General Formulas

Vector Identities and Operator Definitions

(Used in ECE311)

Vector Identities and Operator Definitions
Vector Identities
Notes Identity

$\bold{x}\cdot \left(\bold{y}\times \bold{z}\right)= \left(\bold{x}\times \bold{y}\right)\cdot \bold{z}$

$\bold{x}\times \left(\bold{y}\times \bold{z} \right)=\bold{y}\left(\bold{x} \cdot \bold{z} \right)-\bold{z} \left( \bold{x}\cdot\bold{y}\right)$
$\left( \bold{x}\times \bold{y}\right)\cdot \left(\bold{z}\times \bold{w} \right)=\left( \bold{x}\cdot \bold{z}\right) \left(\bold{y} \cdot \bold{w} \right)- \left(\bold{x}\cdot\bold{w} \right) \left( \bold{y}\cdot\bold{z}\right)$
$\nabla \left( \bold{x}\cdot \bold{y}\right)= \bold{y}\times \left(\nabla\times \bold{x}\right)+ \bold{x} \times \left(\nabla\times \bold{y} \right)+ \left(\bold{y}\cdot\nabla \right)\bold{x} + \left( \bold{x}\cdot\nabla\right) \bold{y}$
$\nabla \left( f+g \right)= \nabla f+ \nabla g$
$\nabla \left( f g \right)= f \nabla g+ g\nabla f$
$\nabla \cdot \left( \bold{x}+\bold{y} \right)= \nabla \cdot \bold{x} + \nabla \cdot \bold{y}$
$\nabla \cdot \left( f \bold{x}\right)= \bold{x} \cdot \nabla f + f \left( \nabla \cdot\bold{x} \right)$
$\nabla \times \left( \bold{x} + \bold{y} \right)= \nabla \times \bold{x} + \nabla \times \bold{y}$
$\nabla \times \left( u \bold{x} \right)= \left( \nabla u \right) \times \bold{x} + u \left( \nabla \times \bold{x} \right)$
$\nabla \cdot \left( \bold{x}\times \bold{y}\right)= \bold{y} \cdot \left( \nabla \times \bold{x}\right) - \bold{x} \cdot \left( \nabla \times \bold{y}\right)$
$\nabla \cdot \left(\nabla\times \bold{x} \right)= 0$
$\nabla \times \left( \bold{x} \times \bold{y} \right) = \left( \nabla \cdot \bold{y} \right) \bold{x} - \left( \nabla \cdot \bold{x} \right) \bold{y} + \left( \bold{y} \cdot \nabla \right) \bold{x} - \left( \bold{x} \cdot \nabla \right) \bold{y}$
$\nabla \times \nabla \bold{x} = 0$
$\nabla ( \bold{C} \cdot \bold{r} ) = \bold{C} \qquad \text{where }\bold{C}\text{ is a constant (real or complex)}$
$\nabla \times \left( \nabla \times \bold{x} \right) = \nabla \left( \nabla \cdot \bold{x} \right) - \nabla^2 \bold{x}$
$\left( \bold{A} \cdot \nabla \right) \bold{B} = \hat{\bold{x}} ( \bold{A}_x \frac{\partial \bold{B}_x}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_x}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_x}{\partial z} ) + \hat{\bold{y}} ( \bold{A}_x \frac{\partial \bold{B}_y}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_y}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_y}{\partial z} ) + \hat{\bold{z}} ( \bold{A}_x \frac{\partial \bold{B}_z}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_z}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_z}{\partial z} )$
$\frac{d \left( \bold{x} \cdot \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\cdot \bold{x} + \frac{d \bold{x}}{d\sigma}\cdot \bold{y}$
$\frac{d \left( \bold{x} \times \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\times \bold{x} + \frac{d \bold{x}}{d\sigma}\times \bold{y}$
$\frac {d ( u \bold{v} )}{d \sigma} = \frac {d u}{d \sigma} \bold{v} + u \frac{d \bold{v}}{d \sigma}$
Vector Operators in Rectangular Coordinates
Notes Operator
$\nabla f(x,y,z) = \mathbf{\hat x} \frac{\partial f}{\partial x}+\mathbf{\hat y}\frac{\partial f}{\partial y}+\mathbf{\hat z} \frac{\partial f}{\partial z}$
$\nabla \cdot \bold{v} = \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+ \frac{\partial v_z}{\partial z}$
$\nabla \times \bold{v} = \mathbf{\hat x} \left( \frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z} \right) + \mathbf{\hat y} \left( \frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x} \right) + \mathbf{\hat z} \left( \frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y} \right)$

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

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

$\nabla f(x,y,z) = {\partial f \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}$
$\nabla \cdot \bold{v} = \frac{1}{r^2} \frac{\partial r^2 v_r}{\partial r} + \frac{1}{r\sin\theta} \frac{\partial \sin\theta v_{\theta}}{\partial \theta} + \frac{1}{r\sin\theta} \frac{\partial v_{\phi}}{\partial \phi}$
$\nabla \times \bold{v} = \frac{\boldsymbol{\hat r } }{r \sin \theta} [ \frac{\partial ( \sin \theta \bold{v}_\phi )}{\partial \theta} - \frac{\partial \bold{v}_\theta}{\partial \phi} ] + \frac { \boldsymbol{\hat \theta} }{r} [ \frac{1}{\sin \theta} \frac{\partial \bold{v}_r}{\partial \phi} - \frac{\partial ( r \bold{v}_\phi )}{\partial r} ] + \frac {\boldsymbol{\hat \phi} }{r} [ \frac{\partial ( r \bold{v}_\theta )}{\partial r} - \frac{\partial \bold{v}_r}{\partial \theta} ]$
$\nabla^2 f = \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{\partial f}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial }{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 f}{\partial \phi^2}$

Vector Integral formulas
Notes Operator

$\oint_S \bold{A} \cdot d \bold{a} = \int_V \nabla \cdot \bold{A} d \tau \qquad \text{(Divergence therorem)}$

$\oint_C \bold{A} \cdot d \bold{s} = \int_S ( \nabla \times \bold{A} ) \cdot d \bold{a} \qquad \textrm{(Stokes' therorem)}$

$\oint_S u d \bold{a} = \int_V \nabla u d \tau$

$\oint_S \bold{A} \times d \bold{a} = - \int_V ( \nabla \times \bold{A} ) d \tau$

$\oint_C u d \bold{s} = - \int_S \nabla u \times d \bold{a}$

$\oint_S u \bold{A} \cdot d \bold{a} = \int_V [ \bold{A} \cdot ( \nabla u ) + u ( \nabla \cdot \bold{A} ) ]d \tau$

$\oint_S \bold{B} ( \bold{A} \cdot d \bold{a} ) = \int_V [( \bold{A} \cdot \nabla ) \bold{B} + \bold{B} ( \nabla \cdot \bold{A}) ] d \tau$

Formulas Involving Relative Coordinates
Notes Operator

$\frac{\partial f ( \bold{R} )}{\partial x} = - \frac{\partial f ( \bold{R} )}{\partial x^{'}}$

$\nabla f ( \bold{R} ) = - \nabla^{'} f ( \bold{R} )$

$\nabla \cdot \bold{A} ( \bold{R} ) = - \nabla^{'} \cdot \bold{A} ( \bold{R} )$

$\nabla \times \bold{A} ( \bold{R} ) = - \nabla^{'} \times \bold{A} ( \bold{R} )$

$\nabla^2 f ( \bold{R} )= \nabla^{'2} f ( \bold{R} )$

$\nabla R = - \nabla^{'} R = \frac{\bold{R}}{R} = \hat{\bold{R}}$

$\nabla ( \frac{1}{R} ) = - \nabla^{'} ( \frac{1}{R} ) = - \frac{\hat{\bold{R}}}{R^2} = - \frac{\bold{R}}{R^3}$

$\nabla^2 ( \frac{1}{R} )= \nabla^{'2} ( \frac{1}{R} ) = 0 \qquad ( \ R \neq 0 \ )$

keywords: double-angle, triple-angle, angle sum

Trigonometric Identities

Trigonometric Identities
Basic Definitions
Definition of tangent $\tan \theta = \frac{\sin \theta}{\cos\theta}$
Definition of cotangent $\cot \theta = \frac{\cos \theta}{\sin\theta} \$
Definition of secant $\sec \theta = \frac{1}{\cos \theta} \$
Definition of cosecant $\csc \theta = \frac{1}{\sin \theta} \$
Definition of versed sine (versine) $\text{versin } \theta = 1- \cos \theta \$
Definition of versed cosine (versine) $\text{vercosin } \theta = 1+ \cos \theta \$
Definition of coversed sine (coversine) $\text{coversin } \theta = \text{cvs } \theta = 1- \sin \theta \$
Definition of coversed cosine (covercosine) $\text{covercosin } \theta = 1+ \sin \theta \$
Definition of haversed sine (haversine) $\text{haversin } \theta = \frac{1- \cos \theta}{2}$
Definition of haversed cosine (havercosine) $\text{havercosin } \theta = \frac{1+ \cos \theta}{2}$
Definition of hacoversed sine (hacoversin) $\text{hacoversin } \theta = \frac{1 - \sin \theta}{2}$
Definition of hacoversed cosine (hacovercosin) $\text{hacovercosin } \theta = \frac{1 + \sin \theta}{2}$
Definition of exterior secant (exsec) $\text{exsec } \theta = \sec \theta - 1 \$
Definition of exterior cosecant (excosec) $\text{excosec } \theta = \csc \theta - 1 \$
Definition of chord (crd) $\text{crd } \theta = 2 \sin(\frac{\theta}{2})$
Pythagorean identity and other related identities
Pythagorean identity $\cos^2 \theta+\sin^2 \theta =1 \$
$\sin^2 \theta = 1-\cos^2 \theta \$
$\cos^2 \theta = 1-\sin^2 \theta \$
$\sec^2 \theta = 1+\tan^2 \theta \$
$\csc^2 \theta = 1+\cot^2 \theta \$
Half-Angle Formulas
Half-angle for sine $\sin \frac{\theta}{2} = \pm \sqrt{ \frac{1-\cos \theta}{2} } \$
Half-angle for cosine $\cos \frac{\theta}{2} = \pm \sqrt{ \frac{1+\cos \theta}{2} } \$
Half-angle for tangent $\tan \frac{\theta}{2} = \csc \theta - \cot \theta \$
Half-angle for tangent $\tan \frac{\theta}{2} =\pm\sqrt{\frac{1-\cos \theta}{ 1+\cos \theta }} \$
Half-angle for tangent $\tan \frac{\theta}{2} =\frac{\sin \theta}{ 1+\cos \theta } \$
Half-angle for tangent $\tan \frac{\theta}{2} =\frac{1-\cos \theta}{ \sin \theta } \$
Half-angle for cotangent $\cot \frac{\theta}{2} = \csc \theta + \cot \theta$
Half-angle for cotangent $\cot \frac{\theta}{2} = \frac{1 + \cos \theta}{\sin \theta}$
Half-angle for cotangent $\cot \frac{\theta}{2} = \pm \sqrt{1 + \cos \theta \over 1 - \cos \theta}$
Half-angle for cotangent $\cot \frac{\theta}{2} = \frac{\sin \theta}{1 - \cos \theta}$
Double-Angle Formulas
double-angle for sine $\sin 2 \theta = 2 \sin \theta \cos \theta \$
double-angle for sine $\sin 2 \theta = \frac{ 2 \tan \theta}{1+ \tan^2 \theta } \$
double-angle for cosine $\cos 2 \theta =\cos^2 \theta - \sin^2 \theta \$
double-angle for cosine $\cos 2 \theta =2 \cos^2 \theta - 1 \$
double-angle for cosine $\cos 2 \theta =1- 2 \sin^2 \theta \$
double-angle for cosine $\cos 2 \theta =\frac{1- \tan^2 \theta}{ 1+\tan^2 \theta } \$
double-angle for tangent $\tan 2\theta = \frac{2 \tan \theta} {1 - \tan^2 \theta}\,$
double-angle for cotangent $\cot 2\theta = \frac{\cot^2 \theta - 1}{2 \cot \theta}\,$
Triple-Angle Formulas
triple-angle for sine \begin{align}\sin 3\theta & = 3 \cos^2\theta \sin\theta - \sin^3\theta \\ & = 3\sin\theta - 4\sin^3\theta \end{align}
triple-angle for cosine \begin{align}\cos 3\theta & = \cos^3\theta - 3 \sin^2 \theta\cos \theta \\ & = 4 \cos^3\theta - 3 \cos\theta\end{align}
triple-angle for tangent $\tan 3\theta = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta}$
tripe-angle for cotangent $\cot 3\theta = \frac{3 \cot\theta - \cot^3\theta}{1 - 3 \cot^2\theta}$
Angle sum and difference identities
Sine $\sin \left( \theta\pm \alpha \right)=\sin \theta \cos \alpha \pm \cos \theta \sin \alpha$
Cosine $\cos \left(\theta\pm \alpha \right)= \cos \theta \cos \alpha \mp \sin \theta \sin \alpha$
Tangent $\tan \left(\theta\pm \alpha \right)= \frac {\tan \theta \pm \tan \alpha}{1 \mp \tan \theta \tan \alpha}$
Arcsine $\arcsin\alpha \pm \arcsin\beta = \arcsin(\alpha\sqrt{1-\beta^2} \pm \beta\sqrt{1-\alpha^2})$
Arccosine $\arccos\alpha \pm \arccos\beta = \arccos(\alpha\beta \mp \sqrt{(1-\alpha^2)(1-\beta^2)})$
Arctangent $\arctan\alpha \pm \arctan\beta = \arctan\left(\frac{\alpha \pm \beta}{1 \mp \alpha\beta}\right)$

keywords: magnitude, conjugate, de Moivre, Euler

Complex Number Identities and 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

(Used in ECE301 and ECE438)

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

(used in ECE301 and ECE438)

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

(used in ECE301 and ECE438)

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

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 $\lambda$ $\,E[X] = \frac{1}{\lambda},\ \ Var(X) = \frac{1}{\lambda^2}\,$

Calculus

keywords:quotient rule, chain rule, Leibniz rule

Derivatives

General Rules
Derivative of a constant $\frac{d}{dx}\left( c \right) = 0, \ \text{ for any constant }c$
$\frac{d}{dx}\left( c x \right) = c, \ \text{ for any constant }c$
Linearity $\frac{d}{dx}\left( c_1 u_1+c_2 u_2 \right) = c_1 \frac{d}{dx}\left( u_1 \right)+c_2 \frac{d}{dx}\left( u_2 \right), \ \text{ for any constants }c_1, c_2$
Quotient rule $\frac{d}{dx} ( \frac{u}{v} ) = \frac{v ( \frac{du}{dx} ) - u ( \frac{dv}{dx} )}{v^2}$
Exponent rule $\frac{d}{dx} ( u^n ) = n u^{n-1} \frac{du}{dx}$
Chain rule $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$
$\frac{du}{dx} = \frac{1}{\frac{dx}{du}}$
$\frac{dy}{dx} = \frac{dy}{du}/\frac{dx}{du}$
Leibnitz Rule for Successive Derivatives of a Product
first order $\frac{d}{dx}\left( u v \right)= u \frac{dv }{dx} + v \frac{du }{dx}$
second order $\frac{d^2}{dx^2}\left( u v \right)= u \frac{d^2v }{dx^2} + 2\frac{du }{dx}\frac{dv }{dx}+ v \frac{d^2u }{dx^2}$
third order $\frac{d^3}{dx^3}\left( u v \right)= u \frac{d^3v }{dx^3} + 3 \frac{du }{dx}\frac{d^2v }{dx^2}+ 3 \frac{du^2 }{dx^2}\frac{d v }{dx}+ v \frac{d^3u }{dx^3}$
n-th order $\frac{d^n}{dx^n}\left( u v \right)= u \frac{d^n v }{dx^n} + \left( \begin{array}{cc}n \\ 1 \end{array}\right) \frac{du }{dx}\frac{d^{n-1}v }{dx^{n-1}} + \left( \begin{array}{cc}n \\ 2 \end{array}\right) \frac{d^2u}{dx^2}\frac{d^{n-2}v }{dx^{n-2}}+ \ldots + v \frac{d^n u }{dx^n}$
Derivatives of trigonometric functions
$\frac {d}{dx} \sin u = \cos u \frac{du}{dx}$
$\frac {d}{dx} \cos u = - \sin u \frac{du}{dx}$
$\frac {d}{dx} \tan u = \frac{1}{\cos^2 u} \frac{du}{dx}$
$\frac {d}{dx} \cot u = - \frac{1}{\sin^2 u} \frac{du}{dx}$
$\frac {d}{dx} \frac{1}{\cos u} = \frac{\tan u}{\cos u} \frac{du}{dx}$
$\frac {d}{dx} \frac{1}{\sin u} = - \frac{\cot u}{\sin u} \frac{du}{dx}$
$\frac {d}{dx} \arcsin u = \frac{1}{\sqrt{1-u^2}} \frac{du}{dx} \qquad ( - \frac{\pi}{2} < \arcsin u < \frac{\pi}{2} )$
$\frac {d}{dx} \arccos u = - \frac{1}{\sqrt{1-u^2}} \frac{du}{dx} \qquad ( 0 < \arccos u < \pi )$
$\frac {d}{dx} \arctan u = \frac{1}{1+u^2} \frac{du}{dx} \qquad ( - \frac{\pi}{2} < \arctan u < \frac{\pi}{2} )$
$\frac {d}{dx} \arccot u = - \frac{1}{1+u^2} \frac{du}{dx} \qquad ( 0 < \arccot u < \pi )$
Derivatives of exponential and logarithm functions
$\frac{d}{dx} \log_a u = \frac{log_a e}{u} \frac{du}{dx} \qquad a \neq 0,1$
$\frac{d}{dx} \ln u = \frac{d}{dx} log_e u = \frac{1}{u} \frac{du}{dx}$
$\frac{d}{dx} a^u = a^u \ln a \frac{du}{dx}$
$\frac{d}{dx} e^u = e^u \frac{du}{dx}$
$\frac{d}{dx} u^v = \frac{d}{dx} e^{v ln u} = e^{v ln u} \frac {d}{dx} [ v ln u ] = v u^{v-1} \frac{du}{dx} + u^v ln u \frac{dv}{dx}$
Derivatives of hyperbolic functions
$\frac{d}{dx} \sinh u = \cosh u \frac{du}{dx}$
$\frac{d}{dx} \cosh u = \sinh u \frac{du}{dx}$
$\frac{d}{dx} \tanh u = \frac{1}{\cosh^2 u} \frac{du}{dx}$
$\frac{d}{dx} \coth u = - \frac{1}{\sinh^2 u} \frac{du}{dx}$
$\frac{d}{dx} \frac{1}{\cosh u} = - \frac{\tanh u}{\cosh u} \frac{du}{dx}$
$\frac{d}{dx} \frac{1}{\sinh u} = - \frac{\coth u}{\sinh u} \frac{du}{dx}$
$\frac{d}{dx}\ \operatorname{arsinh}\ u = \frac{1}{\sqrt{u^2+1}} \frac{du}{dx}$
$\frac{d}{dx}\ \operatorname{arcosh}\ u = \frac{1}{\sqrt{u^2-1}} \frac{du}{dx}$
$\frac{d}{dx}\ \operatorname{artanh}\ u = \frac{1}{1-u^2} \frac{du}{dx} \qquad ( \ -1 < u < 1 \ )$
$\frac{d}{dx}\ \operatorname{arcoth}\ u = \frac{1}{1-u^2} \frac{du}{dx} \qquad ( \ u > 1 \ or \ u < -1 \ )$

Transforms

Table of (bidirectional) Laplace Transform Pairs and Properties

(used in ECE301 and ECE438)

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)} \$
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}}\$

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 $\frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 =$

Useful tricks and techniques

keywords: partial fraction, how to.

A guide to Partial Fraction Expansion

This page is meant as a comprehensive review of partial fraction expansion. Partial fraction expansion allows us to fit functions to the known ones given by the known Fourier Transform pairs table.

First, the denominator must be of a higher degree than the numerator. If this is not the case, then perform long division to make it such. Note: for the remainder of this guide it is assumed that the denominator is of a higher degree than the numerator. There are four cases that arise which one must consider:

Case 1 : Denominator is a product of distinct linear factors.

$\frac{(polynomial)}{(a_1x+b_1)(a_2x+b_2)\cdots(a_kx+b_k)}=\frac{A_1}{a_1x+b_1}+\frac{A_2}{a_2x+b_2}+\cdots+\frac{A_k}{a_kx+b_k}$

Case 2 : Denominator is a product of linear factors, some of which are repeated.

$\frac{(polynomial)}{(a_1x+b_1)^r}=\frac{A_1}{a_1x+b_1}+\frac{A_2}{(a_1x+b_1)^2}+\cdots+\frac{A_r}{(a_1x+b_1)^r}$

Case 3 : Denominator contains irreducible quadratic factors, none of which is repeated.

$\frac{(polynomial)}{ax^2+bx+c}=\frac{Ax+B}{ax^2+bx+c}$

Case 4 : Denominator contains a repeated irreducible quadratic factor.

$\frac{(polynomial)}{(ax^2+bx+c)^r}=\frac{A_1x+B_1}{ax^2+bx+c}+\frac{A_2x+B_2}{(ax^2+bx+c)^2}+\cdots+\frac{A_rx+B_r}{(ax^2+bx+c)^r}$

Example encompassing all of the above cases:

$\frac {(polynomial degree < 10)}{x(x-1)(x^2+x+1)(x^2+1)^3} = \frac {A}{x} + \frac {B}{x^2+x+1} + \frac {Cx+D}{x^2+1} + \frac {Ex+F}{(x^2+1)^2} + \frac {Gx+H}{(x^2+1)^2} + \frac {Ix+J}{(x^2+1)^3}$

General method to find out what the Capital Letters equal:

The most general method is to multiply both sides by the left hand side's denominator. This clears all fractions. Then, expand and collect terms on the right hand side. Equate the right and left hand sides' coefficients. This leads to a system of linear equations which can be solved for obtaining the Capital Letters. Tricks to obtaining the Capital Letters quickly: Capital Letters whose denominator is the highest power of its kind can be found directly as follows:

First, multiply both sides by its denominator.

Second, find the value of x which would make the denominator equal 0.

Third, evaluate the equation for that value of x. This leaves you with the Capital Letter on the right and its value on the left. Example of finding Capital Letters:

$\frac{4x}{(x-1)^2(x+1)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x+1}$

The above trick can be used to find B and C but not A.

$B=\frac{4x}{x+1}\;(evaluated\;at\;x=1)=\frac{4}{2}=2$
$C=\frac{4x}{(x-1)^2}\;(evaluated\;at\;x=-1)=\frac{-4}{4}=-1$

A is then found using the general method given above.

\begin{align} 4x&=A(x-1)(x+1)+B(x+1)+C(x-1)^2 \\ &=A(x-1)(x+1)+2(x+1)-(x-1)^2 \\ &=(A-1)x^2+4x+(-A+1) \end{align}
$A-1=0 \rightarrow A=1$

Note: The four cases for finding the form of the partial fraction expansion as well as the general method of finding the capital letters were adapted from section 7.4 in Calculus Early Transcendentals, 5e. (Our Calc I, II, & III book.) The tricks to obtaining the capital letters quickly are from learning to do the Laplace Transform in ECE 202.

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