{-\lambda}}{k!};\qquad \mu=\lambda,\quad \sigma^2=\lambda;$
* Transformations of probability density functions
Convolution of probability distributions: $\text{let }Z=X+Y,\quad \Pr(Z=z)=\displaystyle\sum_{k}{\Pr(X=k)\Pr(Y=z-k)},\quad p_Z(z)=\displaystyle\int_{-\infty}^{\infty}{p_X(z-t)p_Y(t)}\d{t}=\int_{-\infty}^{\infty}{p_X(t)p_Y(z-t)}\d{t};$
$\displaystyle\sum_{i=1}^{n}\mathrm{Normal}\left(\mu_i,\sigma_i^2\right) \sim\mathrm{Normal}\left(\sum_{i=1}^{n}\mu_i,\sum_{i=1}^{n}\sigma_i^2\right);\quad \sum_{i=1}^{n}\mathrm{Binomial}\left(n_i,p\right) \sim\mathrm{Binomial}\left(\sum_{i=1}^{n}{n_i},p\right);\quad \sum_{i=1}^{n}\text{Poisson}\left(\lambda_i\right) \sim\text{Poisson}\left(\sum_{i=1}^{n}{\lambda_i}\right);$
Miscellaneous Algebra
$a^3\pm b^3=(a\pm b)\left(a^2\mp ab+b^2\right), \quad a^4-b^4=(a-b)\left(a^3+a^2b+ab^2+b^3\right), \quad a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\cdots+a^2b^{n-3}+ab^{n-2}+b^{n-1}\right);$
$\displaystyle \frac{1}{ab}=\frac{1}{b-a}\left(\frac{1}{a}-\frac{1}{b}\right);\quad \frac{1}{\frac{1}{x}+\frac{1}{y}}=\frac{xy}{x+y},\quad \frac{1}{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}}=\frac{xyz}{xy+xz+yz};\quad \dfrac{1}{1-x}-\dfrac{1}{1+x}=\dfrac{2x}{1-x^{2}};$
Solutions of $ax^2+bx+c=0$: $x=\dfrac{\pm\sqrt{b^2-4ac}-b}{2a}$; Solutions of $ax^2+2bx+c=0$: $x=\dfrac{\pm\sqrt{b^2-ac}-b}{a}$;
Vertex of $ax^2+bx+c$: $\left(-\dfrac{b}{2a},\dfrac{4ac-b^{2}}{4a}\right);\quad$ Inflection point of $ax^3+bx^2+cx+d$: $\left(-\dfrac{b}{3a},\dfrac{27a^{2}d-9abc+2b^{3}}{27a^{2}}\right);\quad$
Let $x_1$, $x_2$ be the roots of $ax^2+bx+c$: $x_1+x_2=-\dfrac{b}{a},\quad x_1\cdot x_2=\dfrac{c}{a},\quad x_1^2+x_2^2=\dfrac{b^2-2ac}{a^2};$
Parabola through $(x_1,y_1),(x_2,y_2),(x_3,y_3)$:
$f(x)
=y_1\cdot\dfrac{\left(x-x_2\right)\left(x-x_3\right)}{\left(x_1-x_2\right)\left(x_1-x_3\right)}
+y_2\cdot\dfrac{\left(x-x_1\right)\left(x-x_3\right)}{\left(x_2-x_1\right)\left(x_2-x_3\right)}
+y_3\cdot\dfrac{\left(x-x_1\right)\left(x-x_2\right)}{\left(x_3-x_1\right)\left(x_3-x_2\right)};$
$f(x)=ax^2+bx+c,\quad
a=\dfrac{x_1\left(y_3-y_2\right)+x_2\left(y_1-y_3\right)+x_3\left(y_2-y_1\right)}{\left(x_1-x_2\right)\left(x_1-x_3\right)\left(x_2-x_3\right)}
=\dfrac{y_1\left(x_2-x_3\right)+y_2\left(x_3-x_1\right)+y_3\left(x_1-x_2\right)}{\left(x_1-x_2\right)\left(x_1-x_3\right)\left(x_2-x_3\right)},
\\
b=-\dfrac{x_1^2\left(y_3-y_2\right)+x_2^2\left(y_1-y_3\right)+x_3^2\left(y_2-y_1\right)}{\left(x_1-x_2\right)\left(x_1-x_3\right)\left(x_2-x_3\right)}
=-\dfrac{y_1\left(x_2^2-x_3^2\right)+y_2\left(x_3^2-x_1^2\right)+y_3\left(x_1^2-x_2^2\right)}{\left(x_1-x_2\right)\left(x_1-x_3\right)\left(x_2-x_3\right)},
\\
c=\dfrac{x_1^2\left(x_2y_3-x_3y_2\right)+x_2^2\left(x_3y_1-x_1y_3\right)+x_3^2\left(x_1y_2-x_2y_1\right)}{\left(x_1-x_2\right)\left(x_1-x_3\right)\left(x_2-x_3\right)}
=\dfrac{y_1\left(x_2x_3\left(x_2-x_3\right)\right)+y_2\left(x_3x_1\left(x_3-x_1\right)\right)+y_3\left(x_1x_2\left(x_1-x_2\right)\right)}{\left(x_1-x_2\right)\left(x_1-x_3\right)\left(x_2-x_3\right)};$
Sequence and Series
Arithmetic Sequence: $a_n=a_1+d(n-1)$, $n=\dfrac{a_n-a_1}{d}+1$, $d=\dfrac{a_n-a_1}{n-1}$; $S_n=\dfrac{n(a_1+a_n)}{2}=na_1+\dfrac{n(n-1)}{2}d$, $P_n=d^{n}{\dfrac{\Gamma\left(a_{1}/d+n\right)}{\Gamma\left(a_{1}/d\right)}} \; (a_1/d\notin{N^-})$;
Geometric Sequence: $a_n=a_1\cdot q^{n-1}$, $n=\log_q\dfrac{a_n}{a_1}+1$, $q=\displaystyle\left(\frac{a_n}{a_1}\right)^{\frac{1}{n-1}}$; $S_n=\dfrac{a_1(q^n-1)}{q-1}=\dfrac{a_nq-a_1}{q-1}$; $P_n=\displaystyle a_1^n \cdot q^{n(n-1)\over2}$;
$\displaystyle\sum_{k=0}^{\infty}{r^k}=\begin{cases} +\infty & r\ge1 \\[3pt] \dfrac{1}{1-r} & |r|\lt 1 \\[3pt] \text{diverge} & r\le-1 \end{cases}$; $\displaystyle\sum_{k=1}^n{r^k}=\frac{r(1-r^n)}{(1-r)}, \quad \displaystyle\sum_{k=0}^n{r^k}=\frac{1-r^{n+1}}{1-r};$
$\displaystyle{\sum_{k=1}^{n}k=\frac{1}{2}n\left(n+1\right)}, \quad \displaystyle{\sum_{k=1}^{n}k^{2}=\frac{1}{6}n\left(n+1\right)\left(2n+1\right)}, \quad \displaystyle{\sum_{k=1}^{n}k^{3}=\frac{1}{4}n^{2}\left(n+1\right)^{2}}, \quad \displaystyle{\sum_{k=1}^{n}k^{4}=\frac{1}{30}n\left(n+1\right)\left(2n+1\right)\left(3n^{2}+3n-1\right)}; \quad$
$(a+b)^n=\displaystyle\sum_{k=0}^{n}{\binom{n}{k}a^{k}b^{n-k}}=\sum_{k=0}^n{\binom{n}{k}a^{n-k}b^{k}}, \quad (1+x)^n=\displaystyle\sum_{k=0}^n{\binom{n}{k}x^k}, \quad (1-x)^n=\displaystyle\sum_{k=0}^n{\binom{n}{k}(-1)^{k}x^{k}}, \quad (x-1)^n=\displaystyle\sum_{k=0}^n{\binom{n}{k}(-1)^{n-k}x^{k}};$
Taylor and Maclaurin Series
$\displaystyle\frac{1}{1-x}=\sum_{k=0}^{\infty}x^{k}=1+x+x^{2}+x^{3}+\cdots, \quad$ $\displaystyle\frac{1}{1+x}=\sum_{k=0}^{\infty}\left(-1\right)^{k}x^{k}=1-x+x^{2}-x^{3}+\cdots,\quad$ $\displaystyle\ln\left(1+x\right)=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}x^{k+1}}{k+1}=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots;\quad$ $(|x|\le1)$
$\displaystyle\frac{1}{1-x^{2}}=\sum_{k=0}^{\infty}x^{2k}=1+x^{2}+x^{4}+x^{6}+\cdots,\quad$ $\displaystyle\frac{1}{1+x^{2}}=\sum_{k=0}^{\infty}\left(-1\right)^{k}x^{2k}=1-x^{2}+x^{4}-x^{6}+\cdots,\quad$
$\displaystyle\arctan\left(x\right)=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}x^{2k+1}}{2k+1}=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\cdots,\quad$ $\displaystyle\operatorname{arctanh}\left(x\right)=\sum_{k=0}^{\infty}\frac{x^{2k+1}}{2k+1}=x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\frac{x^{7}}{7}+\cdots;\quad$ $(|x|\le1)$
$\displaystyle\e^{x}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots,\quad$ $\displaystyle\sin\left(x\right)=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}x^{2k+1}}{\left(2k+1\right)!}=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots,\quad$ $\displaystyle\cos\left(x\right)=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}x^{2k}}{\left(2k\right)!}=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\cdots,\quad$
$\displaystyle\sinh\left(x\right)=\sum_{k=0}^{\infty}\frac{x^{2k+1}}{\left(2k+1\right)!}=x+\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\frac{x^{7}}{7!}+\cdots,\quad$ $\displaystyle\cosh\left(x\right)=\sum_{k=0}^{\infty}\frac{x^{2k}}{\left(2k\right)!}=1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\frac{x^{6}}{6!}+\cdots;\quad$
Fourier Series and Integral
Function $f$ with period $T$: $f(t)=\dfrac{a_{0}}{2} +\displaystyle\sum_{k=1}^{\infty}\left(a_{n}\cos(k\omega t) +b_{n}\sin(k\omega t)\right)$, where $a_k=\dfrac{2}{T}\displaystyle\int_{t_0}^{t_0+T}{f(t)\cos(k\omega t)\d{t}},\quad b_k=\dfrac{2}{T}\displaystyle\int_{t_0}^{t_0+T}{f(t)\sin(k\omega t)\d{t}},\quad \omega=\dfrac{2\pi}{T}$;
or $f(t) = \displaystyle\sum_{k=-\infty}^{\infty}c_n\e^{ki\omega t}, \quad c_k=\dfrac{1}{T}\displaystyle\int_{t_0}^{t_0+T}f(t)\e^{-ki\omega t}\d{t},\quad \omega=\dfrac{2\pi}{T}$; $c_{k} = \overline{c}_{-k}$ for real $f$;
When $T=2\pi$: $\displaystyle f(x)=\frac{a_{0}}{2}+\sum_{k=1}^{\infty}\left(a_{k}\cos(kx)+b_{k}\sin(kx)\right),\quad a_{k}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(kx)\d{x}, \quad b_{k}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(kx)\d{x}$; $\displaystyle \dfrac{1}{\pi}\int_{-\pi}^{\pi}f(x)^2\d{x} =\dfrac{a_0^2}{2}+\sum_{k=1}^{\infty}\left(a_k^2+b_k^2\right)$
Fourier integral: $\displaystyle f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\d{\lambda}\int_{-\infty}^{\infty}{f(t)\e^{-i\lambda(t-x)}}\d{t} =\frac{1}{2\pi}\int_{-\infty}^{\infty}C(\lambda)\e^{i\lambda x}\d{\lambda},\quad C(\lambda)=\int_{-\infty}^{\infty}{f(t)\e^{-i\lambda t}\d{t}}$;
or $\displaystyle f(x)=\frac{1}{\pi}\int_{0}^{\infty}\left(A(\lambda)\cos\lambda x + B(\lambda)\sin\lambda x\right)\d\lambda,\quad A(\lambda)=\int_{-\infty}^{\infty}{f(t)\cos\lambda t}\ \d{t},\quad B(\lambda)=\int_{-\infty}^{\infty}{f(t)\sin\lambda t}\ \d{t}$;
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Fourier Transform
Fourier transform and its inverse: $\displaystyle \hat{f}(\xi)=\int_{-\infty}^{\infty}{f(x)\e^{-2\pi i\xi x}\d{x}},\quad f(x)=\int_{-\infty}^{\infty}{\hat{f}(\xi)\e^{2\pi i\xi x}\d{\xi}};$
$\F\{af(x)+bg(x)\}=a\hat{f}(\xi)+b\hat{g}(\xi),\quad \F\{f(x-x_0)\}=\e^{-2\pi i\xi x_0}\hat{f}(\xi),\quad \F\left\{\e^{2\pi i\xi_0x}f(x)\right\}=\hat{f}(\xi-\xi_0),\quad\\ \F\{f(ax)\}=\dfrac{1}{|a|}\hat{f}\left(\dfrac{\xi}{a}\right)\;(a\in\R),\quad \F\{f(-x)\}=\hat{f}(-\xi),\quad \F\left\{\overline{f(x)}\right\}=\overline{\hat{f}(-\xi)};\quad \hat{f}(-\xi)=\overline{\hat{f}(\xi)}\quad(f\in\R);$
$\F\left\{\dfrac{\d}{\d{x}}f(x)\right\}=2\pi i\xi\cdot\hat{f}(\xi),\quad
\F\left\{\dfrac{\d^n}{\d{x^n}}\right\}=(2\pi i\xi)^n\hat{f}(\xi);\quad$
Discrete Fourier Transform
Discrete Fourier transform and its inverse: $\displaystyle X_k=\sum_{n=0}^{N-1}{x_n\exp\left(-2\pi ik\frac{n}{N}\right)},\quad x_n=\frac{1}{N}\sum_{k=0}^{N-1}{X_k\exp\left(2\pi in\frac{k}{N}\right)};$
$\F\left(\left\{ax_n+by_n\right\}\right)_k=aX_k+bY_k,\quad \F\left(\left\{x_{-n}\right\}\right)_k=X_{-k},\quad \F\left(\left\{\overline{x_n}\right\}\right)_k=\overline{X_{-k}},\quad \F\left(\left\{x_{n-t}\right\}\right)_k=\e^{-2\pi it\frac{k}{N}}X_k,\quad \F\left(\left\{\e^{2\pi it\frac{n}{N}}x_{n}\right\}\right)_k=X_{k-t};\quad$
$\displaystyle \F^{-1}\left(\left\{X_{k}Y_{k}\right\}\right)_n =\sum_{t=0}^{N-1}x_{t}\cdot y_{n-t},\quad \F\left(\left\{x_{n}y_{n}\right\}\right)_k =\frac{1}{N}\sum_{t=0}^{N-1}x_{t}\cdot y_{k-t};\quad$
$\displaystyle \sum_{n=0}^{N-1}x_n\overline{y_n} =\frac{1}{N}\sum_{k=0}^{N-1}X_k\overline{Y_k},\quad \sum_{n=0}^{N-1}|x_n|^2 =\dfrac{1}{N}\sum_{k=0}^{N-1}|X_k|^2;\quad$