OpenLLM-France/wikimedia huggingface dataset:
# Government and Binding Theory/DP Hypothesis

Although we have so far treated the determiner and possessor as the possible specifiers of the noun phrase, there are several problems with this:
* Determiners are functional, but possessors are lexical.
* Determiners are words, but possessors are phrases.
* The two are not in complementary distribution in some languages, like Hungarian. Since the two can co-exist, they cannot both fill up the specifier slot.
* There is some evidence suggesting that the determiner is a head. For example, incorporation is a process that whereby two heads are joined to form a complex head. In French, the prepositions à and de can join with the determiners le and les through incorporation:
(1a) *Je suis à le parc → Je suis au parc (1b) *Je demande à les étudiants → Je demande aux étudiants (1c) *la plupart de les étudiants → la plupart des étudiants
Since the determiner seems to be a head, the determiner phrase has been suggested as its maximal projection.

## The Determiner Phrase

The determiner phrase suggested by Abney has an X-bar structure like any other phrase:
This gives us a 'bare-bones' structure of the DP. Let's flesh it out by filling up the specifier slots of the DP and the NP.

### Phrases with possessives

Let's consider the case of DPs with possessives. Under the DP hypothesis, the possessive phrase is assumed to be the specifier of the DP. This leads to an important question, though: What sits in the head position of the DP?
For now, we will insert here an empty pos determiner:

### Phrases with post-determiners

The DP hypothesis also provides a home for post-determiners, like many and several. They have determiner-like properties, but are not exactly determiners, and they have adjunct-like properties, but aren't exactly adjuncts, as we can see here:
(2a) The many people (2b) *The many several people
(2a) shows that they aren't in complementary distribution with determiners, but (2b) shows that they aren't adjuncts. The specifier position of the NP can now accommodate these determiner-like entities:
# Fractals/Iterations in the complex plane/julia/interior

" Stable orbits of polynomials may :
* converge to a (super)-attracting fixed point,
* (coverge) to a parabolic fixed point (where the multiplier is a root of unity),
* belong to a rotation domain (a simply connected domain on which the dynamics is conjugate to a rotation)." Lasse Rempe-Gillen

# Local discrete complex dynamics

* attracting : hyperbolic dynamics
** superattracting : the very fast ( = exponential) convergence to periodic cycle ( fixed point )
* parabolic component = slow ( lazy ) dynamics = slow ( exponential slowdown) convergence to parabolic fixed point ( periodic cycle)
* Siegel disc component = rotation around fixed point and never reach the fixed point
When Julia set is disconnected ther is no interior of Julia set ( critical fixed point is repelling ( or attracting to infinity)

# References

* ↑ mathoverflow : attractive-basins-and-loops-in-julia-sets
# Fractals/Iterations in the complex plane/mset components number

Number of the Mandelbrot set's components
Whole number of all components by period
Counting Hyperbolic Components in the Main Molecule

# See also

* order of hyperbolic components
* centers lists
* shape of components

# References

* ↑ Counting Hyperbolic Components in the Main Molecule by Schinella D'Souza
# Electronics Fundamentals
## Introduction to Electronics

Electronics is the study of electronic components, circuits, devices characteristics to form electronic machines and systems.

## Electronics Components

* Passive Components
* Electromagnetic Devices
* Semi Conductor Devices
* Integrated Cicuit

## Electronics Circuits

* Resistors Networks
* Series Circuit
* Parallel Circuit
* Diode Circuit
* Transistor Circuit
* Operational Amplifier Circuit

## Electronics Devices

* Electronic Filter
* Electronic Oscillator
* Electronic Amplifier
* Electronic Rectifier
* Electronic Converter
* Electronic Regulator

## Further reading

* Electronics, Electronics Handbook, Electronics Communication , Practical Electronics
* Digital Electronics, Circuit Theory, Communication Systems, Signals and Systems
# Electronics Fundamentals/Diode Circuit
## Digital Logic Gate

AND Gate
OR Gate
# Electronics Fundamentals/Electromagnetic Devices/Electric Generator
## Electric Generator

An electric generator is an electromagnetic device that can produce alternate current electric (AC) through the interaction of an electromagnet and a conventional magnet.
# Electronics Communication/Signal Processing/Signal Transformation
## Signal Transformation

* Laplace Transform
* Fourier Transform
* Z Transform

## See Also

* Engineering Tables
# Electronics Communication/Signal Processing/Signal Transformation/Fourier Transform
## Fourier Transform

Fourier Transform is a process to transfor function in Frequency domain to time domain
    $F(j\omega )={\mathcal {F}}\left\{f(t)\right\}=\int _{-\infty }^{\infty }f(t)e^{-j\omega t}dt$

## Inverse Fourier Transform

    ${\mathcal {F}}^{-1}\left\{F(j\omega )\right\}=f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(j\omega )e^{j\omega t}d\omega$

## Table of Fourier Transforms

This table contains some of the most commonly encountered Fourier transforms.
| | Time Domain | Frequency Domain |
| | $x(t)={\mathcal {F}}^{-1}\left\{X(\omega )\right\}$ | $X(\omega )={\mathcal {F}}\left\{x(t)\right\}$ |
| 1 | $X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt$ | $x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega$ |
| 2 | $1\,$ | $2\pi \delta (\omega )\,$ |
| 3 | $-0.5+u(t)\,$ | ${\frac {1}{j\omega }}\,$ |
| 4 | $\delta (t)\,$ | $1\,$ |
| 5 | $\delta (t-c)\,$ | $e^{-j\omega c}\,$ |
| 6 | $u(t)\,$ | $\pi \delta (\omega )+{\frac {1}{j\omega }}\,$ |
| 7 | $e^{-bt}u(t)\,(b>0)$ | ${\frac {1}{j\omega +b}}\,$ |
| 8 | $\cos \omega _{0}t\,$ | $\pi \left[\delta (\omega +\omega _{0})+\delta (\omega -\omega _{0})\right]\,$ |
| 9 | $\cos(\omega _{0}t+\theta )\,$ | $\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})+e^{j\theta }\delta (\omega -\omega _{0})\right]\,$ |
| 10 | $\sin \omega _{0}t\,$ | $j\pi \left[\delta (\omega +\omega _{0})-\delta (\omega -\omega _{0})\right]\,$ |
| 11 | $\sin(\omega _{0}t+\theta )\,$ | $j\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})-e^{j\theta }\delta (\omega -\omega _{0})\right]\,$ |
| 12 | ${\mbox{rect}}\left({\frac {t}{\tau }}\right)\,$ | $\tau {\mbox{sinc}}\left({\frac {\tau \omega }{2\pi }}\right)\,$ |
| 13 | $\tau {\mbox{sinc}}\left({\frac {\tau t}{2\pi }}\right)\,$ | $2\pi {\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,$ |
| 14 | $\left(1-{\frac {2|t|}{\tau }}\right){\mbox{rect}}\left({\frac {t}{\tau }}\right)\,$ | ${\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau \omega }{4\pi }}\right)\,$ |
| 15 | ${\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau t}{4\pi }}\right)\,$ | $2\pi \left(1-{\frac {2|\omega |}{\tau }}\right){\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,$ |
| 16 | $e^{-a|t|},\Re \{a\}>0\,$ | ${\frac {2a}{a^{2}+\omega ^{2}}}\,$ |
| Notes: | ${\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)$ ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function | ${\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)$ ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function |
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# Electronics Fundamentals/Electromagnetic Devices
## Electromagnetic Devices

* Relay
* Transformer
* Motor
* Electric Generator
* Microphone
* Speaker
* Antenna
# Electronics Communication/Speaker
## Speaker

Speaker is an ElectroAcoustic Device using Electromagnetic principle to perform conversion of Electric Signal into original Sound Wave
# Electronics Communication/Signal Processing/Signal Sampling & Desampling

Signal sampling and desampling is the process of sending modulated pulses from a transmitter to a receiver

## Signal Sampling

* PAM which stands for Pulse Amplitude Modulation
* PFM which stands for Pulse Frequency Modulation
* PAM which stands for Pulse Angle Modulation
# Fukushima Aftermath/Wikibooks on related topica

* Physics Study Guide
* This Quantum World
* Radiation Oncology
* Basic Physics of Nuclear Medicine
# Fukushima Aftermath/Subsequent investigations Fukushima Daichi nuclear accidents
## "IAEA Launches Investigation into Japan Nuclear Accident"

In a headline article by MITSURU OBE, the Wall Street Journal highlighted the actions of an international bureaucracy in Geneva. Deprecating the emergency response process, the article opened as follows:
"A team of experts from the International Atomic Energy Agency began its investigation into the Fukushima Daiichi nuclear plant accident on Tuesday, amid allegations that slow decision making by plant operators and regulators may have exacerbated what has become the world's worst nuclear accident since the 1986 Chernobyl disaster."
* ↑ http://online.wsj.com/article/SB10001424052702304066504576342903404364040.html
# Fukushima Aftermath/Who's Who

* Albert Einstein
* John Gofman
* Jim Green
* Edward B. Lewis
* Bertrand Russell
* Karen Silkwood
* Alice Stewart
* Ernest J. Sternglass
* Edward Teller
# Fukushima Aftermath/Section One

"A nuclear power plant on top of three major earthquake faults. Blueprints read backwards. Stuck valves on critical failsafe components.
The Diablo plant had a lot of people worried.
Then came Fukushima: 9.0 earthquake. Tragic monster tsunami. Thousands dead. And things went downhill from there."
Source: Fukushima Aftermath/Diablo Nuclear Renaissance or Industry Meltdown?
The tragic developments at the Fukushima Daiichi nuclear power plant have brought public opinion to the same lows experienced in the aftermath of the Three Mile Island accident. Indeed, the consequences are more severe, and the full impact is yet to be known. Yet new awareness of global warming has already caused many environmentalists to defect from the fold of anti-nuclear advocacy. Thus, everyone is wondering what will indeed be the fate of the nuclear industry.
The nuclear industry inherently requires long range planning for capital formation, licensing and waste disposal. In Collapse of an Industry,: Nuclear Power and the Contradictions of U.S.Policy, John L. Campbell contrasts how regulatory mechanisms in the US and Germany differ from France, which tends to exclude citizen participation in public policy. Sweden, according to Campbell, occupies something of a middle ground, with extensive citizen input in the policy formation phase, unlike the US and Germany wherein citizen input tends to be at the implementation level, or France, where there is virtually no citizen participation.

## Wind power not recognized as the solution

Despite hopes that wind power might solve the putative global climate change problem associated with green house gas emissions, expert opinion asserts that, in, for instance, the US, only 10-15 % of electricity generation is reachable by 2020.
Despite widespread enthusiasm for German use of wind power, it has been asserted by authorities such as Professor Burton Richter, Stanford, that the problems of grid stabilization, intermittency, and power storage are insurmountable and that taxpayer subsidies create a misleading appearance of viability unless storage can be scaled in terms of "gigawatt days not megawatt hours".
Others refute this and maintain that wind power can be scaled up with better results than such sceptics are willing to allow.

## See also

Fukushima Aftermath/Linear no-threshold model

## References

* ↑ Author(s): John A. Hall| American Journal of Sociology, Vol. 94, No. 6 (May, 1989), pp. 1449-1451|Publisher The University of Chicago Press|Stable URL: http://www.jstor.org/stable/2780975
* ↑ Collapse of an Industry: Nuclear Power and the Contradictions of U.S. Policy (Cornell Studies in Political Economy) John L. Campbell
* ↑ Burton Richter, Director Emeritus of the SLAC National Accelerator Laboratory and Matthew Wald, reporter with the Washington Bureau of The New York Times |Videoconference discuss the unfolding of events in the Fukushima Daiichi nuclear disaster|http://www.youtube.com/watch?v=YSvwwfqyguc%7CApril 11, 2011||1:05 approximate location of quote
# Fukushima Aftermath/US public opinion shifts in the aftermath

A new CBS poll found that 50 percent of Americans now oppose building new nuclear plants, up from 34 percent the last time the poll was conducted in 2008.
Only 43 percent, meanwhile, approve of building more nuclear plants - a drop of 14 points from the 2008 poll.
More recently, polling in 2022 found that 35% in US favor and about 25% oppose nuclear power development.

## References

* ↑ http://www.theatlantic.com/technology/archive/2011/03/in-fukushima-aftermath-half-of-americans-now-oppose-building-new-nuclear-plants/72911/
* ↑ http://www.wltx.com/news/national/article/129423/142/Poll-Support-for-New-Nuclear-Plants-Drops
* ↑ Leppert, Rebecca. "Americans continue to express mixed views about nuclear power". Pew Research Center. Retrieved 2023-06-10.
# Engineering Handbook/Motion

    MotionLinearNon Linear Speed$v={\frac {s}{t}}$v(t) Acceleration$a={\frac {v}{t}}$$a={\frac {dv(t)}{dt}}$ Distance$S=vt=at^{2}$$s=\int v(t)dt$ Force$F=ma={\frac {mv}{t}}={\frac {p}{t}}$$a=m{\frac {dv(t)}{dt}}$ Work$W=Fs=Fvt=pv$$F\int v(t)dt$ Energy$E={\frac {W}{t}}=Fv=Fat=pa$${\frac {F}{t}}\int v(t)dt$ Pressure$a={\frac {F}{A}}$
# Engineering Handbook/PhotoElectric
## PhotoElectric

Light of certain Energy shines on some material generates Electricity . This effect is called PhotoElectric
    hf = hfo + ${\frac {1}{2}}mv^{2}$
    f > fo for v > 0
Photon Light of Quantum Energy E = hf has the capability to generate Electricity when shines on Light conducting materials consists
* Visible Light
* UVF Light
* X Ray
* Gamma Ray
# Engineering Handbook/Physics/Force

    ForceFormula Gravity Force$F_{g}=G{\frac {mM}{r^{2}}}$ Motion Force$F_{a}=ma={\frac {mv}{t}}={\frac {p}{t}}$ Elastic Force$F_{s}=F$ $-kx=m{\frac {d^{2}x}{dt^{2}}}$ $-ky=m{\frac {d^{2}y}{dt^{2}}}$ Electrostatic Force$F_{Q}=K{\frac {Q_{+}Q_{-}}{r^{2}}}$ Electromotive Force$F_{E}=QE$ ElectromoMagnetomotive Force$F_{B}=\pm QvB$ ElectroMagnetic Force$F_{E}B=F_{E}+F_{B}=Q(E\pm vB)$
# Engineering Handbook/Mathematics/Fourier Transformation
## Fourier Transform

    $F(j\omega )={\mathcal {F}}\left\{f(t)\right\}=\int _{-\infty }^{\infty }f(t)e^{-j\omega t}dt$

## Inverse Fourier Transform

    ${\mathcal {F}}^{-1}\left\{F(j\omega )\right\}=f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(j\omega )e^{j\omega t}d\omega$

## Table of Fourier Transforms

This table contains some of the most commonly encountered Fourier transforms.
| | Time Domain | Frequency Domain |
| | $x(t)={\mathcal {F}}^{-1}\left\{X(\omega )\right\}$ | $X(\omega )={\mathcal {F}}\left\{x(t)\right\}$ |
| 1 | $X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt$ | $x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega$ |
| 2 | $1\,$ | $2\pi \delta (\omega )\,$ |
| 3 | $-0.5+u(t)\,$ | ${\frac {1}{j\omega }}\,$ |
| 4 | $\delta (t)\,$ | $1\,$ |
| 5 | $\delta (t-c)\,$ | $e^{-j\omega c}\,$ |
| 6 | $u(t)\,$ | $\pi \delta (\omega )+{\frac {1}{j\omega }}\,$ |
| 7 | $e^{-bt}u(t)\,(b>0)$ | ${\frac {1}{j\omega +b}}\,$ |
| 8 | $\cos \omega _{0}t\,$ | $\pi \left[\delta (\omega +\omega _{0})+\delta (\omega -\omega _{0})\right]\,$ |
| 9 | $\cos(\omega _{0}t+\theta )\,$ | $\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})+e^{j\theta }\delta (\omega -\omega _{0})\right]\,$ |
| 10 | $\sin \omega _{0}t\,$ | $j\pi \left[\delta (\omega +\omega _{0})-\delta (\omega -\omega _{0})\right]\,$ |
| 11 | $\sin(\omega _{0}t+\theta )\,$ | $j\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})-e^{j\theta }\delta (\omega -\omega _{0})\right]\,$ |
| 12 | ${\mbox{rect}}\left({\frac {t}{\tau }}\right)\,$ | $\tau {\mbox{sinc}}\left({\frac {\tau \omega }{2\pi }}\right)\,$ |
| 13 | $\tau {\mbox{sinc}}\left({\frac {\tau t}{2\pi }}\right)\,$ | $2\pi {\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,$ |
| 14 | $\left(1-{\frac {2|t|}{\tau }}\right){\mbox{rect}}\left({\frac {t}{\tau }}\right)\,$ | ${\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau \omega }{4\pi }}\right)\,$ |
| 15 | ${\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau t}{4\pi }}\right)\,$ | $2\pi \left(1-{\frac {2|\omega |}{\tau }}\right){\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,$ |
| 16 | $e^{-a|t|},\Re \{a\}>0\,$ | ${\frac {2a}{a^{2}+\omega ^{2}}}\,$ |
| Notes: | ${\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)$ ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function | ${\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)$ ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function |
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# Engineering Handbook/Mathematics/Integral
## Integral
### Indefinite Integral

    $\int f(x)dx$
    Table of Properties of Integrals   Rule Conditions 1 $\int a\,dx=ax$ 2 Homogeniety $\int af(x)\,dx=a\int f(x)\,dx$ 3 Associativity $\int {\left(f\pm g\pm h\pm \cdots \right)\,dx}=\int f\,dx\pm \int g\,dx\pm \int h\,dx\pm \cdots$ 4 Integration by Parts $\int _{a}^{b}fg'\,dx=\left[fg\right]_{a}^{b}-\int _{a}^{b}gf'\,dx$ 4 General Integration by Parts $\int f^{(n)}g\,dx=f^{(n-1)}g'-f^{(n-2)}g''+\ldots +(-1)^{n}\int fg^{(n)}\,dx$ 5 $\int f(ax)\,dx={\frac {1}{a}}\int f(x)\,dx$ 6 Substitution Rule $\int g\{f(x)\}\,dx=\int g(u){\frac {dx}{du}}\,du=\int {\frac {g(u)}{f'(x)}}\,du$ $u=f(x)\,$ 7 $\int x^{n}\,dx={\frac {x^{n+1}}{n+1}}$ $n\neq -1\,$ 8 $\int {\frac {1}{x}}\,dx=\ln |x|$ 9 $\int e^{x}\,dx=e^{x}$ 10 $\int a^{x}\,dx={\frac {a^{x}}{\ln |a|}}$ $a\neq 1$ Notes: f, g, h are functions of x a, n are constants. The constant of integration, C has been omitted from this table. It should be included in the working of the equation if applicable. This box: view • talk •
    Integral Value Remarks 1 $\int c\,dx$ $cx+C\,$ 2 $\int x^{n}\,dx$ ${\frac {x^{n+1}}{n+1}}+C$ $n\neq -1$ 3 $\int {\frac {1}{x}}\,dx$ $\ln {\left|x\right|}+C$ 4 $\int {1 \over {a^{2}+x^{2}}}\,dx$ ${1 \over a}\arctan {x \over a}+C$ 5 $\int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx$ $\arcsin {x \over a}+C$ 6 $\int {-1 \over {\sqrt {a^{2}-x^{2}}}}\,dx$ $\arccos {x \over a}+C$ 7 $\int {1 \over x{\sqrt {x^{2}-a^{2}}}}\,dx$ ${1 \over a}{\mbox{arcsec}}\,{|x| \over a}+C$ 8 $\int \ln {x}\,dx$ $x\ln {x}-x+C\,$ 9 $\int \log _{b}{x}\,dx$ $x\log _{b}{x}-x\log _{b}{e}+C\,$ 10 $\int e^{x}\,dx$ $e^{x}+C\,$ 11 $\int a^{x}\,dx$ ${\frac {a^{x}}{\ln {a}}}+C$ 12 $\int \sin {x}\,dx$ $-\cos {x}+C\,$ 13 $\int \cos {x}\,dx$ $\sin {x}+C\,$ 14 $\int \tan {x}\,dx$ $-\ln {\left|\cos {x}\right|}+C\,$ 15 $\int \cot {x}\,dx$ $\ln {\left|\sin {x}\right|}+C\,$ 16 $\int \sec {x}\,dx$ $\ln {\left|\sec {x}+\tan {x}\right|}+C\,$ 17 $\int \csc {x}\,dx$ $-\ln {\left|\csc {x}+\cot {x}\right|}+C\,$ 18 $\int \sec ^{2}x\,dx$ $\tan x+C\,$ 19 $\int \csc ^{2}x\,dx$ $-\cot x+C\,$ 20 $\int \sec {x}\,\tan {x}\,dx$ $\sec {x}+C\,$ 21 $\int \csc {x}\,\cot {x}\,dx$ $-\csc {x}+C\,$ 22 $\int \sin ^{2}x\,dx$ ${\frac {1}{2}}(x-\sin x\cos x)+C\,$ 23 $\int \cos ^{2}x\,dx$ ${\frac {1}{2}}(x+\sin x\cos x)+C\,$ 24 $\int \sin ^{n}x\,dx$ $-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx$ 25 $\int \cos ^{n}x\,dx$ $-{\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx$ 26 $\int \arctan {x}\,dx$ $x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C$ 27 $\int \sinh x\,dx$ $\cosh x+C\,$ 28 $\int \cosh x\,dx$ $\sinh x+C\,$ 29 $\int \tanh x\,dx$ $\ln |\cosh x|+C\,$ 30 $\int {\mbox{csch}}\,x\,dx$ $\ln \left|\tanh {x \over 2}\right|+C$ 31 $\int {\mbox{sech}}\,x\,dx$ $\arctan(\sinh x)+C\,$ 32 $\int \coth x\,dx$ $\ln |\sinh x|+C\,$

### Definite Integral

    $\int _{a}^{b}f(x),dx$
    $\int _{a}^{b}f(x)\,dx=F(b)-F(a).$
Furthermore, for every x in the interval (a, b),
    ${\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).$

## References

* Integral
# Engineering Handbook/Oscillation

    OscillationFormula -->$-Fx=F$ $-kx=m{\frac {d^{2}x}{dt^{2}}}$ ${\frac {d^{2}x}{dt^{2}}}+{\frac {k}{m}}=0$ $s^{2}+{\frac {k}{m}}=0$ $s=\pm {\sqrt {-{\frac {k}{m}}}}t$ $s=\pm j\omega t$ $x=e^{(\pm j\omega t)}$ $x=e^{(j\omega t)}+e^{(-j\omega t)}$ $x=ASin\omega t$ $-Fy=F$ $-ky=m{\frac {d^{2}y}{dt^{2}}}$ ${\frac {d^{2}y}{dt^{2}}}+{\frac {k}{m}}=0$ $s^{2}+{\frac {k}{m}}=0$ $s=\pm {\sqrt {-{\frac {k}{m}}}}t$ $s=\pm j\omega t$ $y=e^{(\pm j\omega t)}$ $y=e^{(j\omega t)}+e^{(-j\omega t)}$ $y=ASin\omega t$ ${\mathrm {d} ^{2}\theta \over \mathrm {d} t^{2}}+{g \over \ell }\theta =0$ $\theta (t)=\theta _{0}\cos \left({\sqrt {g \over \ell }}t\right)\quad \quad \quad \quad |\theta _{0}|\ll 1$
# Engineering Handbook/Mathematics/Trigonometry
## Trigonometry Identities Table

    $\sin ^{2}+\cos ^{2}=1$$1+\tan ^{2}=\sec ^{2}$ $\sin({\frac {\pi }{2}}-\theta )=\cos \theta$$\cos({\frac {\pi }{2}}-\theta )=\sin \theta$ $\sec({\frac {\pi }{2}}-\theta )=\csc \theta$$\csc({\frac {\pi }{2}}-\theta )=\sec \theta$ $\sin(-\theta )=-\sin \theta$$\cos(-\theta )=\cos \theta$ $\sin 2\theta =2\sin \theta \cos \theta$$\cos 2\theta =\cos ^{2}-\sin ^{2}=2\cos ^{2}\theta -1=1-2\sin ^{2}\theta$ $\sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}$$\cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}$ $\sin \alpha +\sin \beta =2\sin({\frac {\alpha +\beta }{2}})\cos({\frac {\alpha -\beta }{2}})$$\sin \alpha -\sin \beta =2\cos({\frac {\alpha +\beta }{2}})\sin({\frac {\alpha -\beta }{2}})$ $\cos \alpha +\cos \beta =2\cos({\frac {\alpha +\beta }{2}})\cos({\frac {\alpha -\beta }{2}})$$\cos \alpha -\cos \beta =-2\sin({\frac {\alpha +\beta }{2}})\sin({\frac {\alpha -\beta }{2}})$ $\sin \alpha \sin \beta ={\frac {1}{2}}[\cos(\alpha -\beta )-\cos(\alpha +\beta )]$$\cos \alpha \cos \beta ={\frac {1}{2}}[\cos(\alpha -\beta )+\cos(\alpha +\beta )]$ $\sin \alpha \cos \beta ={\frac {1}{2}}[\sin(\alpha +\beta )+\sin(\alpha -\beta )]$$1+\cot ^{2}=\csc ^{2}$ $e^{j\theta }=\cos \theta +j\sin \theta$$\cos \theta ={\frac {e^{j\theta }+e^{-j\theta }}{2}}$ $\tan({\frac {\pi }{2}}-\theta )=\cot \theta$$\cot({\frac {\pi }{2}}-\theta )=\tan \theta$ $\tan(-\theta )=-\tan \theta$$\tan 2\theta ={\frac {2\tan \theta }{1-tan^{2}\theta }}$ $\tan ^{2}\theta ={\frac {1-\cos 2\theta }{1+\cos 2\theta }}$$\sin \theta ={\frac {e^{j\theta }-e^{-j\theta }}{j2}}$

## References

* Trigonometric Identities