OpenLLM-France/wikimedia huggingface dataset:
# PlanetPhysics/Exact Differential Equation

Let $R$ be a region in $\mathbb {R} ^{2}$ and let the functions\, $X\!:R\to \mathbb {R}$,\, $Y\!:R\to \mathbb {R}$ have continuous partial derivatives in $R$.\, The first order differential equation $X(x,\,y)+Y(x,\,y){\frac {dy}{dx}}=0$ or
   ${\begin{matrix}X(x,\,y)dx+Y(x,\,y)dy=0\end{matrix}}$
is called an exact differential equation , if the condition ${\frac {\partial X}{\partial y}}={\frac {\partial Y}{\partial x}}$ is true in $R$.
Then there is a function\, $f\!:R\to \mathbb {R}$\, such that the equation (1) has the form $d\,f(x,\,y)=0,$ whence its general integral is $f(x,\,y)=C.$
The solution function $f$ can be calculated as the line integral
   ${\begin{matrix}f(x,\,y):=\int _{P_{0}}^{P}[X(x,\,y)\,dx+Y(x,\,y)\,dy]\end{matrix}}$
along any curve $\gamma$ connecting an arbitrarily chosen point \,$P_{0}=(x_{0},\,y_{0})$\, and the point\, $P=(x,\,y)$\, in the region $R$ (the integrating factor is now $\equiv 1$).\\
Example. \, Solve the differential equation ${\frac {2x}{y^{3}}}\,dx+{\frac {y^{2}-3x^{2}}{y^{4}}}\,dy=0.$ This equation is exact, since ${\frac {\partial }{\partial y}}{\frac {2x}{y^{3}}}=-{\frac {6x}{y^{4}}}={\frac {\partial }{\partial x}}{\frac {y^{2}-3x^{2}}{y^{4}}}.$ If we use as the integrating way the broken line from\, $(0,\,1)$\, to\, $(x,\,1)$\, and from this to\, $(x,\,y)$,\, the integral (2) is simply $\int _{0}^{x}{\frac {2x}{1^{3}}}\,dx+\!\int _{1}^{y}{\frac {y^{2}-3x^{2}}{y^{4}}}\,dy={\frac {x^{2}}{y^{3}}}-{\frac {1}{y}}+1=x^{2}-{\frac {1}{y}}+{\frac {x^{2}}{y^{3}}}+1-x^{2}={\frac {x^{2}}{y^{3}}}-{\frac {1}{y}}+1.$ Thus we have the general integral ${\frac {x^{2}}{y^{3}}}-{\frac {1}{y}}=C$ of the given differential equation.
# PlanetPhysics/Euler 323 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on Euler Angle Sequences. \\
$R_{323}(\phi ,\theta ,\psi )=R_{3}(\psi )R_{2}(\theta )R_{3}(\phi )$ \\
The rotation matrices are
   $R_{3}(\psi )=\left[{\begin{matrix}c_{\psi }&s_{\psi }&0\\-s_{\psi }&c_{\psi }&0\\0&0&1\end{matrix}}\right]$
   $R_{2}(\theta )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]$
   $R_{3}(\phi )=\left[{\begin{matrix}c_{\phi }&s_{\phi }&0\\-s_{\phi }&c_{\phi }&0\\0&0&1\end{matrix}}\right]$
Carrying out the matrix multiplication from right to left \\
$R_{2}(\theta )R_{3}(\phi )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]\left[{\begin{matrix}c_{\phi }&s_{\phi }&0\\-s_{\phi }&c_{\phi }&0\\0&0&1\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }c_{\phi }&c_{\theta }s_{\phi }&-s_{\theta }\\-s_{\phi }&c_{\phi }&0\\s_{\theta }c_{\phi }&s_{\theta }s_{\phi }&c_{\theta }\end{matrix}}\right]$ \\
Finaly leaving us with the Euler 323 sequence \\
$R_{3}(\psi )R_{2}(\theta )R_{3}(\phi )=\left[{\begin{matrix}c_{\psi }c_{\theta }c_{\phi }-s_{\psi }s_{\phi }&c_{\psi }c_{\theta }s_{\phi }+s_{\psi }c_{\phi }&-c_{\psi }s_{\theta }\\-s_{\psi }c_{\theta }c_{\phi }-c_{\psi }s_{\phi }&-s_{\psi }c_{\theta }s_{\phi }+c_{\psi }c_{\phi }&s_{\psi }s_{\theta }\\s_{\theta }c_{\phi }&s_{\theta }s_{\phi }&c_{\theta }\end{matrix}}\right]$
# PlanetPhysics/Euler Angle Velocity

The velocity of the Euler angles of a rotating coordinate frame can be derived from the angular velocity vector of this frame with respect to a reference frame. The derivation is lengthy for each Euler angle sequence and is given in their respective entries. One useful result is acheived by combining the angular velocity vector, in terms of Euler angles, with Euler's moment equations yielding the differential equations of motion that characterizes rigid body motion for a given sequence.
Notation: $cos(\theta )=c_{\theta },sin(\theta )=s_{\theta }$
Euler angle velocity of 123 Sequence:
$\left[{\begin{matrix}{\dot {\phi }}\\{\dot {\theta }}\\{\dot {\psi }}\end{matrix}}\right]=\left[{\begin{matrix}(\omega _{x}c_{\psi }-\omega _{y}s_{\psi })/c_{\theta }\\\omega _{x}s_{\psi }+\omega _{y}c_{\psi }\\(-\omega _{x}c_{\psi }+\omega _{y}s_{\psi })s_{\theta }/c_{\theta }+\omega _{z}\end{matrix}}\right]$
Euler angle velocity of 321 Sequence:
$\left[{\begin{matrix}{\dot {\phi }}\\{\dot {\theta }}\\{\dot {\psi }}\end{matrix}}\right]=\left[{\begin{matrix}\left(\omega _{y}s_{\psi }+\omega _{z}c_{\psi }\right)sec({\theta })\\\omega _{y}c_{\psi }-\omega _{z}s_{\psi }\\\omega _{x}+\omega _{y}s_{\psi }t_{\theta }+\omega _{z}c_{\psi }t_{\theta }\end{matrix}}\right]$
# Real powers/Comparison/2/Exercise

Compare the thee numbers
    $2^{\sqrt {3}},\,4,\,3^{\sqrt {2}}.$
# Real rational function/Curve behavior/Exercise form

Discuss the following properties of the rational function
    $f\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)={\frac},$
domain, zeros, growth behavior, (local) extrema. Sketch the graph of the function.
# Real power series/Infinitely often differentiable/Fact/Proof

This follows immediately from fact.
# Real powers/Comparison/1/Exercise

Compare the two numbers
    ${\sqrt {3}}^{-{\frac {9}{4}}}\,\,{\text{ and }}\,\,{\sqrt {3}}^{-{\sqrt {5}}}.$
# Real sequence/Arithmetical mean/Dependent on start value/Convergence/Exercise

Let $a\in \mathbb {R}$. For initial value $x_{0}\in \mathbb {R}$, let a real sequence be recursively defined by
    $x_{n+1}={\frac {x_{n}+a}{2}}\,.$
Show the following statements.
(a) For $x_{0}>a$, we have $x_{n}>a$ for all $n\in \mathbb {N}$, and the sequence is strictly decreasing.
(b) For $x_{0}=a$, the sequence is constant.
(c) For $x_{0}<a$, we have $x_{n}<a$ for all $n\in \mathbb {N}$. and the sequence is strictly increasing.
(d) The sequence converges.
(e) The limit is $a$.
# Real sequence/Convergence/Unit fractions/Exercise

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real sequence. Prove that the sequence converges to $x$ if and only if for all $k\in \mathbb {N} _{+}$ a natural number $n_{0}\in \mathbb {N}$ exists, such that for all $n\geq n_{0}$ the estimation $\vert {x_{n}-x}\vert \leq {\frac {1}{k}}$ holds.
# Progress and Prospects in Parkinson's Research/Symptoms/Hypomania

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Hypomania
Pathological elation.
The following references are relevant:-
Zahodne and Fernandez (2010) Parkinson’s Psychosis
Haq et al (2010) A Case of Mania following Deep Brain Stimulation for Obsessive Compulsive Disorder
Copra et al (2012) Underlying Neurobiology and Clinical Correlates of Mania Status-Post Subthalamic Nucleus Deep Brain Stimulation in Parkinson’s disease: A Review of the Literature
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3045857/
Further Research
References
* ↑ Zahodne, Laura B. and Fernandez, Hubert H. (2011) Full Text Curr. Treat. Options Neurol. 12 (3) 200 – 211. Parkinson’s Psychosis http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3045857/
* ↑ Haq, Ihtsham U.; Foote, Kelly D.; Goodman, Wayne K.; Ricciuti, Nicola; Ward, Herbert; Sudhyadhom, Arhar; Jacobson, Charles E.; Siddiqui, Mustafa S. and Okun, Michael S. (2010) Full TextStereotact. Funct. Neurosurg. 88 (5) 322 – 328.A Case of Mania following Deep Brain Stimulation for Obsessive Compulsive Disorder http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2969111/
* ↑ Chopra, Ami; Tye, Susannah J.; Lee, Kendall H.; Sampson, Shirlene; Matsumoto, Joseph; Adams, Andrea; Klassen, Bran; Stead, Matt; Fields, Julie A. and Frye, Mark A. (2012) Full TextJ. Neuropsychiatry Clin. Neurosci. 24 (1) 102 – 110. Underlying Neurobiology and Clinical Correlates of Mania Status-Post Subthalamic Nucleus Deep Brain Stimulation in Parkinson’s disease: A Review of the Literaturehttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC3570815/
# Progress and Prospects in Parkinson's Research/Symptoms/Hyperkinesia

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Hyperkinesia
Increased force, speed, violence, speed of movement.
The following references are relevant:-
Ackermann et al (1989) Palilalia as a symptom of levodopa induced hyperkinesia in Parkinson's disease.
Inzelberg et al (1995) Kinematic properties of upper limb trajectories in idiopathic torsion dystonia
Thomsen et al (2011) Psychomotor stimulation by dopamine D1-like but not D2-like agonists in most mouse strains

## Further Research

References
* ↑ Ackermann, A; Ziegler, W. and Oertel, W. H. (1989) Full Text J. Neurol. Neurosurg. Psychiatry. 56 (6) 805 – 807. Palilalia as a symptom of levodopa induced hyperkinesia in Parkinson's disease. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1032047/
* ↑ Inzelberg, R.; Flash, T.; Schechtmann, E and Korczyn, A.D. (1995) Full Text J. Neurol. Neurosurg. Psychiatry. 58 (3) 312 – 319. Kinematic properties of upper limb trajectories in idiopathic torsion dystonia http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1073367/
* ↑ Thomsen, Morgane; Ralph, Rebecca J. and Caine, S. Barak (2011) Full Text Exp. Clin. Psychopharmacol. 19 (5) 342 – 360.Psychomotor stimulation by dopamine D1-like but not D2-like agonists in most mouse strains http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3319345/
# Progress and Prospects in Parkinson's Research/Symptoms/Hypersexuality

Hypersexuality
Excessive sexual activity
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(Related symptoms: Satyriasis, Sadomachism)
The following references are relevant:-
Witjas et al (2012) Addictive behaviors and Parkinson's disease.
Politis et al (2013) Neural response to visual sexual cues in dopamine treatment-linked hypersexuality in Parkinson's disease.
Pereira et al (2013) Screening hypersexuality in Parkinson's disease in everyday practice.
Further Research
References
* ↑ Witjas, T.; Eusebio, A.; Fluchère, F. and Azulay, J.P. (2012) Full Text Rev. Neurol. (Paris). 168 (8-9) 624 - 633. Addictive behaviors and Parkinson's disease. http://www.ncbi.nlm.nih.gov/pubmed/22921247
* ↑ Politis, M.; Loane, C.; Wu, K.; O'Sullivan, S.S.; Woodhead, Z.; Kiferle, L.; Lawrence, A.D.; Lees, A.J. and Piccini, P. (2013) Full Text Brain. 136 (Pt 2) 400 -411.Neural response to visual sexual cues in dopamine treatment-linked hypersexuality in Parkinson's disease.http://www.ncbi.nlm.nih.gov/pubmed/23378222
* ↑ Pereira, B.; Llorca, P.M.; Durif, F.; Brousse, G.; Blanc, O.; Rieu, I.; Derost, P.; Ulla, M.; Debilly, B. and de Chazeron, I. (2013) Full Text Parkinsonism Relat. Disord. 19 (2) 242 - 246. Screening hypersexuality in Parkinson's disease in everyday practice. http://www.ncbi.nlm.nih.gov/pubmed/23176749
# Progress and Prospects in Parkinson's Research/Symptoms/Hypertonia

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## Hypertonia

Excessive muscle tone
The followng references are relevant:-
Wright et al (2007) Axial hypertonicity in Parkinson’s disease: Direct measurements of trunk and hip torque
Goldenberg(2008) Medical Management of Parkinson’s Disease
Xia et al (2011) Differentiation between the contributions of shortening reaction and stretch-induced inhibition to rigidity in Parkinson’s disease

## Further Research

References
* ↑ Wright, W,G,; Gurfinkel, V.S.; Nutt, J.; Horak, F.B. and Cordo, P.J. (2007) Full Text Exp. Neurol. 208 (1) 38 – 46. Axial hypertonicity in Parkinson’s disease: Direct measurements of trunk and hip torque http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2144734/
* ↑ Goldenberg, Marvin M. (2008) Full Text P.T. 33 (10) 590 – 594, 599 – 606 Medical Management of Parkinson’s Disease http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2730785/
* ↑ Xia, Ruiping,; Powell, Douglas.; Rymer, W. Zev; Hanson, Nicholas;Fang, Xiang; and Threlkeld, A. Joseph (2011) Full TextExp. Brain Res. 209 (4): 609 – 618. Differentiation between the contributions of shortening reaction and stretch-induced inhibition to rigidity in Parkinson’s disease http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3142787/
# Progress and Prospects in Parkinson's Research/Symptoms/Gegenhalten

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## Gegenhalten

Involuntary variable resistance during passive movement
The following references are relevant:-
Bahram et al (2009) Exercise therapy, quality of life, and activities of daily living in patients with Parkinson disease: a small scale quasi-randomised trial
Franzen et al (2009) Reduced Performance in Balance, Walking and Turning Tasks is Associated with Increased Neck Tone in Parkinson's Disease
Xia et al (2011) Differentiation between the contributions of shortening reaction and stretch-induced inhibition to rigidity in Parkinson’s disease
# Progress and Prospects in Parkinson's Research/Symptoms/Hypokinesia

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## Hypokinesia

Reduced force, impetus or speed of movement
The following refences are relevant:-
Ackermann et al (1989) Palilalia as a symptom of levodopa induced hyperkinesia in Parkinson's disease.
Heldman et al (2011) The Modified Bradykinesia Rating Scale for Parkinson’s Disease: Reliability and Comparison with Kinematic Measures #
Espay et al (2011) Differential Response of Speed, Amplitude, and Rhythm to Dopaminergic Medications in Parkinson’s Disease

## Further Research

References
* ↑ Ackermann, H.; Ziegler,W. andOertel, W.H. (1989) Full Text J. Neurol. Neurosurg. Psychiatry. 52 (6) 805 – 807. Palilalia as a symptom of levodopa induced hyperkinesia in Parkinson's disease.http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1032047/
* ↑ Heldman, Dustin A.; Giuffrida, Joseph P.; Robert; Megan; Filomena; Andrew P.; Alok; Sang Jin; Fredy J. and Espay, Alberto J. (2011) Full Text Mov. Disord. 26 (10) 1859 – 1863. The Modified Bradykinesia Rating Scale for Parkinson’s Disease: Reliability and Comparison with Kinematic Measures http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3324112/
* ↑ Espay, Alberto J,;Giuffrida, Joe P.; Chen, Robert; Payne, Megan; Mazzella, Filomena; Dunn, Emily; Vaughan, Jennifer E.; Duker, Andrew P.; Sahay, Alok; Kim, Sang Jin; Revilla, Fredy J. and Heldman, Dustin A. (2011) Full TextMov. Disord. 26 (14) 2504 – 2508. Differential Response of Speed, Amplitude, and Rhythm to Dopaminergic Medications in Parkinson’s Disease http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3318914/
# Progress and Prospects in Parkinson's Research/Symptoms/G

The A-Z of Parkinson's Symptoms
H
< Back to Parkinson's Symptoms
Links to the symptoms of Parkinson's in alphabetical order:
| A | B | C | D | E |
| F - G | H | I - L | M - N | O |
| P - Q | R | S | T | U - Z |
(Click to go to table for editing)
# Progress and Prospects in Parkinson's Research/Symptoms/Glabellar tap

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Glabellar tap
Tapping just above the nose (which can initiate involuntary blinking in PD)
The following references are relevant:-
Klawans Jr. and Goodwin (1969) Reversal of the glabellar reflex in Parkinsonism by L-dopa.
Penders and Delwaide (1971) Blink reflex studies in patients with Parkinsonism before and during therapy
Brodsky et al (2004) Glabellar and palmomental reflexes in Parkinsonian disorders.
Further Research
References
* ↑ Klawans Jr., H.L. and Goodwin, J.A, font color="maroon">(1969) Full Text J. Neurol. Neurosurg. Psychiatry. 32 (5) 423 – 427 Reversal of the glabellar reflex in Parkinsonism by L-dopa. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC496552/
* ↑ Pemders, C.A. and Delwaide, P.J. font color="maroon">(1972) Full Text J. Neurol. Neurosur.g Psychiatry. 34 (6) 674 – 678. Blink reflex studies in patients with Parkinsonism before and during therapy http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1083500/
* ↑ Brodsky, H.; Dat Vuong, K.; Thomas, M. and Jankovic. J. font color="maroon">(2004) Abstract Neurology. 63 (6) 1096 - 1098. Glabellar and palmomental reflexes in Parkinsonian disorders. http://www.ncbi.nlm.nih.gov/pubmed/15452308
# Progress and Prospects in Parkinson's Research/Symptoms/Hallucination

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Hallucination
An experience of something that does not exist outside the mind
The following references are relevant:-
Goetz et al (2010) Hallucinations and sleep disorders in PD
Fenelon et al (2011) Feeling of presence in Parkinson's disease
Bertram and Williams (2012) Visual hallucinations in the differential diagnosis of parkinsonism
Further Research
References
<references>
* ↑ http://medical-dictionary.thefreedictionary.com/
* ↑ Goetz, Christopher G.; Ouyang, Bichun; Negrom, Alive and Stebbins, Glenn T. (1931) Full Text Neurology 75 (20) 1773 – 1779 Hallucinations and sleep disorders in PD http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2995381/
* ↑ Fénelon, Gilles; Soulas, Thierry; De Langavant, Laurent Cleret; Trinklet, Iris and Bachoud-Lévi, Anne-Catherine (2011) Full TextJ. Neurol. Neurosurg. Psychiatry. 82 (11): 1219 – 1224. Feeling of presence in Parkinson's disease http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3382202/
* ↑ Bertram, Kelly and Williams, David R. (201) Full Text J. Neurol. Neurosurg. Psychiatry 83 (4) 448 – 452 Visual hallucinations in the differential diagnosis of parkinsonism http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3297805/
# Fundamental Electrochemistry

Electrochemistry is the branch of chemistry that deals with the interchange of chemical and electrical energy. The electrons in a balanced redox reaction show the relationship between electricity and a redox reaction.

## Galvanic Cells

A redox reaction which occurs in a galvanic cell is a spontaneous reaction. These galvanic cell reactions produce energy which can be directed to do work. This energy can be captured by placing the two half reactions of the redox reaction in separate containers called half-cells. The half-cells are connected usually by a wire which allows a flow of electrons. The flow of electrons spontaneously flows through the outer circuitry in which their energy is extirpated. The oxidation half reaction occurs at the anode and the reduction half reaction occurs at the cathode. The electron flow is always from the anode to the cathode.
A galvanic cell includes the following parts:
* 2 electrodes (1 in each half-cell)
* 2 separate containers (each is a half-cell)
* Salt bridge
* Apparatus (usually a wire)
* Voltmeter

## Electrolytic Cells

Energy input in the form of voltage energy is needed to drive non-spontaneous reaction in an electrolytic cell. Electron flow in this case flows from anode to cathode only because the extra voltage is driving the reaction. There are many applications for the electrolytic cell and the general setup is very similar to galvanic cells.
# Fundamental Mathematics/Arithmetic/Arithmetic Number/Complex Number
## Complex Number

Complex Number
    $Z=A+iB=|Z|\angle \theta ={\sqrt {B^{2}+A^{2}}}\angle Tan^{-1}{\frac {B}{A}}=|Z|e^{i\theta }\,$
Complex conjugate Number
    $Z^{*}=A-iB=|Z|\angle -\theta ={\sqrt {B^{2}+A^{2}}}\angle -Tan^{-1}{\frac {B}{A}}=-|Z|e^{i\theta }\,$

## Mathematical Operations
### Operation on 2 different complex numbers

    Addition$(A+iB)+(C+iD)=(A+C)+i(B+D)$ Subtraction$(A+iB)-(C+iD)=(A-C)+i(B-D)$ Multilication$(A+iB)+(C+iD)=(AC+BD)+i(AD+BC)$ Division${\frac {(A+iB)}{(C+iD)}}={\frac {(A+iB)}{(C+iD)}}{\frac {(C+iD)}{(C+iD)}}={\frac {(AC+BD)+i(AD+BC)}{(C+iD)^{2}}}$

### Operation on complex numbers and its conjugate

    Addition$(A+iB)+(A-iB)=2A$ Subtraction$(A+iB)-(A+iB)=i2B$ Multilication$(A+iB)+(A-iB)=A^{2}-B^{2}$ Division${\frac {(A+iB)}{(A-iB)}}={\frac {(A+iB)}{(A-iB)}}{\frac {(A-iB)}{(A-iB)}}={\frac {A^{2}-B^{2}}{(A-iB)^{2}}}$
In Polar form
    $Z\times Z^{*}=|Z|\angle \theta \times |Z|\angle -\theta =|Z|^{2}\angle (\theta -\theta )=|Z|^{2}$
    ${\frac {Z}{Z^{*}}}={\frac {|Z|\angle \theta }{|Z|\angle -\theta }}=1\angle 2\theta$

## Power of Z

Since
    $Z\times Z=Z^{2}=(|Z|\angle \theta )(|Z|\angle \theta )=|Z|^{2}\angle (\theta +\theta )=|Z|^{2}\angle 2\theta$
Hence
    $Z^{n}=Z\times Z\times Z...=|Z|^{n}\angle n\theta$
Since
    $Z^{*}\times Z^{*}=(Z^{*})^{2}=(|Z|\angle -\theta )(|Z|\angle -\theta )=|Z|^{2}\angle (-\theta -\theta )=|Z|^{2}\angle -2\theta$
Hence
    $(Z^{*})^{n}=Z^{*}\times Z^{*}\times Z^{*}...=|Z^{*}|^{n}\angle -n\theta$

## Euler's formula
### Euler formula

of which there is the famous case (for θ = π):
    $e^{i\pi }=-1\,$
More generally,
Eucleur's power can be expressed as complex number
    $e^{i\theta }=\cos \theta +i\sin \theta \,$
Hence, conjugate of the complex number
    $e^{-i\theta }=\cos \theta -i\sin \theta \,$
Adding complex number and its conjugate
    $e^{i\theta }+e^{-i\theta }=2Cos\theta$
    $Cos\theta ={\frac {1}{2}}(e^{i\theta }+e^{-i\theta })$
Minus complex number and its conjugate
    $e^{i\theta }-e^{-i\theta }=2iSin\theta$
    $Sin\theta ={\frac {1}{2i}}(e^{i\theta }-e^{-i\theta })$

### de Moivre's formula

for any real $x$ and integer $n$. This result is known as
# Fundamental attribution error

This psychology resource is a "stub". Learn how you can help Wikiversity to develop it.
In attribution theory, the fundamental attribution error (also known as correspondence bias or overattribution effect) is the tendency for people to over-emphasize dispositional, or personality-based, explanations for behaviors observed in others while under-emphasizing situational explanations. In other words, people have an unjustified tendency to assume that a person's actions depend on what "kind" of person that person is rather than on the social and environmental forces influencing the person. Overattribution is less likely, perhaps even inverted, when people explain their own behavior; this discrepancy is called the actor-observer bias.

## Activity

Hand out two copies to each student of the attribution scale. Ask them to complete the scale twice – once for themselves and once for George Bush. After they have completed both forms, have them count the number of times they circled “depends on the situation” on each rating sheet. A show of hands should show that a greater tendency to attribute another person’s behaviour to personal disposition, while attributing their own behaviour to the environment. Ask students why this is the case and discuss.
We tend to attribute causation to the focus of our attention, which is different when we are observing than when we are acting. When another person acts, our focus is on that person, thus they are the cause of what is happening. When we act, the environment commands our attention and thus seems to explain our behaviour.
# Fundamental theorem of calculus/Riemann/Fact/Proof

Let $x$ be fixed. The difference quotient is
    ${\frac {F(x+h)-F(x)}{h}}={\frac {1}{h}}{\left(\int _{a}^{x+h}f(t)\,dt-\int _{a}^{x}f(t)\,dt\right)}={\frac {1}{h}}\int _{x}^{x+h}f(t)\,dt\,.$
We have to show that for $h\rightarrow 0$, the limit exists and equals $f(x)$. Because of the Mean value theorem for definite integrals, for every $h$, there exists a $c_{h}\in [x,x+h]$ with
    $f(c_{h})\cdot h=\int _{x}^{x+h}f(t)dt\,,$
and therefore
    $f(c_{h})={\frac {\int _{x}^{x+h}f(t)dt}{h}}\,.$
For $h\rightarrow 0$, $c_{h}$ converges to $x$, and because of the continuity of $f$, also $f(c_{h})$ converges to $f(x)$.
# Fundamental theorem of algebra/Nonconstant polynomial/Zero/Fact

Every nonconstant polynomial $P\in \mathbb {C} [X]$ over the complex numbers
# Fundamental theorem of calculus/Riemann/Fact

Let $I$ denote a real interval, and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a continuous function. Let $a\in I$, and let
    $F(x):=\int _{a}^{x}f(t)\,dt\,$
denote the corresponding integral function.
$F$
differentiable, and the identity
    $F'(x)=f(x)\,$
holds for all
$x\in I$