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All automorphisms of all Calkin algebras

http://www.arxiv.org/abs/1007.4034
...The Proper Forcing Axiom implies all automorphisms of every Calkin algebra associated with an infinite-dimensional complex Hilbert space and the ideal of compact operators are inner. As a means of the proof we introduce the notion of Po...
Ilijas Farah


Default all is dangerous!!

http://www.arxiv.org/abs/1105.4395
...We show that the default-all propagation scheme for database annotations is dangerous. Dangerous here means that it can propagate annotations to the query output which are semantically irrelevant to the query the user asked. Thi...
Wolfgang Gatterbauer, Alexandra Meliou, Dan Suciu


Including All the Lines

http://www.arxiv.org/abs/0912.5371
...I present a progress report on including all the lines in the linelists, including all the lines in the opacities, including all the lines in the model atmosphere and spectrum synthesis calculations, producing high-resolution, high-signal-to-noise at...
Robert L. Kurucz


All CHSH polytopes

http://www.arxiv.org/abs/1402.6914
...or which Bell polytopes the CHSH inequality is also the unique (non-trivial) facet. We prove that the CHSH inequality is the unique facet for all bipartite polytopes where at least one party has a binary choice of dichotomic measurements, irrespectiv...
Stefano Pironio


All Dielectric Optical Nanoantennas

http://www.arxiv.org/abs/1206.5597
...ployed in the Yagi-Uda geometry for creating highly efficient optical nanoantennas. By comparing plasmonic and dielectric nanoantennas, we demonstrate that all-dielectric nanoantennas may exhibit better radiation efficiency also allowing more compact...
Alexander E. Krasnok, Andrey E. Miroshnichenko, Pavel A. Belov, Yuri S. Kivshar


All the Bell inequalities

http://www.arxiv.org/abs/quant-ph/9807017
...Clauser- Horne inequalities corresponding to the faces of a convex polytope. It is plausible that quantum systems whose density matrix has a positive partial transposition satisfy all these inequalities, and therefore are compatible with local object...
Asher Peres


All Superparticles are BPS

http://www.arxiv.org/abs/1401.5116
... for which all supersymmetries are manifest. An example is the N=1 massive D=10 superparticle, which actually has N=2 supersymmetry and is equivalent to the action for a D0-brane of IIA superstring theory....
Luca Mezincescu, Alasdair J. Routh, Paul K. Townsend


Determining All Universal Tilers

http://www.arxiv.org/abs/1110.1715
...A universal tiler is a convex polyhedron whose every cross-section tiles the plane. In this paper, we introduce a certain slight-rotating operation for cross-sections of pentahedra. Based on a selected initial cross-section and by applying the slight...
David G. L. Wang


Generating All Wigner Functions

http://www.arxiv.org/abs/hep-th/0011137
...y to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often ...
Thomas Curtright, Tsuneo Uematsu, Cosmas Zachos


Unifying All Blazars

http://www.arxiv.org/abs/astro-ph/9812158
...We propose a new physical model for the unification of all Blazars. The blazar phenomenology can be accounted for by a sequence in the source power and intensity of the diffuse radiation field surrounding the relativistic jet. This regulates the equi...
G. Fossati



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Limits




$\displaystyle{I}{f}{f{{\left({a}\right)}}}={g{{\left({a}\right)}}}={0}\ \text{ then }\ \lim_{{{x}\to{a}}}\frac{{f{{\left({x}\right)}}}}{{g{{\left({x}\right)}}}}=\frac{{{f}'{\left({a}\right)}}}{{{g}'{\left({a}\right)}}}$

L'Hopital’s rule



$\displaystyle{n}^{{c}}{x}^{{n}}\to{0}\ \text{ as }\ {n}\to\infty{\quad\text{if}\quad}$





$\displaystyle\frac{{x}^{{n}}}{{{n}!}}\to{0}\ \text{ as }\ {n}\to\infty\ \text{ (any fixed x)}$





$\displaystyle{\left({1}+{x}{/}{n}\right)}^{{n}}\to{e}^{{x}}\ \text{ as }\ {n}\to\infty, {x}{\ln{\to}}{0}\ \text{ as }\ {x}\to{0}$



Integration


Differentiation of an integral

$\displaystyle\frac{{d}}{{{d}\alpha}}{\int_{{{a}{\left(\alpha\right)}}}^{{{b}{\left(\alpha\right)}}}}{f{{\left({x},\alpha\right)}}}{\left.{d}{x}\right.}={f{{\left({b},\alpha\right)}}}\frac{{{d}{b}}}{{{d}\alpha}}-{f{{\left({a},\alpha\right)}}}\frac{{{d}{a}}}{{{d}\alpha}}+{\int_{{{a}{\left(\alpha\right)}}}^{{{b}{\left(\alpha\right)}}}}\frac{\partial}{{\partial\alpha}}{f{{\left({x},\alpha\right)}}}{\left.{d}{x}\right.}$

If f (x, α) is a function of x containing a parameter α and the limits of integration a and b are functions of α then

Dirac delta-function

$\displaystyle\delta{\left({t}-\tau\right)}=\frac{{1}}{{{2}\pi}}{\int_{{-{\infty}}}^{{+\infty}}}{\exp{{\left[{i}\omega{\left({t}-\tau\right)}\right]}}}{d}\omega$



Integration by parts

$\displaystyle{\int_{{a}}^{{b}}}{u}{d}{v}={u}{v}$



Vector Algebra


Equation of a line

$\displaystyle{A}{p}\oint{r}\equiv{\left({x},{y},{z}\right)}\ \text{ lies on a line passing through a point }\ {a}\ \text{ and parallel to vector }\ \vec{{b}}\ \text{ if r=a+lambda vec b, with }\ \lambda\ \text{ a real number.}$



Vector product

$\displaystyle{A}\times{B}={n}$

Vector multiplication (not commutative. It's a antisymetric operator.)

Vector triple product

$\displaystyle{A}\times{\left({B}\times{C}\right)}={\left({A}\cdot{C}\right)}{B}-{\left({A}\cdot{B}\right)}{C},     {\left({A}\times{B}\right)}\times{C}={\left({A}\cdot{C}\right)}{B}-{\left({B}\cdot{C}\right)}{A}$



Non-orthogonal basis

$\displaystyle{A}={A}_{{1}}{e}_{{1}}+{A}_{{2}}{e}_{{2}}+{A}_{{3}}{e}_{{3}}, {A}_{{1}}={e}'\cdot{A}\ \text{ where }\ {e}'=\frac{{{e}_{{2}}\times{e}_{{3}}}}{{{e}_{{1}}\cdot{\left({e}_{{2}}\times{e}_{{3}}\right)}}}\ \text{  , and similarly for }\ {A}_{{2}}{\quad\text{and}\quad}{A}_{{3.}}$



Scalar product

$\displaystyle{A}\cdot{B}={\left|{A}\right|}{\mid}{B}{\cos{\theta}}={A}_{{x}}{B}_{{x}}+{A}_{{y}}{B}_{{y}}+{A}_{{z}}{B}_{{z}}={\left[{A}_{{x}},{A}_{{y}},{A}_{{z}}\right]}{\left[\matrix{{B}_{{x}}\\{B}_{{y}}\\{B}_{{z}}}\right]}$

Scalar mutiplication (commutative)

Summation convention

$\displaystyle{a}={a}_{{i}}{e}_{{i}}=\sum{a}_{{i}}{e}_{{i}}$

expression in a basis, summation over i=1..3



$\displaystyle{a}\cdot{b}={a}_{{i}}{b}_{{i}}$

expression of scalar multiplication, summation over i=1..3

Module of a vector

$\displaystyle{I}{f}{i},{j},{k}\ \text{ are orthonomormal vectors and }\ {A}={A}_{\xi}+{A}_{{y}}{j}+{A}_{{z}}{k},\ \text{ then }\ $

in orthonormal basis

Equation of a plane

$\displaystyle\text{A point }\ {r}\equiv{\left({x},{y},{z}\right)}\text{}{i}{s}{o}{n}{a}{p}{l}{a}\ne{\quad\text{if}\quad}{e}{i}{t}{h}{e}{r}: $





$\displaystyle{\left({a}\times{b}\right)}_{{i}}=\epsilon_{{{i}{j}{k}}}{a}_{{j}}{b}_{{k}}\ \text{ , where }\ \epsilon_{{123}}={1}; \epsilon_{{{i}{j}{k}}}=-\epsilon_{{{i}{k}{j}}}; \epsilon_{{{i}{j}{k}}}\epsilon_{{{k}{l}{m}}}=\delta_{{{i}{l}}}\delta_{{{j}{m}}}-\delta_{{{i}{m}}}\delta_{{{j}{l}}}$

expression of vector multiplication, summation over i=1..3

Scalar triple product

$\displaystyle{A}\times{B}\cdot{C}={A}\cdot{B}\times{C}={\left|\matrix{{A}_{{x}}&{A}_{{y}}&{A}_{{z}}\\{B}_{{x}}&{B}_{{y}}&{B}_{{z}}\\{C}_{{x}}&{C}_{{y}}&{C}_{{z}}}\right|}=-{A}\times{C}\cdot{B}$



Complex form of Fourier series




$\displaystyle{y}{\left({x}\right)}\approx{\sum_{{{m}={1}}}^{{M}}}{C}_{{m}}{e}^{{{i}{m}{x}}},  {C}_{{m}}=\frac{{1}}{{{2}\pi}}{\int_{{-\pi}}^{\pi}} {y}{\left({x}\right)}{e}^{{-{i}{m}{x}}}{\left.{d}{x}\right.}$

Range 2pi

Fourier transforms




$\displaystyle{y}{\left({t}\right)}={\sum_{{{m}=-\infty}}^{\infty}}\delta{\left({t}-{m}\tau\right)}  ,    \hat{{y}}{\left(\omega\right)}={\sum_{{{n}=-\infty}}^{\infty}}\delta{\left(\omega-{2}\pi{n}{/}\tau\right)}$

sampling function



$\displaystyle{y}{\left({t}\right)}=\frac{{1}}{{{2}\pi}}{\int_{{-{\infty}}}^{\infty}}\hat{{y}}{\left({f}\right)}{e}^{{{i}{2}\pi{f}{t}}}{d}{f},  \hat{{y}}{\left({f}\right)}={\int_{{-{\infty}}}^{\infty}}{y}{\left({t}\right)}{e}^{{-{i}{2}\pi{f}{t}}}{\left.{d}{t}\right.}$

Replace omega by 2pi*f



$\displaystyle{y}{\left({t}\right)}=\frac{{1}}{{\pi}}{\int_{{0}}^{\infty}}\hat{{y}}{\left(\omega\right)}{\cos{\omega}}{t} {d}\omega,  \hat{{y}}{\left(\omega\right)}={2}{\int_{{0}}^{\infty}}{y}{\left({t}\right)}{\cos{\omega}}{t} {\left.{d}{t}\right.}$

If y is even



$\displaystyle{y}{\left({t}\right)}=\frac{{1}}{{\pi}}{\int_{{0}}^{\infty}}\hat{{y}}{\left(\omega\right)}{\sin{\omega}}{t} {d}\omega,  \hat{{y}}{\left(\omega\right)}={2}{\int_{{0}}^{\infty}}{y}{\left({t}\right)}{\sin{\omega}}{t} {\left.{d}{t}\right.}$

If y is odd



$\displaystyle{y}{\left({t}\right)}={\left\lbrace{\left({a},  \right.}\right.}$

Top Hat <-> Sine cardinal



$\displaystyle{y}{\left({t}\right)}={\exp{{\left(-{t}^{{2}}{/}{{t}_{{0}}^{{2}}}\right)}}}  ,    \hat{{y}}{\left(\omega\right)}={t}_{{0}}\sqrt{\pi}{\exp{{\left(-\omega^{{2}}{{t}_{{o}}^{{2}}}{/}{4}\right)}}}$

Gaussian <-> Gaussian in frequency



$\displaystyle{y}{\left({t}\right)}={\left\lbrace{\left({a}{\left({1}-\right.}\right.}\right.}$

Saw-tooth <-> Sine cardinal square



$\displaystyle{y}{\left({t}\right)}={f{{\left({t}\right)}}}{e}^{{{i}\omega_{{0}}{t}}}  ,    \hat{{y}}{\left(\omega\right)}=\hat{{f}} {\left(\omega-\omega_{{0}}\right)}$

modulated function

Fourier transforms

$\displaystyle{y}{\left({t}\right)}=\frac{{1}}{{{2}\pi}}{\int_{{-{\infty}}}^{\infty}}\hat{{y}}{\left(\omega\right)}{e}^{{{i}\omega{t}}}{d}\omega,  \hat{{y}}{\left(\omega\right)}={\int_{{-{\infty}}}^{\infty}}{y}{\left({t}\right)}{e}^{{-{i}\omega{t}}}{\left.{d}{t}\right.}$

Range R

Numerical Analysis


Finding the zeros of equations

$\displaystyle\text{If the equation is }\ {y}={f{{\left({x}\right)}}}\ \text{ and }\ {x}_{{n}}\ \text{ is an approximation to the root, then }\ $

Newton

Finding the zeros of equations

$\displaystyle\text{If the equation is }\ {y}={f{{\left({x}\right)}}}\ \text{ and }\ {x}_{{n}}\ \text{ is an approximation to the root, then }\ $

Linear interpolation



$\displaystyle\text{If }\ \frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}\right.}}={f{{\left({x},{y}\right)}}}\ \text{ then }\ $

Euler method

Integration Reduction formulae


Reduction formulae

$\displaystyle{F}{\quad\text{or}\quad}{a}{n}{y}{p}\gt-{1},{\int_{{0}}^{\infty}}{x}^{{p}}{e}^{{-{{x}}}}{\left.{d}{x}\right.}=\pi{n}{{t}_{{o}}^{\infty}}{x}^{{{p}-{1}}}{e}^{{-{{x}}}}{\left.{d}{x}\right.}={p}!.   {\left(-{1}{/}{2}\right)}!=\sqrt{\pi},  {\left({1}{/}{2}\right)}!=\sqrt{\pi}{/}{2},\ \text{  etc.}$

Factorials



$\displaystyle{F}{\quad\text{or}\quad}{a}{n}{y}{p},{q}\gt-{1}, {\int_{{0}}^{{1}}} {x}^{{p}}{\left({1}-{x}\right)}^{{q}}{\left.{d}{x}\right.}=\frac{{{p}!{q}!}}{{{\left({p}+{q}+{1}\right)}!}}$





$\displaystyle{I}{f}{m},{n}\ \text{ are integers, }{\int_{{0}}^{{\pi{/}{2}}}} {{\sin}^{{m}}\theta} {{\cos}^{{n}}\theta}{d}\theta=\frac{{{m}-{1}}}{{{m}+{n}}} {\int_{{0}}^{{\pi{/}{2}}}}{{\sin}^{{{m}-{2}}}\theta} {{\cos}^{{n}}\theta}{d}\theta=\frac{{{n}-{1}}}{{{m}+{n}}} {\int_{{0}}^{{\pi{/}{2}}}}{{\sin}^{{m}}\theta} {{\cos}^{{{n}-{2}}}\theta}{d}\theta$

Trigonometrical



$\displaystyle{I}{f}{I}_{{n}}={\int_{{0}}^{\infty}} {x}^{{n}} {\exp{{\left(-\alpha{x}^{{2}}\right)}}} {\left.{d}{x}\right.}\ \text{ then }\ {I}_{{n}}=\frac{{{n}-{1}}}{{{2}\alpha}}{I}_{{{n}-{2}}}, {I}_{{o}}=\frac{{1}}{{2}}\sqrt{{\frac{\pi}{\alpha}}}, {I}_{{1}}=\frac{{1}}{{{2}\alpha}}$

Other

Relations of Hyperbolic functions: Inverse functions




$\displaystyle{{\sech}^{{-{{1}}}}{\left(\frac{{x}}{{a}}\right)}}={\ln{{\left(\frac{{a}}{{x}}+\sqrt{{\frac{{a}^{{2}}}{{x}^{{2}}}-{1}}}\right)}}} \ \text{ for }\ {0}\lt{x}&\leq{a}$





$\displaystyle{{\text{cosh}}^{{-{{1}}}}{\left(\frac{{x}}{{a}}\right)}}={\ln{{\left(\frac{{{x}+\sqrt{{{x}^{{2}}-{a}^{{2}}}}}}{{a}}\right)}}} \ \text{ for }\ {x}\ge{a}$





$\displaystyle{c}{o}{{\sech}^{{-{{1}}}}{\left(\frac{{x}}{{a}}\right)}}={\ln{{\left(\frac{{a}}{{x}}+\sqrt{{\frac{{a}^{{2}}}{{x}^{{2}}}+{1}}}\right)}}} \ \text{ for }\ {x}\ne{0}$





$\displaystyle{{\text{tanh}}^{{-{{1}}}}{\left(\frac{{x}}{{a}}\right)}}=\frac{{1}}{{2}}{\ln{{\left(\frac{{{a}+{x}}}{{{a}-{x}}}\right)}}} \ \text{ for }\ {x}^{{2}}\lt{a}^{{2}}$





$\displaystyle{{\coth}^{{-{{1}}}}{\left(\frac{{x}}{{a}}\right)}}=\frac{{1}}{{2}}{\ln{{\left(\frac{{{x}+{a}}}{{{x}-{a}}}\right)}}} \ \text{ for }\ {x}^{{2}}\gt{a}^{{2}}$



Inverse functions

$\displaystyle{{\text{sinh}}^{{-{{1}}}}{\left(\frac{{x}}{{a}}\right)}}={\ln{{\left(\frac{{{x}+\sqrt{{{x}^{{2}}+{a}^{{2}}}}}}{{a}}\right)}}} \ \text{ for }\ -\infty\lt{x}\lt+\infty$



Combination of errors


Sum of Errors

$\displaystyle{I}{f}{Z}={A}\pm{B}\pm{C}, {\left(\sigma_{{Z}}\right)}^{{2}}={\left(\sigma_{{A}}\right)}^{{2}}+{\left(\sigma_{{B}}\right)}^{{2}}+{\left(\sigma_{{C}}\right)}^{{2}}$



Power of Errors

$\displaystyle{I}{f}{Z}={A}^{{m}}, \frac{\sigma_{{Z}}}{{Z}}={m}\frac{\sigma_{{A}}}{{A}}$



Ln of errors

$\displaystyle{I}{f}{Z}={\ln{{A}}}, \sigma_{{Z}}=\frac{\sigma_{{A}}}{{A}}$



Exp of errors

$\displaystyle{I}{f}{Z}={\exp{{A}}}, \frac{\sigma_{{Z}}}{{Z}}=\sigma_{{A}}$



Combination of errors

$\displaystyle\text{If }\ {Z}={Z}{\left({A},{B},\ldots\right)}\ \text{ (with }\ {A},{B},\text{etc. independent) then }\ $



Product of Errors

$\displaystyle{I}{f}{Z}={A}{B}{\quad\text{or}\quad}{A}{/}{B}, {\left(\frac{\sigma_{{Z}}}{{Z}}\right)}^{{2}}={\left(\frac{\sigma_{{A}}}{{A}}\right)}^{{2}}+{\left(\frac{\sigma_{{B}}}{{B}}\right)}^{{2}}$





$\displaystyle{I}{f}{Z}={\exp{{A}}}, \frac{\sigma_{{Z}}}{{Z}}=\sigma_{{A}}$

(v) exp

Relations for the plane triangle




$\displaystyle{\cos{{A}}}=\frac{{{b}^{{2}}+{c}^{{2}}-{a}^{{2}}}}{{{2}{b}{c}}}$





$\displaystyle{\tan{{\left(\frac{{{A}-{B}}}{{2}}\right)}}}=\frac{{{a}-{b}}}{{{a}+{b}}}{\cot{{\left(\frac{{C}}{{2}}\right)}}}$





$\displaystyle\text{area}=\frac{{1}}{{2}}{a}{b}{\sin{{C}}}=\frac{{1}}{{2}}{b}{c}{\sin{{A}}}=\frac{{1}}{{2}}{c}{a}{\sin{{B}}}=\sqrt{{{s}{\left({s}-{a}\right)}{\left({s}-{b}\right)}{\left({s}-{c}\right)}}}\text{, where }\ {s}=\frac{{1}}{{2}}{\left({a}+{b}+{c}\right)}\text{area}=\frac{{1}}{{2}}{\left|\in\right|}{C}=\frac{{1}}{{2}}{b}{c}{\sin{{A}}}=\frac{{1}}{{2}}{c}{a}{\sin{{B}}}=\sqrt{{{s}{\left({s}-{a}\right)}{\left({s}-{b}\right)}{\left({s}-{c}\right)}}}\text{, where }\ {s}=\frac{{1}}{{2}}{\left({a}+{b}+{c}\right)}$





$\displaystyle\frac{{a}}{{\sin{{A}}}}=\frac{{b}}{{\sin{{B}}}}=\frac{{c}}{{\sin{{C}}}}=\text{diameter of circumscribed circle}$

In a plane triangle with angles A,B,C and sides opposite a,b, and c respectively



$\displaystyle{a}^{{2}}={b}^{{2}}+{c}^{{2}}-{2}{b}{\mathcal{{o}}}{s}{A}$





$\displaystyle{a}={b}{\cos{{C}}}+{\mathcal{{o}}}{s}{B}$



Relations for the spherical triangle




$\displaystyle\frac{{\sin{{a}}}}{{\sin{{A}}}}=\frac{{\sin{{b}}}}{{\sin{{B}}}}=\frac{{\sin{{c}}}}{{\sin{{C}}}}$

In a spherical triangle with angles A,B,C and sides opposite a,b, and c respectively



$\displaystyle{\cos{{a}}}={\cos{{b}}}{\cos{{c}}}+{\sin{{b}}}{\sin{{c}}}{\cos{{A}}}$





$\displaystyle{\cos{{A}}}=-{\cos{{B}}}{\cos{{C}}}+{\sin{{B}}}{\sin{{C}}}{\cos{{a}}}$



Power series with real variables




$\displaystyle{e}^{{x}}={1}+{x}+\frac{{x}^{{2}}}{{{2}!}}+\ldots+\frac{{x}^{{n}}}{{{n}!}}+…$

exponential



$\displaystyle{\tan{{x}}}={x}+\frac{{1}}{{3}}{x}^{{3}}+\frac{{2}}{{15}}{x}^{{5}}+\ldots$

tangent



$\displaystyle{\sin{{x}}}=\frac{{{e}^{{{i}{x}}}-{e}^{{-{i}{x}}}}}{{{2}{i}}}={x}-\frac{{x}^{{3}}}{{{3}!}}+\frac{{x}^{{5}}}{{{5}!}}+\ldots$

sine



$\displaystyle{\cos{{x}}}=\frac{{{e}^{{{i}{x}}}+{e}^{{-{i}{x}}}}}{{2}}={1}-\frac{{x}^{{2}}}{{{2}!}}+\frac{{x}^{{4}}}{{{4}!}}-\frac{{x}^{{6}}}{{{6}!}}+\ldots$

cosine



$\displaystyle{{\sin}^{{-{{1}}}}{x}}={x}+\frac{{1}}{{2}}\frac{{x}^{{3}}}{{3}}+\frac{{{1}\cdot{3}}}{{{2}\cdot{4}}}\frac{{x}^{{5}}}{{5}}+\ldots$

arcsine

Logarithm

$\displaystyle{\ln{{\left({1}+{x}\right)}}}={x}-\frac{{x}^{{2}}}{{2}}+\frac{{x}^{{3}}}{{3}}+\ldots+{\left(-{1}\right)}^{{{n}+{1}}}\frac{{x}^{{n}}}{{n}}+\ldots$

ln



$\displaystyle{{\tan}^{{-{{1}}}}{x}}={x}-\frac{{1}}{{3}}{x}^{{3}}+\frac{{1}}{{5}}{x}^{{5}}-\ldots$

arctangent

Basic Trigonometric Fomulae




$\displaystyle{\sin{{A}}}+{\sin{{B}}}={2}{\sin{{\left(\frac{{{A}+{B}}}{{2}}\right)}}}{\cos{{\left(\frac{{{A}-{B}}}{{2}}\right)}}}$





$\displaystyle{\cos{{A}}}-{\cos{{B}}}=-{2}{\sin{{\left(\frac{{{A}+{B}}}{{2}}\right)}}}{\sin{{\left(\frac{{{A}-{B}}}{{2}}\right)}}}$





$\displaystyle{\sin{{\left({A}\pm{B}\right)}}}={\sin{{A}}}{\cos{{B}}}\pm{\cos{{A}}}{\sin{{B}}}$





$\displaystyle{{\sec}^{{2}}{A}}-{{\tan}^{{2}}{A}}={1}$





$\displaystyle{\cos{{A}}}{\cos{{B}}}=\frac{{{\cos{{\left({A}+{B}\right)}}}+{\cos{{\left({A}-{B}\right)}}}}}{{2}}$





$\displaystyle{\cos{{\left({A}\pm{B}\right)}}}={\cos{{A}}}{\cos{{B}}}\pm{\left(-{1}\right)}{\sin{{A}}}{\sin{{B}}}$





$\displaystyle{{\sin}^{{3}}{A}}=\frac{{{3}{\sin{{A}}}-{\sin{{3}}}{A}}}{{4}}$





$\displaystyle{{\cos}^{{2}}{A}}=\frac{{{1}+{\cos{{2}}}{A}}}{{2}}$





$\displaystyle{c}{o}{{\sec}^{{2}}{A}}-{{\cot}^{{2}}{A}}={1}$





$\displaystyle{\sin{{A}}}-{\sin{{B}}}={2}{\cos{{\left(\frac{{{A}+{B}}}{{2}}\right)}}}{\sin{{\left(\frac{{{A}-{B}}}{{2}}\right)}}}$





$\displaystyle{\sin{{A}}}{\sin{{B}}}=\frac{{{\cos{{\left({A}-{B}\right)}}}-{\cos{{\left({A}+{B}\right)}}}}}{{2}}$





$\displaystyle{\sin{{2}}}{A}={2}{\sin{{A}}}{\cos{{A}}}$





$\displaystyle{{\sin}^{{2}}{A}}=\frac{{{1}-{\cos{{2}}}{A}}}{{2}}$





$\displaystyle{{\cos}^{{3}}{A}}=\frac{{{3}{\cos{{A}}}={\cos{{3}}}{A}}}{{4}}$





$\displaystyle{\tan{{\left({A}\pm{B}\right)}}}=\frac{{{\tan{{A}}}\pm{\tan{{B}}}}}{{{1}\pm{\left(-{1}\right)}{\tan{{A}}}{\tan{{B}}}}}$





$\displaystyle{\cos{{2}}}{A}={{\cos}^{{2}}{A}}-{{\sin}^{{2}}{A}}$





$\displaystyle{\cos{{A}}}+{\cos{{B}}}={2}{\cos{{\left(\frac{{{A}+{B}}}{{2}}\right)}}}{\cos{{\left(\frac{{{A}-{B}}}{{2}}\right)}}}$





$\displaystyle{\sin{{A}}}{\cos{{B}}}=\frac{{{\sin{{\left({A}+{B}\right)}}}+{\sin{{\left({A}-{B}\right)}}}}}{{2}}$





$\displaystyle{\tan{{2}}}{A}=\frac{{{2}{\tan{{A}}}}}{{{1}-{{\tan}^{{2}}{A}}}}$





$\displaystyle{{\cos}^{{2}}{A}}+{{\sin}^{{2}}{A}}={1}$



Relations of Hyperbolic functions




$\displaystyle{\text{cosh}{{x}}}+{\text{cosh}{{y}}}={2}{\text{cosh}{{\left(\frac{{1}}{{2}}{\left({x}+{y}\right)}\right)}}}{\text{cosh}{{\left(\frac{{1}}{{2}}{\left({x}-{y}\right)}\right)}}}$





$\displaystyle{\text{cosh}{{x}}}-{\text{cosh}{{y}}}={2}{\text{sinh}{{\left(\frac{{1}}{{2}}{\left({x}+{y}\right)}\right)}}}{\text{sinh}{{\left(\frac{{1}}{{2}}{\left({x}-{y}\right)}\right)}}}$





$\displaystyle{\text{tanh}{{x}}}=\sqrt{{{1}-{{\sech}^{{2}}{x}}}}$





$\displaystyle{\coth{{x}}}=\sqrt{{{\cos{{e}}}{c}{h}^{{2}}{\left({x}\right)}+{1}}}$





$\displaystyle{\text{sinh}{{\left({x}\pm{y}\right)}}}={\text{sinh}{{x}}}{\text{cosh}{{y}}}\pm{\text{cosh}{{x}}}{\text{sinh}{{y}}}$





$\displaystyle{\text{tanh}{{\left({x}{/}{2}\right)}}}=\frac{{{\text{cosh}{{x}}}-{1}}}{{\text{sinh}{{x}}}}=\frac{{\text{sinh}{{x}}}}{{{\text{cosh}{{x}}}+{1}}}$



Relations of the functions

$\displaystyle{\text{sinh}{{x}}}=-{\text{sinh}{{\left(-{x}\right)}}}$





$\displaystyle{\text{tanh}{{x}}}=-{\text{tanh}{{\left(-{x}\right)}}}$





$\displaystyle{\cos{{e}}}{c}{h}{x}=\sqrt{{{{\coth}^{{2}}{x}}-{1}}}$





$\displaystyle{\text{sinh}{{\left({2}{x}\right)}}}={2}{\text{sinh}{{x}}}{\text{cosh}{{x}}}$





$\displaystyle{\text{cosh}{{\left({x}\pm{y}\right)}}}={\text{cosh}{{x}}}{\text{cosh}{{y}}}\pm{\text{sinh}{{x}}}{\text{sinh}{{y}}}$





$\displaystyle{\text{sinh}{{x}}}-{\text{sinh}{{y}}}={2}{\text{cosh}{{\left(\frac{{1}}{{2}}{\left({x}+{y}\right)}\right)}}}{\text{sinh}{{\left(\frac{{1}}{{2}}{\left({x}-{y}\right)}\right)}}}$





$\displaystyle{\text{sinh}{{x}}}\pm{\text{cosh}{{x}}}=\frac{{{1}\pm{\text{tanh}{{\left({x}{/}{2}\right)}}}}}{{{1}\pm{\left(-{1}\right)}{\text{tanh}{{\left({x}{/}{2}\right)}}}}}={e}^{{\pm{x}}}$





$\displaystyle{\sech{{x}}}={\sech{{\left(-{x}\right)}}}$





$\displaystyle{\text{tanh}{{x}}}\pm{\text{tanh}{{y}}}=\frac{{\text{sinh}{{\left({x}\pm{y}\right)}}}}{{{\text{cosh}{{x}}}{\text{cosh}{{y}}}}}$





$\displaystyle{\text{tanh}{{\left({x}\pm{y}\right)}}}=\frac{{{\text{tanh}{{x}}}\pm{\text{tanh}{{y}}}}}{{{1}\pm{\text{tanh}{{x}}}{\text{tanh}{{y}}}}}$





$\displaystyle{\text{sinh}{{\left({x}{/}{2}\right)}}}=\sqrt{{\frac{{{\text{cosh}{{x}}}-{1}}}{{2}}}}$





$\displaystyle{\text{sinh}{{\left({3}{x}\right)}}}={3}{\text{sinh}{{x}}}+{4}{{\text{sinh}}^{{3}}{x}}$





$\displaystyle{\text{tanh}{{\left({2}{x}\right)}}}=\frac{{{2}{\text{tanh}{{x}}}}}{{{1}+{{\text{tanh}}^{{2}}{x}}}}$





$\displaystyle{\coth{{x}}}=-{\coth{{\left(-{x}\right)}}}$





$\displaystyle{\text{cosh}{{x}}}={\text{cosh}{{\left(-{x}\right)}}}$





$\displaystyle{\text{sinh}{{x}}}=\frac{{{2}{\text{tanh}{{\left({x}{/}{2}\right)}}}}}{{{1}-{{\text{tanh}}^{{2}}{\left({x}{/}{2}\right)}}}}=\frac{{\text{tanh}{{x}}}}{\sqrt{{{1}-{{\text{tanh}}^{{2}}{x}}}}}$





$\displaystyle{\text{cosh}{{\left({x}{/}{2}\right)}}}=\sqrt{{\frac{{{\text{cosh}{{x}}}+{1}}}{{2}}}}$





$\displaystyle{\text{cosh}{{\left({2}{x}\right)}}}={{\text{cosh}}^{{2}}{x}}+{{\text{sinh}}^{{2}}{x}}={2}{{\text{cosh}}^{{2}}-}{1}={1}+{2}{{\text{sinh}}^{{2}}{x}}$





$\displaystyle{\text{cosh}{{3}}}{x}={4}{{\text{cosh}}^{{3}}{x}}-{3}{\text{cosh}{{x}}}$





$\displaystyle{\coth{{x}}}\pm{\coth{{y}}}=\pm\frac{{\text{sinh}{{\left({x}\pm{y}\right)}}}}{{{\text{sinh}{{x}}}{\text{sinh}{{y}}}}}$





$\displaystyle{\cos{{e}}}{c}{h}{x}=-{\cos{{e}}}{c}{h}{\left(-{x}\right)}$





$\displaystyle{\text{cosh}{{x}}}=\frac{{{1}+{{\text{tanh}}^{{2}}{\left({x}{/}{2}\right)}}}}{{{1}-{{\text{tanh}}^{{2}}{\left({x}{/}{2}\right)}}}}=\frac{{1}}{\sqrt{{{1}-{{\text{tanh}}^{{2}}{x}}}}}$





$\displaystyle{\sech{{x}}}=\sqrt{{{1}-{{\text{tanh}}^{{2}}{x}}}}$





$\displaystyle{\text{tanh}{{\left({3}{x}\right)}}}=\frac{{{3}{\text{tanh}{{x}}}+{{\text{tanh}}^{{3}}{x}}}}{{{1}+{3}{{\text{tanh}}^{{2}}{x}}}}$





$\displaystyle{\text{sinh}{{x}}}+{\text{sinh}{{y}}}={2}{\text{sinh}{{\left(\frac{{1}}{{2}}{\left({x}+{y}\right)}\right)}}}{\text{cosh}{{\left(\frac{{1}}{{2}}{\left({x}-{y}\right)}\right)}}}$



Hyperbolic Functions




$\displaystyle{\text{sinh}{{x}}}=\frac{{1}}{{2}}{\left({e}^{{x}}-{e}^{{-{{x}}}}\right)}={x}+\frac{{x}^{{3}}}{{{3}!}}+\frac{{x}^{{5}}}{{{5}!}}+\ldots$

sine hyperbolic



$\displaystyle{\coth{{x}}}=\frac{{\text{cosh}{{x}}}}{{\text{sinh}{{x}}}}$

cotangent hyperbolic



$\displaystyle{\text{tanh}{{x}}}\to{1}$





$\displaystyle{\text{cosh}{{x}}}\approx-{\text{sinh}{{x}}}\to\frac{{e}^{{-{{x}}}}}{{2}}$

For large negative x



$\displaystyle{\text{cosh}{{i}}}{x}={\cos{{x}}}$





$\displaystyle{\text{tanh}{{x}}}\to-{1}$





$\displaystyle{\cos{{e}}}{c}{h}{x}=\frac{{1}}{{\text{sinh}{{x}}}}$

cosecant hyperbolic



$\displaystyle{\cos{{i}}}{x}={\text{cosh}{{x}}}$





$\displaystyle{\text{sinh}{{i}}}{x}={i}{\sin{{x}}}$





$\displaystyle{\text{tanh}{{x}}}=\frac{{\text{sinh}{{x}}}}{{\text{cosh}{{x}}}}$

tangent hyperbolic



$\displaystyle{{\text{cosh}}^{{2}}{x}}-{{\text{sinh}}^{{2}}{x}}={1}$





$\displaystyle{\text{cosh}{{x}}}=\frac{{1}}{{2}}{\left({e}^{{x}}+{e}^{{-{{x}}}}\right)}={1}+\frac{{x}^{{2}}}{{{2}!}}+\frac{{x}^{{4}}}{{{4}!}}+\ldots$

cosine hyperbolic



$\displaystyle{\sin{{i}}}{x}={i}{\text{sinh}{{x}}}$





$\displaystyle{\sech{{x}}}=\frac{{1}}{{\text{cosh}{{x}}}}$

secant hyperbolic



$\displaystyle{\text{cosh}{{x}}}\approx{\text{sinh}{{x}}}\to\frac{{e}^{{x}}}{{2}}$

For large positive x

Random Errors




$\displaystyle\overline{{x}}=\frac{{1}}{{n}}{\left({x}_{{1}}+{x}_{{2}}+\ldots+{x}_{{n}}\right)}$

Sample mean



$\displaystyle\sigma_{{m}}\approx\frac{\sigma}{\sqrt{{{n}}}}=\frac{{1}}{\sqrt{{{n}{\left({n}-{1}\right)}}}}{\left({{d}_{{1}}^{{2}}}+{{d}_{{2}}^{{2}}}+,,,+{{d}_{{n}}^{{2}}}\right)}^{{{1}{/}{2}}}=\frac{{1}}{\sqrt{{{n}{\left({n}-{1}\right)}}}}{\left[\sum{{x}_{{i}}^{{2}}}-\frac{{1}}{{n}}{\left(\sum{x}_{{i}}\right)}^{{2}}\right]}^{{{1}{/}{2}}}$

Standard deviation of mean



$\displaystyle{d}={x}-\overline{{x}}$

Residual



$\displaystyle\overline{{x}}\pm\sigma_{{m}}$

Result of n measurements is quoted as:

Range method

$\displaystyle\sigma\approx\frac{{r}}{\sqrt{{n}}}$

A quick but crude method of estimating Standard deviation of distribution by finding a rang r of a set of n readings



$\displaystyle{s}=\frac{{1}}{\sqrt{{n}}}{\left({{d}_{{1}}^{{2}}}+{{d}_{{2}}^{{2}}}+,,,+{{d}_{{n}}^{{2}}}\right)}^{{{1}{/}{2}}}$

Standard deviation of sample



$\displaystyle\sigma\approx\frac{{1}}{\sqrt{{{n}-{1}}}}{\left({{d}_{{1}}^{{2}}}+{{d}_{{2}}^{{2}}}+,,,+{{d}_{{n}}^{{2}}}\right)}^{{{1}{/}{2}}}$

Standard deviation of distribution

Discrete Fourier series




$\displaystyle{y}{\left({x}_{{n}}\right)}={c}_{{0}}+{c}_{{1}}{{\cos{{x}}}_{{n}}+}{c}_{{2}}{\cos{{2}}}{x}_{{n}}+\ldots+{c}_{{{N}-{1}}}{x}_{{n}}+{c}_{{N}}{\cos{{N}}}{x}_{{n}} $

for a function defined in [-pi,pi] which is sampled in the 2N equally spaced points x_n=nx/N



$\displaystyle{c}_{{0}}=\frac{{1}}{{{2}{N}}}\sum{y}{\left({x}_{{n}}\right)},  $

where the coefficients are

Convolution theorem


Convolution theorem

$\displaystyle{I}{f}{z}{\left({t}\right)}={\int_{{-{\infty}}}^{\infty}}{x}{\left(\tau\right)}{y}{\left({t}-\tau\right)}{d}\tau={\int_{{-{\infty}}}^{\infty}}{x}{\left({t}-\tau\right)}{y}{\left(\tau\right)}{d}\tau\equiv{x}{\left({t}\right)}\ast{y}{\left({t}\right)}\ \text{ , then }\ \hat{{z}}{\left(\omega\right)}=\hat{{x}}{\left(\omega\right)}\hat{{y}}{\left(\omega\right)}\text{. Conversely, }\ \hat{{{x}{y}}}=\hat{{x}}\ast\hat{{y}}.$



Spherical harmonics




$\displaystyle{\left[\frac{{1}}{{\sin{\theta}}}\frac{\partial}{{\partial\theta}}{\left({\sin{\theta}}\frac{\partial}{{\partial\theta}}\right)}+\frac{{1}}{{{\sin}^{{2}}\theta}}\frac{\partial^{{2}}}{{\partial\varphi^{{2}}}}\right]}{{Y}_{{l}}^{{m}}}+{l}{\left({l}+{1}\right)}{{Y}_{{l}}^{{m}}}={0}$

The normalized solutionsYlm(θ,φ) of the equation



$\displaystyle{{Y}_{{l}}^{{m}}}{\left(\theta,\varphi\right)}=\sqrt{{\frac{{{2}{l}+{1}}}{{{4}\pi}}\frac{{{\left({l}-{\left|{m}\right|}\right)}!}}{{{\left({l}+{\left|{m}\right|}\right)}!}}}}{{P}_{{l}}^{{m}}}{\left({\cos{\theta}}\right)}{e}^{{{i}{m}\varphi}}\times{\left\lbrace\matrix{{\left(-{1}\right)}^{{m}}\ \text{ for }\ {m}\ge{0}\\{1}\ \text{ for }\ {m}\lt{0}}\right.}$

are called spherical harmonics, and have values given by



$\displaystyle{{Y}_{{0}}^{{0}}}=\sqrt{{\frac{{1}}{{{4}\pi}}}}, {{Y}_{{1}}^{{0}}}=\sqrt{{\frac{{3}}{{{4}\pi}}}}{\cos{\theta}}, {{Y}_{{1}}^{{\pm{1}}}}=\pm{\left(-{1}\right)}\sqrt{{\frac{{3}}{{{8}\pi}}}}{\sin{\theta}} {e}^{{\pm{i}\varphi}},\ \text{ etc.}$

i.e.

Functions of Several Variables


Taylor series for two variables

$\displaystyle\phi{\left({x},{y}\right)}=\phi{\left({a}+{u},{b}+{v}\right)}=\phi{\left({a},{b}\right)}+{u}\frac{{\partial\phi}}{{\partial{x}}}+{v}\frac{{\partial\phi}}{{\partial{y}}}+\frac{{1}}{{{2}!}}{\left({u}^{{2}}\frac{{\partial^{{2}}\phi}}{{\partial{x}^{{2}}}}+{2}{u}{v}\frac{{\partial^{{2}}\phi}}{{\partial{x}\partial{y}}}+{v}^{{2}}\frac{{\partial^{{2}}\phi}}{{\partial{y}^{{2}}}}\right)}+\ldots\ \text{ where }\ {x}={a}+{u}, {y}={b}+{v}\ \text{ and the differential coefficients are evaluated at }\ {x}={a}, {y}={b}.$

phi(x,y)is well-behaved in the vicinity of x=a, y=b then it has a Taylor series

Parseval's theorem


Parseval's theorem

$\displaystyle{\int_{{-{\infty}}}^{\infty}}{y}^{\ast}{\left({t}\right)}{y}{\left({t}\right)}{\left.{d}{t}\right.}=\frac{{1}}{{{2}\pi}}{\int_{{-{\infty}}}^{\infty}}\hat{{y}}^{\ast}{\left(\omega\right)}\hat{{y}}{\left(\omega\right)}{d}\omega$



Taylor and Maclaurin Series




$\displaystyle{y}{\left({x}\right)}={y}{\left({a}+{u}\right)}={y}{\left({a}\right)}+{u}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}+\frac{{u}^{{2}}}{{{2}!}}\frac{{{d}^{{2}}{y}}}{{{\left.{d}{x}\right.}^{{2}}}}+\frac{{u}^{{3}}}{{{3}!}}\frac{{{d}^{{3}}{y}}}{{{\left.{d}{x}\right.}^{{3}}}}+…$

y has a Taylor series if y is well-behaved in the vicinity of a



$\displaystyle{y}{\left({x}\right)}={y}{\left({0}\right)}+{x}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}+\frac{{x}^{{2}}}{{{2}!}}\frac{{{d}^{{2}}{y}}}{{{\left.{d}{x}\right.}^{{2}}}}+\frac{{x}^{{3}}}{{{3}!}}\frac{{{d}^{{3}}{y}}}{{{\left.{d}{x}\right.}^{{3}}}}+\ldots$

Maclaurin series is a Taylor series with a=0

Fourier transforms in two dimensions


Fourier transforms in two dimensions

$\displaystyle\hat{{V}}{\left({k}\right)}=\int{V}{\left({r}\right)}{e}^{{-{i}{k}\cdot{r}}}{d}^{{2}}{r}={\int_{{0}}^{\infty}}{2}\pi{r}{V}{\left({r}\right)}{J}_{{0}}{\left({k}{r}\right)}{d}{r}\ \text{ if azimuthally symmetric}$



Dirac delta-function




$\displaystyle\delta{\left({t}-\tau\right)}=\frac{{1}}{{{2}\pi}}{\int_{{-{\infty}}}^{{+\infty}}}{\exp{{\left[{i}\omega{\left({t}-\tau\right)}\right]}}}{d}\omega$





$\displaystyle{\int_{{-{\infty}}}^{{+\infty}}}\delta{\left({t}-\tau\right)}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}={f{{\left(\tau\right)}}}$

for an arbitrary function f

Fourier transforms in three dimensions


Fourier transforms in three dimensions

$\displaystyle\hat{{V}}{\left({k}\right)}=\int{V}{\left({r}\right)}{e}^{{-{i}{k}\cdot{r}}}{d}^{{3}}{r}=\frac{{{4}\pi}}{{k}}{\int_{{0}}^{\infty}}{V}{\left({r}\right)}{r} {\sin{{k}}}{r} {d}{r}\ \text{ if spherically symmetric}\ \text{ and }\ {V}{\left({r}\right)}=\frac{{1}}{{\left({2}\pi\right)}^{{3}}} \int\hat{{V}}{\left({k}\right)}{e}^{{{i}{k}\cdot{r}}}{d}^{{3}}{k}$





Projects blah