General Formulas
| Various Formulas | |
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| place note here | place formula here credit |
| place note here | place formula here |
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keywords: vector, gradient, curl, laplacian
Vector Identities and Operator Definitions
(Used in ECE311)
| Vector Identities and Operator Definitions | |
|---|---|
| Vector Identities | |
| Notes | Identity |
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$ \bold{x}\cdot \left(\bold{y}\times \bold{z}\right)= \left(\bold{x}\times \bold{y}\right)\cdot \bold{z} $ | |
| $ \bold{x}\times \left(\bold{y}\times \bold{z} \right)=\bold{y}\left(\bold{x} \cdot \bold{z} \right)-\bold{z} \left( \bold{x}\cdot\bold{y}\right) $ | |
| $ \left( \bold{x}\times \bold{y}\right)\cdot \left(\bold{z}\times \bold{w} \right)=\left( \bold{x}\cdot \bold{z}\right) \left(\bold{y} \cdot \bold{w} \right)- \left(\bold{x}\cdot\bold{w} \right) \left( \bold{y}\cdot\bold{z}\right) $ | |
| $ \nabla \left( \bold{x}\cdot \bold{y}\right)= \bold{y}\times \left(\nabla\times \bold{x}\right)+ \bold{x} \times \left(\nabla\times \bold{y} \right)+ \left(\bold{y}\cdot\nabla \right)\bold{x} + \left( \bold{x}\cdot\nabla\right) \bold{y} $ | |
| $ \nabla \left( f+g \right)= \nabla f+ \nabla g $ | |
| $ \nabla \left( f g \right)= f \nabla g+ g\nabla f $ | |
| $ \nabla \cdot \left( \bold{x}+\bold{y} \right)= \nabla \cdot \bold{x} + \nabla \cdot \bold{y} $ | |
| $ \nabla \cdot \left( f \bold{x}\right)= \bold{x} \cdot \nabla f + f \left( \nabla \cdot\bold{x} \right) $ | |
| $ \nabla \times \left( \bold{x} + \bold{y} \right)= \nabla \times \bold{x} + \nabla \times \bold{y} $ | |
| $ \nabla \times \left( u \bold{x} \right)= \left( \nabla u \right) \times \bold{x} + u \left( \nabla \times \bold{x} \right) $ | |
| $ \nabla \cdot \left( \bold{x}\times \bold{y}\right)= \bold{y} \cdot \left( \nabla \times \bold{x}\right) - \bold{x} \cdot \left( \nabla \times \bold{y}\right) $ | |
| $ \nabla \cdot \left(\nabla\times \bold{x} \right)= 0 $ | |
| $ \nabla \times \left( \bold{x} \times \bold{y} \right) = \left( \nabla \cdot \bold{y} \right) \bold{x} - \left( \nabla \cdot \bold{x} \right) \bold{y} + \left( \bold{y} \cdot \nabla \right) \bold{x} - \left( \bold{x} \cdot \nabla \right) \bold{y} $ | |
| $ \nabla \times \nabla \bold{x} = 0 $ | |
| $ \nabla ( \bold{C} \cdot \bold{r} ) = \bold{C} \qquad \text{where }\bold{C}\text{ is a constant (real or complex)} $ | |
| $ \nabla \times \left( \nabla \times \bold{x} \right) = \nabla \left( \nabla \cdot \bold{x} \right) - \nabla^2 \bold{x} $ | |
| $ \left( \bold{A} \cdot \nabla \right) \bold{B} = \hat{\bold{x}} ( \bold{A}_x \frac{\partial \bold{B}_x}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_x}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_x}{\partial z} ) + \hat{\bold{y}} ( \bold{A}_x \frac{\partial \bold{B}_y}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_y}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_y}{\partial z} ) + \hat{\bold{z}} ( \bold{A}_x \frac{\partial \bold{B}_z}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_z}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_z}{\partial z} ) $ | |
| $ \frac{d \left( \bold{x} \cdot \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\cdot \bold{x} + \frac{d \bold{x}}{d\sigma}\cdot \bold{y} $ | |
| $ \frac{d \left( \bold{x} \times \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\times \bold{x} + \frac{d \bold{x}}{d\sigma}\times \bold{y} $ | |
| $ \frac {d ( u \bold{v} )}{d \sigma} = \frac {d u}{d \sigma} \bold{v} + u \frac{d \bold{v}}{d \sigma} $ | |
| Vector Operators in Rectangular Coordinates | ||
|---|---|---|
| Notes | Operator | |
| $ \nabla f(x,y,z) = \mathbf{\hat x} \frac{\partial f}{\partial x}+\mathbf{\hat y}\frac{\partial f}{\partial y}+\mathbf{\hat z} \frac{\partial f}{\partial z} $ | ||
| $ \nabla \cdot \bold{v} = \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+ \frac{\partial v_z}{\partial z} $ | ||
| $ \nabla \times \bold{v} = \mathbf{\hat x} \left( \frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z} \right) + \mathbf{\hat y} \left( \frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x} \right) + \mathbf{\hat z} \left( \frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y} \right) $ | ||
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$ \nabla^2 f = \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z^2} $ | |
| Vector Operators in Cylindrical Coordinates | |
|---|---|
| Notes | Operator |
| $ \nabla f(\rho,\phi,z) = {\partial f \over \partial \rho}\boldsymbol{\hat \rho} + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} + {\partial f \over \partial z}\boldsymbol{\hat z} $ | |
| $ \nabla \cdot \bold{v} = \frac{1}{\rho} \frac{\partial \rho v_{\rho}}{\partial \rho} + \frac{1}{\rho} \frac{\partial v_{\phi}}{\partial \phi} + \frac{\partial v_z}{\partial z} $ | |
| $ \nabla \times \bold{v} = \boldsymbol{\hat \rho} ( \frac{1}{\rho} \frac{\partial \bold{v}_z }{\partial \phi} - \frac{\partial \bold{v}_\phi}{\partial z} ) + \boldsymbol{\hat \phi} ( \frac{\partial \bold{v}_\rho}{\partial z} - \frac{\partial \bold{v}_z}{\partial \rho} ) + \hat{\bold{z}} ( \frac{1}{\rho} \frac{\partial ( \rho \bold{v}_\phi )}{\partial \rho} - \frac{1}{\rho} \frac{\partial \bold{v}_\rho}{\partial \phi} ) $ | |
| $ \nabla^2 f = \frac{1}{\rho} \frac{\partial }{\partial \rho} \left( \rho \frac{\partial f}{\partial \rho}\right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2} $ | |
| Vector Operators in Spherical Coordinates | |
|---|---|
| Notes | Operator |
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$ \nabla f(x,y,z) = {\partial f \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} $ |
| $ \nabla \cdot \bold{v} = \frac{1}{r^2} \frac{\partial r^2 v_r}{\partial r} + \frac{1}{r\sin\theta} \frac{\partial \sin\theta v_{\theta}}{\partial \theta} + \frac{1}{r\sin\theta} \frac{\partial v_{\phi}}{\partial \phi} $ | |
| $ \nabla \times \bold{v} = \frac{\boldsymbol{\hat r } }{r \sin \theta} [ \frac{\partial ( \sin \theta \bold{v}_\phi )}{\partial \theta} - \frac{\partial \bold{v}_\theta}{\partial \phi} ] + \frac { \boldsymbol{\hat \theta} }{r} [ \frac{1}{\sin \theta} \frac{\partial \bold{v}_r}{\partial \phi} - \frac{\partial ( r \bold{v}_\phi )}{\partial r} ] + \frac {\boldsymbol{\hat \phi} }{r} [ \frac{\partial ( r \bold{v}_\theta )}{\partial r} - \frac{\partial \bold{v}_r}{\partial \theta} ] $ | |
| $ \nabla^2 f = \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{\partial f}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial }{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 f}{\partial \phi^2} $ | |
| Vector Integral formulas | |
|---|---|
| Notes | Operator |
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$ \oint_S \bold{A} \cdot d \bold{a} = \int_V \nabla \cdot \bold{A} d \tau \qquad \text{(Divergence therorem)} $ |
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$ \oint_C \bold{A} \cdot d \bold{s} = \int_S ( \nabla \times \bold{A} ) \cdot d \bold{a} \qquad \textrm{(Stokes' therorem)} $ |
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$ \oint_S u d \bold{a} = \int_V \nabla u d \tau $ |
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$ \oint_S \bold{A} \times d \bold{a} = - \int_V ( \nabla \times \bold{A} ) d \tau $ |
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$ \oint_C u d \bold{s} = - \int_S \nabla u \times d \bold{a} $ |
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$ \oint_S u \bold{A} \cdot d \bold{a} = \int_V [ \bold{A} \cdot ( \nabla u ) + u ( \nabla \cdot \bold{A} ) ]d \tau $ |
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$ \oint_S \bold{B} ( \bold{A} \cdot d \bold{a} ) = \int_V [( \bold{A} \cdot \nabla ) \bold{B} + \bold{B} ( \nabla \cdot \bold{A}) ] d \tau $ |
| Formulas Involving Relative Coordinates | |
|---|---|
| Notes | Operator |
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$ \frac{\partial f ( \bold{R} )}{\partial x} = - \frac{\partial f ( \bold{R} )}{\partial x^{'}} $ |
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$ \nabla f ( \bold{R} ) = - \nabla^{'} f ( \bold{R} ) $ |
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$ \nabla \cdot \bold{A} ( \bold{R} ) = - \nabla^{'} \cdot \bold{A} ( \bold{R} ) $ |
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$ \nabla \times \bold{A} ( \bold{R} ) = - \nabla^{'} \times \bold{A} ( \bold{R} ) $ |
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$ \nabla^2 f ( \bold{R} )= \nabla^{'2} f ( \bold{R} ) $ |
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$ \nabla R = - \nabla^{'} R = \frac{\bold{R}}{R} = \hat{\bold{R}} $ |
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$ \nabla ( \frac{1}{R} ) = - \nabla^{'} ( \frac{1}{R} ) = - \frac{\hat{\bold{R}}}{R^2} = - \frac{\bold{R}}{R^3} $ |
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$ \nabla^2 ( \frac{1}{R} )= \nabla^{'2} ( \frac{1}{R} ) = 0 \qquad ( \ R \neq 0 \ ) $ |
keywords: magnitude, conjugate, de Moivre, Euler
Complex Number Identities and Formulas
click here for more formulas
| Complex Number Identities and Formulas (info) | |
|---|---|
| Basic Definitions | |
| imaginary number | $ i=\sqrt{-1} \ $ |
| electrical engineers' imaginary number | $ j=\sqrt{-1}\ $ |
| (info) conjugate of a complex number | $ \text{if}\ z=a+ib,\ \text{for}\ a,\ b \in {\mathbb R},\ \text{then} \ \bar{z}=a-ib $ |
| (info) magnitude of a complex number | $ \| z \| = \sqrt{ z \bar{z} } $ |
| (info) magnitude of a complex number | $ \| z \| = \sqrt{\left(Re(z)\right)^2+\left(Im(z)\right)^2} $ |
| (info) magnitude of a complex number | $ \| a+ib \| = \sqrt{a^2+b^2},\ \text{for}\ a,b\in {\mathbb R} $ |
| (info) magnitude of a complex number | $ \| r e^{i \theta} \| = r,\ \text{for}\ r,\theta\in {\mathbb R} $ |
| Complex Number Operations | |
| addition | $ (a+ib)+(c+id)=(a+c) + i (b+d) \ $ |
| multiplication | $ (a+ib) (c+id)=(ac-bd) + i (ad+bc) \ $ |
| multiplication in polar form | $ \left( r_1 (\cos \theta_1 + i \sin \theta_1) \right) \left( r_2 (\cos \theta_2 + i \sin \theta_2) \right)= r_1 r_2 \left( \cos (\theta_1+\theta_2)+i \sin (\theta_1-\theta_2) \right)\ $ |
| division | $ \frac{a+ib} {c+id}=\frac{ac+bd} {c^2+d^2}+ i \frac{bc-ad} {c^2+d^2} \ $ |
| division in polar form | $ \frac{ r_1 (\cos \theta_1 + i \sin \theta_1)}{ r_2 (\cos \theta_2 + i \sin \theta_2) }= \frac{r_1}{ r_2} \left( \cos (\theta_1-\theta_2)+i \sin (\theta_1+\theta_2) \right)\ $ |
| exponentiation | $ i^n =\left\{ \begin{array}{ll}1,& \text{when }n\equiv 0\mod 4 \\ i,& \text{when }n\equiv 1\mod 4 \\-1,& \text{when }n\equiv 2\mod 4 \\-i,& \text{when }n\equiv 3\mod 4 \end{array} \right. \ $ |
| Euler's Formula and Related Equalities (info) | |
| (info) Euler's formula | $ e^{iw_0t}=\cos w_0t+i\sin w_0t \ $ |
| A really cute formula | $ e^{i\pi}=-1 \ $ |
| Cosine function in terms of complex exponentials | $ \cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2} $ |
| Sine function in terms of complex exponentials | $ \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i} $ |
| Other Formulas | |
| De Moivre's theorem | $ \left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right).\, $ |
| Root of a complex number | $ \left( r (\cos x+i\sin x) \right)^{\frac{1}{n}}=r^{\frac{1}{n}} \cos\left(\frac{x+2 k \pi}{n}\right) +i\sin\left(\frac{x+2 k \pi}{n} \right), k=0,1,\ldots, n-1.\, $ |
keywords: Taylor, Geometric, Binomial
Power Series
| Taylor Series Formulas | ||
|---|---|---|
| Series in symbolic forms | ||
| $ \text{Taylor Series in one variable } = \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n} $ (info) | ||
| $ \text{Taylor Series in } d \text{ variables } =\sum_{n_1=0}^{\infin} \cdots \sum_{n_d=0}^{\infin} \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d).\! $ | ||
| Taylor Series to remember | ||
| $ \text{Exponential } e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, \text{ for all } x\in {\mathbb C}\ $ | ||
| $ \text{Logarithm } \ln (1+x) = \sum^{\infin}_{n=1} (-1)^{n+1}\frac{x^n}n,\text{ when }-1<x\leq 1 $ | ||
| $ \sin x \ = \ x \ - \ \frac{x^3}{3!} \ + \ \frac{x^5}{5!} \ - \ \frac{x^7}{7!} \ + \ \cdots, \quad \text{ for } - \infty < x < \infty $ | ||
| $ \cos x \ = \ 1 \ - \ \frac{x^2}{2!} \ + \ \frac{x^4}{4!} \ - \ \frac{x6}{6!} \ + \ \cdots, \quad \text{ for } - \infty < x < \infty $ | ||
| Geometric Series and related series | ||
| (info) $ \text{Finite Geometric Series Formula } \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &, \text{ else}\end{array}\right. $ | ||
| (info) $ \text{Infinite Geometric Series Formula } \sum_{k=0}^\infty x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } |x|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right. $ | ||
| $ \frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n, \quad\mbox{ for }|x| < 1 \text{ and } m\in\mathbb{N}_0\! $ | ||
| $ \frac{x}{(1-x)^2} = \sum^{\infin}_{n=1}n x^n, \quad\text{ for }|x| < 1\! $ | ||
| Taylor series of Single Variable Functions | ||
| $ \,f(x) \ = \ f(a) \ + \ f'(a)(x \ - \ a) \ + \ \frac{f''(a)(x-a)^2}{2!} \ + \ \cdot \cdot \cdot \ + \ \frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!} \ + \ R_n \, $ | ||
| $ \text{Rest of Lagrange } \qquad R_n = \frac {f^{(n)}(\zeta)(x-a)^n}{n!} $ | ||
| $ \text{Rest of Cauchy } \qquad R_n = \frac {f^{(n)}(\zeta)(x-\zeta)^{n-1}(x-a)}{(n-1)!} $ | ||
| Binomial Series | ||
| For any positive integer n: | ||
| $ \begin{align} (a+x)^n & = \sum_{k=0}^n \left( \begin{array}{ll}n\\k \end{array}\right) x^k a^{n-k}\\ & = a^n + \binom{n}{1} a^{n-1}x + \binom{n}{2} a^{n-2}x^2 + \binom{n}{3} a^{n-3}x^3 + \ldots + x^n \\ \end{align} $ | ||
| For any complex number z: | ||
| $ \begin{align} (a+x)^z & = a^z + za^{z-1}x + \frac {z(z-1)}{2!} a^{z-2}x^2 + \frac {z(z-1)(z-2)}{3!} a^{z-3}x^3 + \ldots \\ & = a^z + \binom{z}{1} a^{z-1}x + \binom{z}{2} a^{z-2}x^2 + \binom{z}{3} a^{z-3}x^3 + \ldots \\ \end{align} $ | ||
| Some particular Cases: | ||
| $ (a+x)^2 \ = \ a^2 \ + \ 2ax \ + \ x^2 $ | ||
| $ (a+x)^3 \ = \ a^3 \ + \ 3a^2x \ + \ 3ax^2 \ + \ x^3 $ | ||
| $ (a+x)^4 \ = \ a^4 \ + \ 4a^3x \ + \ 6a^2x^2 \ + \ 4ax^3 \ + \ x^4 $ | ||
| $ (1+x)^{-1} \ = \ 1 \ - \ x \ + \ x^2 \ - \ x^3 \ + \ x^4 \ - \ \cdots $ | $ -1 < x < 1 \qquad $ | |
| $ (1+x)^{-2} \ = \ 1 \ - \ 2x \ + \ 3x^2 \ - \ 4x^3 \ + \ 5x^4 \ - \ \cdots $ | $ -1 < x < 1 \qquad $ | |
| $ (1+x)^{-3} \ = \ 1 \ - \ 3x \ + \ 6x^2 \ - \ 10x^3 \ + \ 15x^4 \ - \ \cdots $ | $ -1 < x < 1 \qquad $ | |
| $ (1+x)^{-1/2} \ = \ 1 \ - \ \frac{1}{2}x \ + \ \frac{1 \cdot 3}{2 \cdot 4}x^2 \ - \ \frac {1 \cdot 3 \cdot 5 }{2 \cdot 4 \cdot 6} x^3 \ + \ \cdots $ | $ -1 < x \leqq 1 \qquad $ | |
| $ (1+x)^{1/2} \ = \ 1 \ + \ \frac{1}{2}x \ - \ \frac{1 }{2 \cdot\ 4}x^2 \ + \ \frac {1 \cdot 3}{2 \cdot 4 \cdot 6} x^3 \ - \ \cdots $ | $ -1 < x \leqq 1 \qquad $ | |
| $ (1+x)^{-1/3} \ = \ 1 \ - \ \frac{1}{3}x \ + \ \frac{1 \cdot 4}{3 \cdot 6}x^2 \ - \ \frac {1 \cdot 4 \cdot 7 }{3 \cdot 6 \cdot 9} x^3 \ + \ \cdots $ | $ -1 < x \leqq 1 \qquad $ | |
| $ (1+x)^{1/3} \ = \ 1 \ + \ \frac{1}{3}x \ - \ \frac{2}{3 \cdot 6}x^2 \ + \ \frac {2 \cdot 5 }{3 \cdot 6 \cdot 9} x^3 \ - \ \cdots $ | $ -1 < x \leqq 1 \qquad $ | |
| Series Expansion of Exponential functions and Logarithms | ||
| $ e^x \ = \ 1 \ + \ x \ + \ \frac{x^2}{2!} \ + \ \frac{x^3}{3!} \ + \ \cdots $ | $ - \infty < x < \infty \qquad $ | |
| $ a^x \ = \ e^{x \ln a} \ = \ 1 \ + \ x \ln a \ + \ \frac{(x \ln a)^2}{2!} \ + \ \frac{(x \ln a)^3}{3!} \ + \ \cdots $ | $ - \infty < x < \infty \qquad $ | |
| $ \ln(1+x) \ = \ x \ - \ \frac{x^2}{2} \ + \ \frac{x^3}{3} \ - \ \frac{x^4}{4} \ + \ \cdots $ | $ -1 < x \leqq 1 \qquad $ | |
| $ \frac{1}{2} \ln \left ( \frac {1+x}{1-x} \right ) \ = \ x \ + \ \frac{x^3}{3} \ + \ \frac {x^5}{5} \ + \ \frac{x^7}{7} \ + \ \cdots \ $ | $ -1 < x < 1 \qquad $ | |
| $ \ln x \ = \ 2 \left \{ \left ( \frac {x-1}{x+1} \right ) \ + \ \frac{1}{3} \left ( \frac {x-1}{x+1} \right ) ^3 \ + \ \frac{1}{5} \left ( \frac{x-1}{x+1} \right ) ^ 5 \ + \ \cdots \ \right \} $ | $ x > 0 \qquad $ | |
| $ \ln x \ = \ \left ( \frac {x-1}{x} \right ) \ + \ \frac{1}{2} \left ( \frac {x-1}{x} \right ) ^2 \ + \ \frac{1}{3} \left ( \frac{x-1}{x} \right ) ^ 3 \ + \ \cdots \ $ | $ x \geqq \frac {1}{2} \qquad $ | |
| Series Expansion of Circular functions | ||
| $ \sin x \ = \ x \ - \ \frac{x^3}{3!} \ + \ \frac{x^5}{5!} \ - \ \frac{x^7}{7!} \ + \ \cdots $ | $ - \infty < x < \infty $ | |
| $ \cos x \ = \ 1 \ - \ \frac{x^2}{2!} \ + \ \frac{x^4}{4!} \ - \ \frac{x6}{6!} \ + \ \cdots $ | $ - \infty < x < \infty $ | |
| $ \cot x \ = \ \frac{1}{x} \ - \ \frac {x}{3} \ - \ \frac{x^3}{45} \ - \ \frac{2x^5}{945} \ - \ \cdots \ - \ \frac{2^{2n}B_n x^{2n-1}}{(2n)!} \ - \ \cdots $ | $ 0 < \left \vert x \right \vert < \pi \qquad $ | |
| $ \frac{1}{\cos x} \ = \ 1 \ + \ \frac {x^2}{2} \ + \ \frac{x^4}{24} \ + \ \frac{61x^6}{720} \ + \ \cdots \ - \ \frac{E_n x^{2n}}{(2n)!} \ + \ \cdots $ | $ \left \vert x \right \vert < \frac {\pi}{2} \qquad $ | |
| $ \frac{1}{\sin x} \ = \ \frac{1}{x} \ + \ \frac {x}{6} \ + \ \frac{7x^3}{360} \ + \ \frac{31x^5}{15120} \ + \ \cdots \ + \ \frac{2(2^{2n-1}-1)B_n x^{2n-1}}{(2n)!} \ + \ \cdots $ | $ 0 < \left \vert x \right \vert < \pi \qquad $ | |
| $ \arcsin x = x + {1 \over 2}{x^3 \over 3} + \frac{1 \cdot 3}{ 2 \cdot 4} {x^5 \over 5} + \frac {1 \cdot 3 \cdot 5}{ 2 \cdot 4 \cdot 6}{x^7 \over 7} + \cdots $ | $ \left \vert x \right \vert < 1 \qquad $ | |
| $ \arccos x = {\pi \over 2} - \sin ^{-1} x = {\pi \over 2} - \left ( x + {1 \over 2}{x^3 \over 3} +\frac{1 \cdot 3}{2 \cdot 4} {x^5 \over 5} + \cdots \ \right ) $ | $ \left \vert x \right \vert < 1 \qquad $ | |
| $ \arctan x = \begin{cases} x - {x^3 \over 3} + {x^5 \over 5} - { x^7 \over 7} + \cdots, & \left \vert x \right \vert < 1 \\ {\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots, &\mbox{ if } x \geqq 1 \\ -{\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots, &\mbox{ if } x \leqq -1 \end{cases} $ | ||
| $ \arccot x = {\pi \over 2} - \arctan x = \begin{cases} {\pi \over 2} - \left ( x - {x^3 \over 3} + {x^5 \over 5} - \cdots \right ), &\left \vert x \right \vert < 1 \\ {\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots, & \mbox{ if } x > 1\\ -{\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots, & \mbox{ if } x < -1 \end{cases} $ | ||
| $ \arccos ({1 \over x}) = {\pi \over 2} - \left ( {1 \over x} + \frac{1}{2 \cdot 3 x^3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 x^5} + \cdots \right ) $ | $ \left \vert x \right \vert > 1 \qquad $ | |
| $ \arcsin ({1 \over x}) = {1 \over x} + {1 \over 2 \cdot 3 x^3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 x^5} + \cdots $ | $ \left \vert x \right \vert > 1 $ | |
| Series Expansion of Hyperbolic functions | ||
| $ \, \sinh x = x + {x^3 \over 3!} + {x^5 \over 5!} + { x^7 \over 7!} + \cdots\, $ | $ - \infty < x < \infty \qquad $ | |
| $ \, \cosh x = 1 + {x^2 \over 2!} + {x^4 \over 4!} + { x^6 \over 6!} + \cdots\, $ | $ - \infty < x < \infty \qquad $ | |
| $ \, \tanh x = x - {x^3 \over 3} + {2x^5 \over 15} - { 17x^7 \over 315} + \cdots \ \frac{(-1)^{n-1}2^{2n}(2^{2n} -1)B_nx^{2n-1}}{(2n)!} + \cdots\, $ | $ \vert x \vert < {\pi \over 2} \qquad $ | |
| $ \, \coth x = {1 \over x} + {x \over 3} - {x^3 \over 45} + { 2x^5 \over 945} + \cdots \frac{(-1)^{n-1}2^{2n}b_nx^{2n-1}}{(2n)!} + \cdots\, $ | $ 0 < \vert x \vert < \pi \qquad $ | |
| $ \frac {1}{\cosh x} = 1 - {x2 \over 2} + {5x^4 \over 24} -{61x^6 \over 720} + \cdots \frac{(-1)^nE_nx^{2n}}{(2n)!} + \cdots $ | $ \vert x \vert < {\pi \over 2} $ | |
| $ \frac{1}{\sinh x} = {1 \over x} - {x \over 6} + {7x^3 \over 360} - {31x^5 \over 15,120} + \cdots \frac{(-1)^n2(2^{2n-1}-1)B_nx^{2n-1}}{(2n)!} + \cdots $ | $ 0 < \vert x \vert < \pi $ | |
| $ \operatorname{arsinh}\,x = \begin{cases} x - {x^3 \over 2 \cdot 3} + {1 \cdot 3 x^5 \cdot 2 \cdot 4 \cdot 5} - {1 \cdot 3 \cdot 5 x^7 \over 2 \cdot 4 \cdot 6 \cdot 7} + \cdots, & \left \vert x \right \vert < 1 \\ \left ( \ln \vert 2x \vert + {1 \over 2 \cdot 2 x^2} - {1 \cdot 3 \over 2 \cdot 4 \cdot 4x^4} + {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6 \cdot 6x^6} - \cdots \right ), & x \geqq 1\\ -\left ( \ln \vert 2x \vert + {1 \over 2 \cdot 2 x^2} - {1 \cdot 3 \over 2 \cdot 4 \cdot 4x^4} + {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6 \cdot 6x^6} - \cdots \right ), & x \leqq -1 \end{cases} $ | ||
| $ \operatorname{arcosh} \,x = \begin{cases} \{ \ln (2x) - ( \frac{1}{2 \cdot 2x^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 4x^4} + \frac { 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 6x^6} + \cdots ) \}, & \operatorname{arsinh}\,x > 0, x \geqq 1 \\ - \{ \ln (2x) - ( \frac{1}{2 \cdot 2x^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 4x^4} + \frac { 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 6x^6} + \cdots ) \}, & \operatorname{arsinh} \,x < 0, x \geqq 1 \end{cases} $ | ||
| $ \operatorname{argth} \,x = x + { x^3 \over 5} + {x^5 \over 5 } + {x^7 \over 7 }+ \cdots $ | $ \vert x \vert < 1 \qquad $ | |
| $ \operatorname{argcoth} \,x = {1 \over x} + { 1 \over 3x^3} + {1 \over 5x^5 } + {1 \over 7x^7 }+ \cdots $ | $ \vert x \vert > 1 \qquad $ | |
| Various Series | ||
| $ \, e^{\sin x} = 1 + x + {x^2 \over 2} - {x^4 \over 8} - {x^5 \over 15} + \cdots\, $ | $ - \infty < x < \infty $ | |
| $ \, e^{\cos x} = e \left ( 1 - {x^2 \over 2} + {x^4 \over 6} - {31x^6 \over 720} + \cdots \right ) \, $ | $ - \infty < x < \infty $ | |
| $ \, e^{\tan x} = 1 + x + {x^2 \over 2} + {x^3 \over 2} + {3x^4 \over 8} + \cdots \, $ | $ \vert x \vert < { \pi \over 2} $ | |
| $ e^x \sin x = x + x^2 + {2x^3 \over 3 } - {x^5 \over 30} - {x^6 \over 90} + \cdots + \frac{2^{n/2} \sin (n \pi /4)\ x^n}{n!} + \cdots $ | $ - \infty < x < \infty $ | |
| $ e^x \cos x = 1 + x - {x^3 \over 3 } - {x^4 \over 6} + \cdots + \frac{2^{n/2} \cos (n \pi /4)\ x^n}{n!} + \cdots $ | $ - \infty < x < \infty $ | |
| $ \ln \vert \sin x \vert = \ln \vert x \vert - {x^2 \over 6} - {x^4 \over 180} - {x^6 \over 2835} - \cdots - \frac{2^{2n-1}B_nx^{2n}}{n(2n)!} + \cdots $ | $ 0 < \vert x \vert < \pi $ | |
| $ \ln \vert \cos x \vert = - {x^2 \over 2} - {x^4 \over 12} - {x^6 \over 45} - {17x^8 \over 2520} - \cdots - \frac{2^{2n-1}(2^{2n}-1)B_nx^{2n}}{n(2n)!} + \cdots $ | $ \vert x \vert < {\pi \over 2} $ | |
| $ \ln \vert \tan x \vert = \ln \vert x \vert + {x^2 \over 3} + {7x^4 \over 90} + {62x^6 \over 2835}+ \cdots + \frac{2^{2n}(2^{2n-1}-1)B_nx^{2n}}{n(2n)!} + \cdots $ | $ 0 < \vert x \vert < {\pi \over 2} $ | |
| $ \frac{\ln (1+x)}{1+x} = x - (1+ {1 \over 2})^{x^2} + (1 + {1 \over 2} + {1 \over 3})^{x^3} - \cdots $ | $ \vert x \vert < 1 $ | |
| Series of Reciprocal Power Series | ||
| $ \text{if }\ y = c_1x +c_2x^3 +c_3x^3 + c_4x^4 + c_5x^5 + c_6x^6 + \cdots\,\qquad \text{then }\ x = C_1y+C_2y^2+C_3y^3+C_4y^4+C_5y^5+C_6y^6+\cdots $ | ||
| $ \text{where }\ c_1C_1 = 1, \qquad c_1^3C_2= -c_2, \qquad c_1^7C_3 = 2c_2^2 - c_1c_3 $ | ||
| $ c_1^7C_4 = 5c_1c_2c_3 - 5c_2^3 - c_2^2c_4, \qquad c_1^9C_5 = 6c_1^2c_2c_4 + $ | ||
| $ c_1^{11}C_6 = 7 c_1^3c_2 c_5 + 84 c_1 c_2^3c_3 + 7c_1^3c_3c_4 - 28c_1^2c_2c_3^2 - c_1^4c/-6 - 28c_1^2c_2^2c_4 - 42c_2^5 $ | ||
| Taylor Series of Two Variables function | ||
| $ \, f(x,y) = f(a,b) + (x-a)f_x(a,b) + (y-b)f_y(a,b) + $ | ||
| $ {1 \over 2!} \left \{ (x-a)^2f_{xx}(a,b) + 2(x-a)(y-b)f_{xy}(a,b)+(y-b)^2f_{yy}(a,b) \right \} + \cdots\, $ | ||
| $ f_x(a,b),f_y(a,b) , \cdots \text {denote the partial derivatives with respect to } x ,\ y \cdots $ | ||
Basic Signals and Functions
| Basic Signals and Functions in one variable | |
|---|---|
| Continuous-time signals. | |
| sinc function | $ sinc(t )=\frac{sin(\pi t )}{\pi t}, \text{ where }t\in {\mathbb R} $ |
| rect function | $ rect (t) = \left\{ \begin{array}{ll}1, & \text{ for } |t|\leq \frac{1}{2} \\ 0, & \text{ else}\end{array}\right., \text{ where }t\in {\mathbb R} $ |
| CT unit step function | $ u(t)=\left\{ \begin{array}{ll}1, & \text{ for } t\geq 0 \\ 0, & \text{ else}\end{array}\right., \text{ where }t\in {\mathbb R} $ |
| Discrete-time signals | |
| DT delta function | $ \delta[n]=\left\{ \begin{array}{ll}1, & \text{ for } n=1 \\ 0, & \text{ else}\end{array}\right., \text{ where }n\in {\mathbb Z} $ |
| DT unit step function | $ u[n]=\left\{ \begin{array}{ll}1, & \text{ for } n\geq 0 \\ 0, & \text{ else}\end{array}\right., \text{ where }n\in {\mathbb Z} $ |
| Basic Signals and Functions in two variables | |
| Continuous-time | |
|
2D Dirac delta |
$ \delta(x,y)=\delta(x) \delta(y), \text{ where }x,y\in {\mathbb R} $ |
|
2D sinc function |
$ sinc(x,y)=\frac{sin(\pi x)sin(\pi y)}{\pi^2 x y }, \text{ where }x,y\in {\mathbb R} $ |
|
(info) 2D rect function |
$ rect(x,y)= \left\{ \begin{array}{ll}1, & \text{ for } |x|\leq \frac{1}{2} \text{ and } |y|\leq \frac{1}{2} \\ 0, & \text{ else}\end{array}\right., \text{ where }x,y\in {\mathbb R} $ |
keywords: energy, power, signal
Signal Metrics Definitions and Formulas
| Metrics for Continuous-time Signals | |
|---|---|
| (info) CT signal energy | $ E_\infty=\int_{-\infty}^\infty | x(t) |^2 dt $ |
| (info) CT signal (average) power | $ P_\infty = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} \left | x (t) \right |^2 \, dt $ |
| CT signal area | $ A_x = \int_{-\infty}^{\infty} x (t) \, dt $ |
| Average value of a CT signal | $ \bar{x} = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} x (t) \, dt $ |
| CT signal magnitude | $ M_x = \max_{-\infty<t<\infty} \left | x (t) \right | $ |
| Metrics for Discrete-time Signals | |
| DT signal energy | $ E_\infty=\sum_{n=-\infty}^\infty | x[n] |^2 $ |
| DT signal average power | $ P_\infty = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} \left | x [n] \right |^2 \, $ |
| DT signal area | $ A_x = \sum_{n=-\infty}^{\infty} x [n] \, $ |
| Average value of a DT signal | $ \bar{x} = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} x [n] \, $ |
| DT signal magnitude | $ M_x = \max_{-\infty<t<\infty} \left | x [n] \right | $ |
Probability Formulas
click here for more formulas
| Probability Formulas | |
|---|---|
| Properties of Probability Functions | |
| The complement of an event A (i.e. the event A not occurring) | $ \,P(A^c) = 1 - P(A)\, $ |
| The intersection of two independent events A and B | $ \,P(A \mbox{ and }B) = P(A \cap B) = P(A) P(B)\, $ |
| The union of two events A and B (i.e. either A or B occurring) | $ \,P(A \mbox{ or } B) = P(A) + P(B) - P(A \mbox{ and } B)\, $ |
| The union of two mutually exclusive events A and B | $ \,P(A \mbox{ or } B) = P(A \cup B)= P(A) + P(B)\, $ |
| Event A occurs given that event B has occurred | $ \,P(A \mid B) = \frac{P(A \cap B)}{P(B)}\, $ |
| Total Probability Law | $ \,P(B) = P(B|A_1)P(A_1) + \dots + P(B|A_n)P(A_n)\, $
$ \mbox{ where } \{A_1,\dots,A_n\} \mbox{ is a partition of sample space } S, B \mbox{ is an event }. $ |
| Bayes Theorem | $ \,P(A_j|B) = \frac{P(B|A_j)P(A_j)}{\sum_{i=1}^{n}P(B|A_i)P(A_i)},\ \{A_i\} \mbox{ and } B \mbox{ are as above }. $ |
| Expectation and Variance of Random Variables | |
| Binomial random variable with parameters n and p | $ \,E[X] = np,\ \ Var(X) = np(1-p)\, $ |
| Poisson random variable with parameter $ \lambda $ | $ \,E[X] = \lambda,\ \ Var(X) = \lambda\, $ |
| Geometric random variable with parameter p | $ \,E[X] = \frac{1}{p},\ \ Var(X) = \frac{1-p}{p^2}\, $ |
| Uniform random variable over (a,b) | $ \,E[X] = \frac{a+b}{2},\ \ Var(X) = \frac{(b-a)^2}{12}\, $ |
| Gaussian random variable with parameter $ \mu \mbox{ and } \sigma^2 $ | $ \,E[X] = \mu,\ \ Var(X) = \sigma^2\, $ |
| Exponential random variable with parameter $ \lambda $ | $ \,E[X] = \frac{1}{\lambda},\ \ Var(X) = \frac{1}{\lambda^2}\, $ |
Relevant Courses
Transforms
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Table of (bidirectional) Laplace Transform Pairs and Properties
| Laplace Transform Pairs and Properties | |||
|---|---|---|---|
| Definition | |||
| (bidirectional) Laplace Transform | $ F(s)=\int_{-\infty}^\infty f(t) e^{-st}dt, \ s\in {\mathbb C} \ $ | ||
| Properties of the Laplace Transform | |||
| Function | Laplace Transform | ROC | |
| $ f(t) \ $ | $ F(s) \ $ | ROC $ R $ | |
| $ af_1(t)+bf_2(t) \ $ | $ aF_1(s)+bF_2(s) \ $ | at least $ R_1 \cap R_2 $ | |
| $ af(at) \ $ | $ F\left( \frac{s}{a} \right) $ | ||
| $ e^{at}f(t) \ $ | $ F(s-a) \ $ | ||
| $ u(t-a) = \begin{cases} f(t-a) & t>a \\ 0 & t<a \end{cases} $ | $ e^{-as}F(s) \ $ | ||
| $ f'(t) \ $ | $ sF(s)-f(0) \ $ | ||
| $ f''(t) \ $ | $ s^2F(s)-sf(0)-f'(0) \ $ | ||
| $ f^{(n)}(t) \ $ | $ s^{n}F(s)-\sum_{k=1}^ns^{n-k}f^{(k)}(0) \ $ | ||
| $ -tf(t) \ $ | $ F'(s) \ $ | ||
| $ t^2f(t) \ $ | $ F''(s) \ $ | ||
| $ (-1)^{(ntn)}f(t) \ $ | $ F^{(n)}(s) \ $ | ||
| $ \int_{0}^{t} f(u) du \ $ | $ \frac{F(s)}s \ $ | ||
| $ \int_{0}^{t}...\int_{0}^{t}f(u)du^n = \int_{0}^{t}\frac{{(t-u)}^{n-1}}{(n-1)!} f(u)du \ $ | $ \frac{F(s)}{s^n} \ $ | ||
| $ \int_{0}^{t}f(u)g(t-u)du \ $ | $ F(s)G(s) \ $ | ||
| $ \frac{f(t)}t \ $ | $ \int_{s}^{\infty}F(u)du \ $ | ||
| $ f(t)=f(t+T) \ $ | $ \frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}f(u)du \ $ | ||
| $ \frac{1}{\sqrt{{\pi}t}}\int_{0}^{\infty}e^{-\frac{u^2}4t}f(u)du $ | $ \frac{F(\sqrt{s})}s \ $ | ||
| $ \int_{0}^{\infty}J_0(2\sqrt{ut})f(u)du \ $ | $ \frac1sF\left(\frac1s\right) \ $ | ||
| $ t^{\frac{n}2}\int_{0}^{\infty}u^{-\frac{n}2}J_n(2\sqrt{ut})f(u)du \ $ | $ \frac1{s^{n+1}}F\left(\frac1s\right) \ $ | ||
| $ \int_{0}^{t}J_0(2\sqrt{u(t-u)})f(u)du \ $ | $ \frac{F(s+\frac1s)}{s^2+1} \ $ | ||
| $ f(t^2) \ $ | $ \frac1{2\sqrt\pi}\int_{0}^{\infty}u^{-\frac32}e^{-\frac{s^2}{4u}}F(u)du \ $ | ||
| $ \int_{0}^{\infty}\frac{t^uf(u)}{\Gamma(u+1)}du \ $ | $ \frac{F(\ln s)}{s\ln s} \ $ | ||
| $ \sum_{k=1}^N \frac{P(\alpha_k)}{Q'(\alpha_k)}e^{\alpha_kt} \ $ | $ \frac{P(s)}{Q(s)} \ $ | ||
| please continue | place formula here | ||
| Laplace Transform Pairs | ||||||
|---|---|---|---|---|---|---|
| Signal | Laplace Transform | ROC | ||||
| unit impulse/Dirac delta | $ \,\!\delta(t) $ | 1 | $ \text{All}\, s \in {\mathbb C} $ | |||
| unit step function | $ \,\! u(t) $ | $ \frac{1}{s} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | |||
| $ \,\! -u(-t) $ | $ \frac{1}{s} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $ | ||||
| $ \frac{t^{n-1}}{(n-1)!}u(t) $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | ||||
| $ -\frac{t^{n-1}}{(n-1)!}u(-t) $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $ | ||||
| $ \,\!e^{-\alpha t}u(t) $ | $ \frac{1}{s+\alpha} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ | ||||
| $ \,\! -e^{-\alpha t}u(-t) $ | $ \frac{1}{s+\alpha} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $ | ||||
| $ \frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t) $ | $ \frac{1}{(s+\alpha )^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ | ||||
| $ -\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t) $ | $ \frac{1}{(s+\alpha )^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $ | ||||
| $ \,\!\delta (t - T) $ | $ \,\! e^{-sT} $ | $ \text{All}\,\, s\in {\mathbb C} $ | ||||
| $ \,\cos( \omega_0 t)u(t) $ | $ \frac{s}{s^2+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | ||||
| $ \, \sin( \omega_0 t)u(t) $ | $ \frac{\omega_0}{s^2+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | ||||
| $ \,e^{-\alpha t}\cos( \omega_0 t) u(t) $ | $ \frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ | ||||
| $ \, e^{-\alpha t}\sin( \omega_0 t)u(t) $ | $ \frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ | ||||
| $ u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}} $ | $ \,\!s^{n} $ | $ All\,\, s $ | ||||
| $ u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times} $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | $ 1 \ $ | $ \frac{1}{s} \ $ | ||
| $ t \ $ | $ \frac1{s^2} \ $ | |||||
| $ \frac{t^{n-1}}{(n-1)!}, \ 0!=1 \ $ | $ \frac1{s^n}, \ n=1,2,3,... \ $ | |||||
| $ \frac{t^{n-1}}{\Gamma(n)} \ $ | $ \frac1{s^n}, \ n>0 \ $ | |||||
| $ e^{at}\ $ | $ \frac1{s-a}\ $ | |||||
| $ \frac{t^{n-1}e^{at}}{(n-1)!}, \ 0!=1\ $ | $ \frac1{(s-a)^n}, \ n=1,2,3,...\ $ | |||||
| $ \frac{t^{n-1}e^{at}}{\Gamma(n)}\ $ | $ \frac1{(s-a)^n}, \ n>0\ $ | |||||
| $ \frac{\sin {at}}{a} \ $ | $ \frac1{s^2+a^2}\ $ | |||||
| $ \cos {at} \ $ | $ \frac{s}{s^2+a^2} \ $ | |||||
| $ \frac{e^{bt}\sin{at}}{a} \ $ | $ \frac1{(s-b)^2+a^2}\ $ | |||||
| $ e^{bt}\cos{at}\ $ | $ \frac{s-b}{(s-b)^2+a^2}\ $ | |||||
| $ \left(\frac{{sh}\ {at}}{a}\right)\ $ | $ \frac{1}{s^2-a^2} \ $ | |||||
| $ {ch}\ {at}\ $ | $ \frac{s}{s^2-a^2}\ $ | |||||
| $ \frac{e^{bt}{sh}\ {at}}a\ $ | $ \frac1{(s-b)^2-a^2}\ $ | |||||
| $ e^{bt} {ch}\ {at}\ $ | $ \frac{s-b}{(s-b)^2-a^2} \ $ | |||||
| $ \frac{e^{bt}-e^{at}}{b-a}\ $ | $ \frac1{(s-a)(s-b)},\ a \ne b\ $ | |||||
| $ \frac{be^{bt}-ae^{at}}{b-a}\ $ | $ \frac{s}{(s-a)(s-b)},\ a \ne b \ $ | |||||
| $ \frac{\sin {at}-at\cos{at}}{2a^3}\ $ | $ \frac1{(s^2+a^2)^2}\ $ | |||||
| $ \frac{t\sin {at}}{2a}\ $ | $ \frac{s}{(s^2+a^2)^2}\ $ | |||||
| $ \frac{\sin {at}+at\cos {at}}{2a}\ $ | $ \frac{s^2}{(s^2+a^2)^2}\ $ | |||||
| $ \cos {at}-\frac12at\sin {at}\ $ | $ \frac{s^3}{(s^2+a^2)^2}\ $ | |||||
| $ t\cos {at}\ $ | $ \frac{s^2-a^2}{(s^2+a^2)^2}\ $ | |||||
| $ \frac{at\ {ch}\ {at}-{sh}\ {at}}{2a^3}\ $ | $ \frac{1}{(s^2-a^2)^2}\ $ | |||||
| $ \frac{t\ {sh}\ {at}}{2a}\ $ | $ \frac{s}{(s^2-a^2)^2}\ $ | |||||
| $ \frac{{sh}\ {at}+at\ {ch}\ {at}}{2a}\ $ | $ \frac{s^2}{(s^2-a^2)^2}\ $ | |||||
| $ {ch}\ {at}+\frac12at\ {sh}\ {at} \ $ | $ \frac{s^3}{(s^2-a^2)^2}\ $ | |||||
| $ t\ {ch}\ {at}\ $ | $ \frac{s^2+a^2}{(s^2-a^2)^2}\ $ | |||||
| $ \frac{(3-a^2t^2)\sin {at}-3at\cos {at}}{8a^5}\ $ | $ \frac{1}{(s^2+a^2)^3}\ $ | |||||
| $ \frac{t\sin {at}-at^2\cos {at}}{8a^3}\ $ | $ \frac{s}{(s^2+a^2)^3}\ $ | |||||
| $ \frac{(1+a^2t^2)\sin {at}-at\cos {at}}{8a^3}\ $ | $ \frac{s^2}{(s^2+a^2)^3}\ $ | |||||
| $ \frac{3t\sin {at}+at^2\cos {at}}{8a}\ $ | $ \frac{s^3}{(s^2+a^2)^3}\ $ | |||||
| $ \frac{(3-a^2t^2)\sin {at}+5at\cos {at}}{8a}\ $ | $ \frac{s^4}{(s^2+a^2)^3}\ $ | |||||
| $ \frac{(8-a^2t^2)\cos {at}-7at\sin {at}}{8}\ $ | $ \frac{s^5}{(s^2+a^2)^3}\ $ | |||||
| $ \frac{t^2\sin {at}}{2a}\ $ | $ \frac{3s^2-a^2}{(s^2+a^2)^3}\ $ | |||||
| $ \frac12t^2\cos {at}\ $ | $ \frac{s^3-3a^2s}{(s^2+a^2)^3}\ $ | |||||
| $ \frac16t^3\cos {at}\ $ | $ \frac{s^4-6a^2s^2+a^4}{(s^2+a^2)^4}\ $ | |||||
| $ \frac{t^3\sin {at}}{24a}\ $ | $ \frac{s^3-a^2s}{(s^2+a^2)^4}\ $ | |||||
| $ \frac{3+a^2t^2\ {sh}\ {at}-3at\ {ch}\ {at}}{8a^5}\ $ | $ \frac{1}{(s^2-a^2)^3}\ $ | |||||
| $ \frac{at^2\ {ch}\ {at}-t\ {sh}\ {at}}{8a^3}\ $ | $ \frac{s}{(s^2-a^2)^3}\ $ | |||||
| $ \frac{at\ {ch}\ {at}+(a^2t^2-1)\ {sh}\ {at}}{8a^3}\ $ | $ \frac{s^2}{(s^2-a^2)^3}\ $ | |||||
| $ \frac{3t\ {sh}\ {at}+at^2\ {ch}\ {at}}{8a}\ $ | $ \frac{s^3}{(s^2-a^2)^3}\ $ | |||||
| $ \frac{(3+a^2t^2)\ {sh}\ {at}+5at\ {ch}\ {at}}{8a}\ $ | $ \frac{s^4}{(s^2-a^2)^3}\ $ | |||||
| $ \frac{(8+a^2t^2)\ {ch}\ {at}+7at\ {sh}\ {at}}{8}\ $ | $ \frac{s^5}{(s^2-a^2)^3}\ $ | |||||
| $ \frac{t^2\ {sh}\ {at}}{2a}\ $ | $ \frac{3s^2+a^2}{(s^2-a^2)^3}\ $ | |||||
| $ \frac12t^2\ {ch}\ {at}\ $ | $ \frac{s^3+3a^2s}{(s^2-a^2)^3}\ $ | |||||
| $ \frac16t^3\ {ch}\ {at}\ $ | $ \frac{s^4+6a^2s^2+a^4}{(s^2-a^2)^4}\ $ | |||||
| $ \frac{t^3\ {sh}\ {at}}{24a}\ $ | $ \frac{s^3+a^2s}{(s^2-a^2)^4}\ $ | |||||
| $ \frac{e^{at/2}}{3a^2} \left \{ \sqrt{3} \sin {\frac{\sqrt{3}at}{2}}-\cos {\frac{\sqrt{3}at}{2}}+e^{-3at/2} \right \}\ $ | $ \frac{1}{s^3+a^3}\ $ | |||||
| $ \frac{e^{at/2}}{3a^2} \left \{ \cos {\frac{\sqrt{3}at}{2}}+ \sqrt{3}\sin {\frac{\sqrt{3}at}{2}}-e^{-3at/2} \right \}\ $ | $ \frac{s}{s^3+a^3}\ $ | |||||
| $ \frac13 \left \{ e^{-at}+ 2e^{at/2} \cos {\frac{\sqrt{3}at}{2}} \right \}\ $ | $ \frac{s^2}{s^3+a^3}\ $ | |||||
| $ \frac{e^{-at/2}}{3a^2} \left \{e^{3at/2}- \cos {\frac{\sqrt{3}at}{2}}- \sqrt{3}\sin {\frac{\sqrt{3}at}{2}} \right \}\ $ | $ \frac{1}{s^3-a^3}\ $ | |||||
| $ \frac{e^{-at/2}}{3a} \left \{ \sqrt{3}\sin {\frac{\sqrt{3}at}{2}}-\cos {\frac{\sqrt{3}at}{2}}+e^{3at/2} \right \}\ $ | $ \frac{s}{s^3-a^3}\ $ | |||||
| $ \frac13 \left \{ e^{at}+ 2e^{-at/2} \cos {\frac{\sqrt{3}at}{2}} \right \}\ $ | $ \frac{s^2}{s^3-a^3}\ $ | |||||
| $ \frac1{4a^3} \left (\sin {at}\ {ch}\ {at}-\cos {at}\ {sh}\ {at} \right )\ $ | $ \frac{1}{s^4+4a^4}\ $ | |||||
| $ \frac{\sin {at}\ {sh}\ {at}}{2a^2}\ $ | $ \frac{s}{s^4+4a^4}\ $ | |||||
| $ \frac1{2a} \left (\sin {at}\ {ch}\ {at}+\cos {at}\ {sh}\ {at} \right )\ $ | $ \frac{s^2}{s^4+4a^4}\ $ | |||||
| $ \cos {at}\ {ch}\ {at}\ $ | $ \frac{s^3}{s^4+4a^4}\ $ | |||||
| $ \frac1{2a^3} \left (\ {sh}\ {at}-\sin {at} \right )\ $ | $ \frac{1}{s^4-a^4}\ $ | |||||
| $ \frac1{2a^2} \left (\ {ch}\ {at}-\cos {at} \right )\ $ | $ \frac{s}{s^4-a^4}\ $ | |||||
| $ \frac1{2a} \left (\ {sh}\ {at}+\sin {at} \right )\ $ | $ \frac{s^2}{s^4-a^4}\ $ | |||||
| $ \frac12 \left (\ {ch}\ {at}+\cos {at} \right )\ $ | $ \frac{s^3}{s^4-a^4}\ $ | |||||
| $ \frac{e^{-bt}-e^{-at}}{2(b-a)\sqrt{\pi t^3}}\ $ | $ \frac1{\sqrt{s+a}+\sqrt{s+b}}\ $ | |||||
| $ \frac{erf\ \sqrt{at}}{\sqrt{a}}\ $ | $ \frac1{s\sqrt{s+a}}\ $ | |||||
| $ \frac{e^{at}\ {erf}\ \sqrt{at}}{\sqrt{a}}\ $ | $ \frac1{\sqrt{s}(s-a)}\ $ | |||||
| $ e^{at} \left \{\frac1{\sqrt{\pi t}}-be^{b^{2}t}\ {erfc}\ (b\sqrt{t}) \right \}\ $ | $ \frac1{\sqrt{s-a}+b}\ $ | |||||
| $ J_0(at)\ $ | $ \frac1{\sqrt{s^2+a^2}}\ $ | |||||
| $ I_0(at)\ $ | $ \frac1{\sqrt{s^2-a^2}}\ $ | |||||
| $ a^nJ_n(at)\ $ | $ \frac{{\left (\sqrt{s^2+a^2}-s \right )}^n}{\sqrt{s^2+a^2}},\quad n>-1 \ $ | |||||
| $ a^nI_n(at)\ $ | $ \frac{{\left (s- \sqrt{s^2-a^2} \right )}^n}{\sqrt{s^2-a^2}},\quad n>-1 \ $ | |||||
| $ J_0(a\sqrt{t(t+2b)})\ $ | $ \frac{e^{b \left (s- \sqrt{s^2+a^2} \right )}}{\sqrt{s^2+a^2}} \ $ | |||||
| $ \begin{cases} J_0(a\sqrt{t^2-b^2}) & t>b \\ 0 &t<b \end{cases} \ $ | $ \frac{e^{-b\sqrt{s^2+a^2}}}{\sqrt{s^2+a^2}} \ $ | |||||
| $ tJ_0(at)\ $ | $ \frac1{(s^2+a^2)^{3/2}}\ $ | |||||
| $ J_0(at)-atJ_1(at)\ $ | $ \frac{s^2}{(s^2+a^2)^{3/2}}\ $ | |||||
| $ \frac{tI_1(at)}{a}\ $ | $ \frac1{(s^2-a^2)^{3/2}}\ $ | |||||
| $ I_0(at)+atI_1(at)\ $ | $ \frac{s}{(s^2+a^2)^{3/2}}\ $ | |||||
| $ f(t)=n,\ n \leqq t\ <n+1,\ n=0,1,2,... \ $ | $ \frac1{s(e^s-1)}\ =\ \frac{e^{-s}}{s(1-e^{-s})}\ $ | |||||
| $ f(t)= \sum_{k=1}^{[t]} r^k\ $ | $ \frac1{s(e^s-r)}\ =\ \frac{e^{-s}}{s(1-re^{-s})}\ $ | |||||
| $ f(t)= r^n,\ n\leqq t<n+1,\ n=0,1,2,...\ $ | $ \frac{s^s-1}{s(e^s-r)}\ =\ \frac{1-e^{-s}}{s(1-re^{-s})}\ $ | |||||
| $ \frac{\cos {2\sqrt{at}}}{\sqrt{ \pi t}}\ $ | $ \frac{s^{-a/s}}{\sqrt{s}}\ $ | |||||
| $ \frac{\sin {2\sqrt{at}}}{\sqrt{ \pi a}}\ $ | $ \frac{e^{-a/s}}{s^{3/2}}\ $ | |||||
| $ \left ( \frac{t}{a} \right )^{n/2}J_n(2\sqrt{at})\ $ | $ \frac{e^{-a/s}}{s^n+1} \quad n>-1 \ $ | |||||
| $ \frac{e^{-a^2/4t}}{\sqrt{ \pi t}}\ $ | $ \frac{e^{-a\sqrt{s}}}{\sqrt{s}}\ $ | |||||
| $ \frac{a}{2\sqrt{ \pi t^3}}e^{-a^2/4t}\ $ | $ e^{-a\sqrt{s}}\ $ | |||||
| $ erf(a/2\sqrt{t})\ $ | $ \frac{1-e^{-a\sqrt{s}}}{s}\ $ | |||||
| $ erfc(a/2\sqrt{t})\ $ | $ \frac{e^{-a\sqrt{s}}}{s}\ $ | |||||
| $ e^{b(bt+a)}erfc \left ( b\sqrt{t}+\frac{a}{2\sqrt{t}} \right )\ $ | $ \frac{e^{-a\sqrt{s}}}{\sqrt{s}(\sqrt{s}+b)}\ $ | |||||
| $ \frac1{\sqrt{\pi t}a^{2n+1}}\int_{0}^{\infty}u^ne^{-u^2/4a^2t}J_{2n}(2\sqrt{u})du \ $ | $ \frac{e^{-a\sqrt{s}}}{s^{n+1}} \quad n>-1\ $ | |||||
| $ \frac{e^{-bt}-e^{-at}}{t}\ $ | $ \ln \left ( \frac{s+a}{s+b} \right )\ $ | |||||
| $ Ci(at)\ $ | $ \frac{\ln [(s^2+a^2)/a^2]}{2s}\ $ | |||||
| $ Ei(at)\ $ | $ \frac{\ln [(s+a)/a]}{s}\ $ | |||||
| $ \ln t\ $ | $ \begin{array}{lcl} -\frac{(\gamma+\ln s)}{s} \\ \gamma = \text{Eular constant}=0.5772156... \end{array} \ $ | |||||
| $ \frac{2(\cos {at}-\cos {bt})}{t}\ $ | $ \ln \left ( \frac{s^2+a^2}{s^2+b^2} \right )\ $ | |||||
| $ \ln^2 t\ $ | $ \begin{array}{lcl} \frac{{\pi}^2}{6s}+\frac{ \left (\gamma+\ln s \right )^2}{s} \\ \gamma = \text{Eular constant}=0.5772156... \end{array} \ $ | |||||
| $ \begin{array}{lcl} - \left (\ln t+\gamma \right ) \\ \gamma = \text{Eular constant}=0.5772156... \end{array} \ $ | $ \frac{\ln s}{s}\ $ | |||||
| $ \begin{array}{lcl} \left ( \ln t+\gamma \right )^2-\frac16{\pi}^2 \\ \gamma = \text{Eular constant}=0.5772156... \end{array} \ $ | $ \frac{\ln^2 s}{s}\ $ | |||||
| $ t^n\ln t\ $ | $ \frac{\Gamma'(n+1)-\Gamma(n+1)\ln s}{s^{n+1}} \quad n>-1\ $ | |||||
| $ \frac{\sin {at}}{t}\ $ | $ {Arc}\ {tg}\ (a/s)\ $ | |||||
| $ Si(at)\ $ | $ \frac{{Arc}\ {tg}\ (a/s)}{s}\ $ | |||||
| $ \frac{e^{-2\sqrt{at}}}{\sqrt{\pi t}} \ $ | $ \frac{e^{a/s}}{\sqrt{s}}\ erfc(\sqrt{a/s})\ $ | |||||
| $ \frac{2a}{\sqrt{\pi }}e^{-a^2t^2}\ $ | $ e^{s^2/4a^2}\ erfc(s/2a)\ $ | |||||
| $ erf(at)\ $ | $ \frac{e^{s^2/4a^2}\ erfc(s/2a)}{s}\ $ | |||||
| $ \frac1{\sqrt{\pi (t+a)}}\ $ | $ \frac{e^{as}erfc\sqrt{as}}{\sqrt{s}}\ $ | |||||
| $ \frac1{t+a}\ $ | $ e^{as}Ei(as)\ $ | |||||
| $ \frac1{t^2+a^2}\ $ | $ \frac1a \left [ \cos {as} \left \{ \frac{\pi }{2}-Si(as) \right \}-\sin {as}\ Ci(as) \right ]\ $ | |||||
| $ \frac{t}{t^2+a^2}\ $ | $ \sin {as} \left \{ \frac{\pi }{2}-Si(as) \right \}+\cos {as}\ Ci(as)\ $ | |||||
| $ {Arc}\ {tg}(t/a)\ $ | $ \frac{\cos {as} \left \{ \frac{\pi }{2}-Si(as) \right \}-\sin {as}\ Ci(as)}{s}\ $ | |||||
| $ \frac12\ln \left (\frac{t^2+a^2}{a^2} \right )\ $ | $ \frac{\sin {as} \left \{ \frac{\pi }{2}-Si(as) \right \}+\cos {as}\ Ci(as)}{s}\ $ | |||||
| $ \frac1t \ln \left ( \frac{t^2+a^2}{a^2} \right )\ $ | $ \left [ \frac{\pi}{2}-Si(as) \right ]^2 + Ci^2(as)\ $ | |||||
| $ \mathcal{N}(t)\ =\ fonction nulle\ $ | $ 0\ $ | |||||
| $ \delta(t)\ =\ fonction delta\ $ | $ 1\ $ | |||||
| $ \delta(t-a)\ $ | $ e^{-as}\ $ | |||||
| $ \mu(t-a)\ $ | $ \frac{e^{-as}}{s}\ $ | |||||
| $ \frac xa+\frac2{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin {\frac{n \pi x}{a}} \cos {\frac{n\pi t}{a}}\ $ | $ \frac{{sh}\ sx}{s\ {sh}\ sa}\ $ | |||||
| $ \frac4{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1} \sin {\frac{(2n-1) \pi x}{2a}} \sin {\frac{(2n-1)\pi t}{2a}}\ $ | $ \frac{{sh}\ sx}{s\ {ch}\ sa}\ $ | |||||
| $ |fracta+\frac2{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \cos {\frac{n \pi x}{a}} \sin {\frac{n\pi t}{a}}\ $ | $ \frac{{ch}\ sx}{s\ {sh}\ as}\ $ | |||||
| $ 1+\frac4{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1} \cos {\frac{n \pi x}{a}} \cos {\frac{(2n-1)\pi t}{2a}}\ $ | $ \frac{{ch}\ sx}{s\ {ch}\ as}\ $ | |||||
| $ \frac {xt}a+\frac{2a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \sin {\frac{n \pi x}{a}} \sin {\frac{n\pi t}{a}}\ $ | $ \frac{{sh}\ sx}{s^2\ {sh}\ sa}\ $ | |||||
| $ x+\frac{8a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^2} \sin {\frac{(2n-1) \pi x}{2a}} \cos {\frac{(2n-1) \pi t}{2a}}\ $ | $ \frac{{sh}\ sx}{s^2\ {ch}\ sa}\ $ | |||||
| $ \frac{t^2}{2a}+\frac{2a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos {\frac{n \pi x}{a}} \left ( 1-\cos {\frac{n \pi t}{a}} \right )\ $ | $ \frac{{ch}\ sx}{s^2\ {sh}\ sa}\ $ | |||||
| $ t+\frac{8a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^2} \cos {\frac{(2n-1) \pi x}{2a}} \sin {\frac{(2n-1) \pi t}{2a}}\ $ | $ \frac{{ch}\ sx}{s^3\ {sh}\ sa}\ $ | |||||
| $ \frac12(t^2+x^2-a^2)-\frac{16a^2}{{\pi}^3 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^3} \cos {\frac{(2n-1) \pi x}{2a}} \cos {\frac{(2n-1) \pi t}{2a}}\ $ | $ \frac{{ch}\ sx}{s^3\ {ch}\ sa}\ $ | |||||
| $ \frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi x}{a}}\ $ | $ \frac{{ch}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\ $ | |||||
| $ \frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi x}{a}}\ $ | $ \frac{{ch}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\ $ | |||||
| $ \frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi x}{a}}\ $ | $ \frac{{sh}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\ $ | |||||
| $ \frac{\pi}{a^2} \sum_{n=1}^{\infty} (-1)^{n-1}(2n-1)e^{-(2n-1)^2{\pi}^2t/4a^2}\cos {\frac{(2n-1) \pi x}{2a}}\ $ | $ \frac{{ch}\ x\sqrt{s}}{{ch}\ a\sqrt{s}}\ $ | |||||
| $ \frac{2}{a} \sum_{n=1}^{\infty} (-1)^{n-1}e^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{(2n-1) \pi x}{2a}}\ $ | $ \frac{{sh}\ x\sqrt{s}}{\sqrt{s}{ch}\ a\sqrt{s}}\ $ | |||||
| $ \frac1a+\frac2a\sum_{n=1}^{\infty} (-1)^ne^{-n^2{\pi}^2t/a^2}\cos {\frac{n \pi x}{a}}\ $ | $ \frac{{ch}\ x\sqrt{s}}{\sqrt{s}{sh}\ a\sqrt{s}}\ $ | |||||
| $ \frac{x}{a}+\frac{2}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n}e^{-n^2{\pi}^2t/a^2} \sin {\frac{n \pi x}{a}}\ $ | $ \frac{{sh}\ x\sqrt{s}}{s{sh}\ a\sqrt{s}}\ $ | |||||
| $ 1+\frac4{\pi}\sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1}e^{-(2n-1)^2{\pi}^2t/a^2}\cos {\frac{(2n-1) \pi x}{2a}}\ $ | $ \frac{{ch}\ x\sqrt{s}}{s{ch}\ a\sqrt{s}}\ $ | |||||
| $ \frac{xt}{a}+\frac{2a^2}{{\pi}^3}\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}(1-e^{-n^2{\pi}^2t/a^2})\sin {\frac{n \pi x}{a}}\ $ | $ \frac{{sh}\ x\sqrt{s}}{s^2{sh}\ a\sqrt{s}}\ $ | |||||
| $ \frac12(x^2+a^2)+t-\frac{16a^2}{{\pi}^3}\sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^3}e^{-{(2n-1)}^2{\pi}^2t/a^2}\cos {\frac{(2n-1) \pi x}{2a}}\ $ | $ \frac{{ch}\ x\sqrt{s}}{s^2{ch}\ a\sqrt{s}}\ $ | |||||
Go to Relevant Course Page: ECE 301
Go to Relevant Course Page: ECE 438
Go to Relevant Course Page: ECE 538
Table of Continuous-time (CT) Fourier Transform Pairs and Properties
as a function of $ \omega $ in radians per time unit
(used in ECE301)
| Definition CT Fourier Transform and its Inverse | |
|---|---|
| (info) CT Fourier Transform | $ \mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt $ |
| (info) Inverse CT Fourier Transform | $ \, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{i\omega t} d \omega\, $ |
| CT Fourier Transform Pairs | ||||||
|---|---|---|---|---|---|---|
| |
signal (function of t) | $ \longrightarrow $ | Fourier transform (function of $ \omega $) | |||
| 1 | CTFT of a unit impulse | $ \delta (t)\ $ | $ 1 \ $ | |||
| 2 | CTFT of a shifted unit impulse | $ \delta (t-t_0)\ $ | $ e^{-iwt_0} $ | |||
| 3 | CTFT of a complex exponential | $ e^{iw_0t} $ | $ 2\pi \delta (\omega - \omega_0) \ $ | |||
| 4 | $ e^{-at}u(t),\ $ $ a\in {\mathbb R}, a>0 $ | $ \frac{1}{a+i\omega} $ | ||||
| 5 | $ te^{-at}u(t),\ $ $ a\in {\mathbb R}, a>0 $ | $ \left( \frac{1}{a+i\omega}\right)^2 $ | ||||
| 6 | CTFT of a cosine | $ \cos(\omega_0 t) \ $ | $ \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ $ | |||
| 7 | CTFT of a sine | $ sin(\omega_0 t) \ $ | $ \frac{\pi}{i} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right] $ | |||
| 8 | CTFT of a rect | $ \left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ $ | $ \frac{2 \sin \left( T \omega \right)}{\omega} \ $ | |||
| 9 | CTFT of a sinc | $ \frac{\sin \left( W t \right)}{\pi t } \ $ | $ \left\{\begin{array}{ll}1, & \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right. \ $ | |||
| 10 | CTFT of a periodic function | $ \sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t} $ | $ 2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \ $ | |||
| 11 | CTFT of an impulse train | $ \sum^{\infty}_{n=-\infty} \delta(t-nT) \ $ | $ \frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T}) $ | |||
| 12 | $ 1 \ $ | $ 2\pi \delta (\omega) \ $ | ||||
| 13 | CTFT of a Periodic Square Wave | $ x(t+T)=x(t)=\left\{\begin{array}{ll}1, & |t|\leq T_1,\\ 0, & T_1<|t|\leq T/2 \end{array} \right. $ | $ \sum^{\infty}_{k=-\infty}\frac{2 \sin(k\frac{2\pi}{T}T_1)}{k}\delta(\omega-k\frac{2\pi}{T}) $ | |||
| 14 | CTFT of a Step Function | $ u(t) \ $ | $ \frac{1}{j\omega}+\pi\delta(\omega) $ | |||
| 15 | $ e^{-\alpha |t|} \ $ | $ \frac{2\alpha}{\alpha^{2}+\omega^{2}} $ | ||||
| CT Fourier Transform Properties | |||||||
|---|---|---|---|---|---|---|---|
| $ x(t) \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |||||
| 16 | (info) multiplication property | $ x(t)y(t) \ $ | $ \frac{1}{2\pi} \mathcal{X}(\omega)*\mathcal{Y}(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\theta)\mathcal{Y}(\omega-\theta)d\theta $ | ||||
| 17 | convolution property | $ x(t)*y(t) \ $ | $ \mathcal{X}(\omega)\mathcal{Y}(\omega) \! $ | ||||
| 18 | time reversal | $ \ x(-t) $ | $ \ \mathcal{X}(-\omega) $ | ||||
| 19 | Frequency Shifting | $ e^{j\omega_0 t}x(t) $ | $ \mathcal{X} (\omega - \omega_0) $ | ||||
| 20 | Conjugation | $ x^{*}(t) \ $ | $ \mathcal{X}^{*} (-\omega) $ | ||||
| 21 | Time and Frequency Scaling | $ x(at) \ $ | $ \frac{1}{|a|} \mathcal{X} (\frac{\omega}{a}) $ | ||||
| 23 | Differentiation in Frequency | $ tx(t) \ $ | $ j\frac{d}{d\omega} \mathcal{X} (\omega) $ | ||||
| 24 | Symmetry | $ x(t)\ \text{ real and even} $ | $ \mathcal{X} (\omega) \ \text{ real and even} $ | ||||
| 25 | $ x(t) \ \text{ real and odd} $ | $ \mathcal{X} (\omega) \ \text{ purely imaginary and odd} $ | |||||
| 26 | Duality | $ \mathcal{X} (-t) $ | $ 2 \pi x (\omega) \ $ | ||||
| 27 | Differentiation | $ \frac{d^{n}x(t)}{dt^{n}} $ | $ (j \omega)^{n} \mathcal{X} (\omega) $ | ||||
| 28 | Linearity | $ ax(t) + by(t) \ $ | $ a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) $ | ||||
| 29 | Time Shifting | $ x(t-t_0) \ $ | $ e^{-j\omega t_0}X(\omega) $ | ||||
| Other CT Fourier Transform Properties | |
|---|---|
| Parseval's relation | $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw $ |
Continuous-time Fourier Transform Pairs and Properties
as a function of frequency f in hertz
(used in ECE438)
| CT Fourier Transform and its Inverse | |
|---|---|
| CT Fourier Transform | $ X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt $ |
| Inverse DT Fourier Transform | $ \, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \, $ |
| CT Fourier Transform Pairs | ||||
|---|---|---|---|---|
| signal (function of t) | $ \longrightarrow $ | Fourier transform (function of f) | ||
| CTFT of a unit impulse | $ \delta (t)\ $ | $ 1 \ $ | ||
| CTFT of a shifted unit impulse | $ \delta (t-t_0)\ $ | $ e^{-i2\pi ft_0} $ | ||
| CTFT of a complex exponential | $ e^{iw_0t} $ | $ \delta (f - \frac{\omega_0}{2\pi}) \ $ | ||
| $ e^{-at}u(t), \ \text{ where } a\in {\mathbb R}, a>0 $ | $ \frac{1}{a+i2\pi f} $ | |||
| $ te^{-at}u(t), \ \text{ where } a\in {\mathbb R}, a>0 $ | $ \left( \frac{1}{a+i2\pi f}\right)^2 $ | |||
| CTFT of a cosine | $ \cos(\omega_0 t) \ $ | $ \frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \ $ | ||
| CTFT of a sine | $ sin(\omega_0 t) \ $ | $ \frac{1}{2i} \left[\delta (f - \frac{\omega_0}{2\pi}) - \delta (f + \frac{\omega_0}{2\pi})\right] $ | ||
| CTFT of a rect | $ \left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ $ | $ \frac{\sin \left(2\pi Tf \right)}{\pi f} \ $ | ||
| CTFT of a sinc | $ \frac{ \sin \left( W t \right)}{\pi t } \ $ | $ \left\{\begin{array}{ll}1, & \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right. \ $ | ||
| CTFT of a periodic function | $ \sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t} $ | $ \sum^{\infty}_{k=-\infty}a_{k}\delta(f-\frac{kw_{0}}{2\pi}) \ $ | ||
| CTFT of an impulse train | $ \sum^{\infty}_{n=-\infty} \delta(t-nT) \ $ | $ \frac{1}{T}\sum^{\infty}_{k=-\infty}\delta(f-\frac{k}{T}) \ $ | ||
| CT Fourier Transform Properties | |||
|---|---|---|---|
| $ x(t) \ $ | $ \longrightarrow $ | $ X(f) \ $ | |
| multiplication property | $ x(t)y(t) \ $ | $ X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta $ | |
| time shifting property | $ x(t-t_0) \ $ | $ X(f)e^{-j 2 \pi f t_0} \ $ | |
| frequency shifting (also called "modulation") property | $ x(t) e^{j 2 \pi f_0 t} \ $ | $ X(f-f_0) \ $ | |
| scaling and shifting property | $ x\left( \frac{ t- t_0}{a} \right) \ $ | $ |a| X(af) e^{-j 2 \pi f t_0} \ $ | |
| convolution property | $ x(t)*y(t) \ $ | $ X(f)Y(f) \ $ | |
| time reversal | $ \ x(-t) $ | $ \ X(-f) $ | |
| Other CT Fourier Transform Properties | |
|---|---|
| Parseval's relation | $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df $ |
| Discrete-time Fourier Transform Pairs and Properties | |
|---|---|
| DT Fourier transform and its Inverse | |
| DT Fourier Transform | $ \,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\, $ |
| Inverse DT Fourier Transform | $ \,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\, $ |
| DT Fourier Transform Pairs | |||
|---|---|---|---|
| $ x[n] \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) \ $ | |
| DTFT of a complex exponential | $ e^{jw_0n} \ $ | $ \pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ $ | |
| (info) DTFT of a rectangular window | $ w[n]= \ $ | add formula here | |
| $ a^{n} u[n], |a|<1 \ $ | $ \frac{1}{1-ae^{-j\omega}} \ $ | ||
| $ \sin\left(\omega _0 n\right) u[n] \ $ | $ \frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right) $ | ||
| DT Fourier Transform Properties | |||
|---|---|---|---|
| $ x[n] \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) \ $ | |
| multiplication property | $ x[n]y[n] \ $ | $ \frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta $ | |
| convolution property | $ x[n]*y[n] \! $ | $ X(\omega)Y(\omega) \! $ | |
| time reversal | $ \ x[-n] $ | $ \ X(-\omega) $ | |
| Other DT Fourier Transform Properties | |
|---|---|
| Parseval's relation | $ \frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = $ |
