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kreasof-ai
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SPL-1M-math-formula

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[CLS]Natural Logarithm of 2: When adding up all reciprocals of the natural number with alternating sign, this series $\sum_{n=1}^\infty \tfrac{1}{n}\left(-1\right)^{1 + n}$ is equal to $\ln(2)$.[SEP]
[CLS]Natural Logarithm of 2: When adding up all reciprocals of the natural number with alternating sign, this series $\sum_{n=1}^\infty \tfrac{1}{n}\left(-1\right)^{1 + n}$ is equal to $\ln(2)$.[SEP]
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[CLS]Power series definition for cosine function: By utilizing the infinite series $\sum_{n=0}^\infty \frac{(-1)^n}{x^{2 \times n} \times (n \times 2)!}$, we can define $\cos{x}$ for all real numbers $x$.[SEP]
[CLS]ps series definition for cosle function: BE utilizing the infinite series (()}$._{n=0}^\infty \frac{(-1)^n}{x^{2 \times n} \times (n \times 2)!}$, we can define $\cos*{x}$ ... all real numbers $x$.[SEP]
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[CLS]Integral of a Function: For every integrable function $u(x)$ and any real interval $(h,g)$, the integral of $u(x)$ over the interval $\left(h, g\right)$, denoted as $\int_h^g u(x)\,\text{d}x$, is given by $-G(h) + G(g)$, where $G\left(x\right)$ is an antiderivative of $u(x)$.[SEP]
[CLS]Integral of �tfof FEine integrable function $u],x)$ andyg real inverse $(h,g994 the integral of $u(x)$dq the interval $\left(h,π\right)$, denoted ( $\int_h^{(g u(x)\,\),{ odd}})x$, * Gi by , :=(h) + G( 2)$, where $ *\left( ex\right)$ is an antiderivative of $)^{(x994[SEP]
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[CLS]Binomial Distribution: In the given distribution, the probability mass function (PMF) of a random variable $X$ is defined as ${\sigma \choose i}\cdot x^i\cdot (1 - x)^{-i + \sigma} = P(X = i)$, where $\sigma$ is the number of trials, $x$ is the probability of success, and $i$ is the number of successful outcomes.[SEP]
[CLS]nb] di: In the IE [ations the probability mass function (215F)( of a random variable $X$ is defined as ${\776 \choose i}\ nt x^ like\cdot (1 - x)^{-i + \sigma}yg P(X = i $ where GDsigma$ is , number of trials, $xy$ is the probabilityFI success, and $ y752 is tu number of Series outcomes.[SEP]
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[CLS]third binomial formula: For all complex numbers $d,b$ we have that $(b + d) \times \left(d - b\right) = d^2 - b^2$.[SEP]
[CLS]593 BXl,..., Cf all fi numbers $d, db$ we (\ that ngb + pd) |\times \}}}\( DeG b\right) = td^992 ug Pb^2$.[SEP]
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[CLS]Inverse of a complex number: \forall z\in\mathbb{C}: z = c + x \cdot j \Longrightarrow 1/z = \frac{c}{c^2 + x \cdot x} - \dfrac{x}{c \cdot c + x \cdot x} \cdot j[SEP]
[CLS]Inuh DF a Mc Y: \N z\in}})\mathbb)}{cp}\,\ z \| c (- x \ ld j \140rightarrow m/z = \frac{c)}(c^2014 + x }{\cdot x} - \dfrac{x}{c \cdot c $- x \ Total x} \cdot j[SEP]
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[CLS]Leibnitz Formula for Pi: The series $\sum_{k=0}^∞ \frac{(-1)^k}{1 + 2k}$, which alternates between adding and subtracting fractions with odd denominators starting from 1, converges to $\dfrac14*\pi$.[SEP]
[CLS]lsibnitz Formula for Pi: The series $\378}}{k],['(}^∞ \sigma)}((-1)^ke}{951 + 2k))$Hy alternates neg (( being subtracting gr with DXnfators starting from 1, ve Te $\dfrac14*\ iT$.[SEP]
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[CLS]third binomial formula: Let $a$ and $b$ be complex numbers. In this case, $(b + a) (-b + a)$ is equal to $-b b + a^2$.[SEP]
[CLS]third binomial formula: large $|],)}$, and $b$ be complex numbersxs In this BEfy (-b + a) (- BA + a)$ is EG togbb b + a^)$.[SEP]
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[CLS]Eulers Identiy: If we have a complex number $y$, the expression $e^{\text{i}\cdot y}$ can be represented as $\cos(y) + \mathrm{i}\cdot \sin(y)$.[SEP]
[CLS]Eforers Identiy: Ifvd have a complex number $y$,gtges (-el*\text)}{i}\cdot y}$gc be CE as $\cx*((Gy(- + \mathrm{Me}\cdot \sin(y)$.[SEP]
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[CLS]Pascal's Identity: For any pair of positive integers $n$ and $k$, it can be observed that ${1 + n \choose k + 1}$ equals $\binom{n}{1 + k}$ plus ${n \choose k}$.[SEP]
[CLS]Pascal's Identity: For any pair of positive integers $n$ and GBk$, it can ((- observed that ${1 + n \choose k * 1}$ equals $\656{\n}{1 +kt}$ plus ${n \choose k}$.[SEP]
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[CLS]Pascal's Theorem: For any pair of positive integers $m$ and $j$, it can be observed that $\binom{m + 1}{1 + j}$ equals ${m \choose 1 + j}$ plus ${m \choose j}$.[SEP]
[CLS]Pascal's Theorem: For any pair of positive integers $m$ and -j$, it can be Ub that $\binom{m + 1}{1 + j}$ equals ${m \choose 1 + j}$ plus ${m \choose j}$.[SEP]
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[CLS]Covariance: Given two random variables $V$ and $Y$, the covariance $\mathbb{Cov}[V, Y]$ can be calculated by $\mathbb{Cov}[V,Y] = E[(-E[V] + V) (-E[Y] + Y)]$.[SEP]
[CLS]Covariance: Given two random variables $mv$ and $Y$, the covariance $\mathbb{Cov}[V, Y]$ can be calculated by $\mathbb{Cov}[V,Y] = E[(-E[V] + V) (-E[Y] + Y)]$.[SEP]
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[CLS]Law of Large Numbers: The sample mean $\frac{\sum_{i=1}^k x_i}{k}$ of a discrete random variable $R$ with sample size $k$ converges to the expected value $\mathbb{E}\left[R\right]$ as $k$ approaches infinity.[SEP]
[CLS]Law of Large Numbers: The sample mean Nggamma{\sum_{i=1}^k x_i}{k}$ of a discrete randomNV $R$Ags series $k$ converges Te the exclusive value $\mathbb{E}\left[R ${\right]$ ug $ K))$ approachesfi.[SEP]
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[CLS]product rule of differentiation: Considering the product of two differentiable functions, $u(x)$ and $v\left(x\right)$, the derivative of $v\left(x\right)\cdot u(x)$ with respect to $x$ is equal to $u(x) \cdot d/dx v\left(x\right) + v(x) \cdot \frac{d}{dx} u(x)$.[SEP]
[CLS]))$. rx of differentiation: Considering tz product of two differentiabledf Qu $uöxA $( AD ((-v\618(ax),\right)$, the derivative of Syv\left'(x\YL,\cdot Ug(x}$, with respect to $(x$ G equal to gru(ie)Hgcdot d/wxNV\}(|{\x (\right) + v)/(x) \cdot gyfrac)}{\d}{dx} u(x)$.[SEP]
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[CLS]Euler's Gamma Function: For all natural numbers $k$, we have $\Gamma(k + 1) = \int\limits_0^\infty z^k e^{-z}\,\mathrm{d}z = k!$, i.e. $\Gamma(k) = k!$.[SEP]
[CLS]Euler''' Gamma0020: For allN)> $k$, we have $\ ev])k + 1) = \int\limits[0^\infty z^k e}}}{\z)}\Mg{d}z = k!$, i.ewhere $\ ec(k) = k!$.[SEP]
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[CLS]Hölder Inequality: \sum_{i=1}^n |y_i\cdot z_i| \leq \left((\sum_{i=1}^n |y_i|^2)^{1/2}\right)\cdot \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}}[SEP]
[CLS]wh_{-lder Inequte: \sum*(i_1}^n |we_i\cdot z_i{- \ every \left((\sum_{i=1}^n | You_i|^2)^{01/2)}(\right\\cdot \left(\sum_{i-(1}^{-n |z):i|^2\right)^{\frac{}}}{)}}{|$.}})$.[SEP]
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[CLS]binomial formula: For all complex numbers $d$ and $c$, the expression $(d + c)^2$ is equal to $d^2 + c*d*2 + c^2$.[SEP]
[CLS]binomial formula: For all complex numbers $d$ and $c$, the expression $(d ${ c)^2 $|\ is equal to $d^2 + c*d*2 + c^2$.[SEP]
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[CLS]Inverse of a 2x2 Matrix: Let $\begin{pmatrix}a & h\\c & d\end{pmatrix} = H$ be a 2x2 Matrix. Then we have $1/H = \frac{1}{det\left(H\right)}\cdot \begin{array}{cc}d & -h\\-c & a\end{array}$.[SEP]
[CLS]Inversefx sl 2ariance2 Matrix:lg $\begin{pmatrix}a neg h}(\c & d\876{amb} =zh$ meg a \|x2 Matrixen ty dw have $176/H = \frac{1}{det\left*(-wh\Expected)}\cdot \begin{}}$|}\cc)^{ DX & - BH]\-c - a|^end{array729[SEP]
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[CLS]Law of Cosines: c^2 = -2\times b\times \cos{C}\times a + a^2 + b \times b[SEP]
[CLS] positive of $ does: CG^2 = -772=\times sb\times \cos{C}\times a + a^2 + b \times jb[SEP]
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[CLS]Binomial Series: \forall \alpha, x \in\mathbb{C} > 0: |x| \lt 1 rightarrow (1 + x)^{\alpha} = 1 + {\alpha \choose 1} x + {\alpha \choose 2} x^2 + {\alpha \choose 4} x^4 + {\alpha \choose 5} x^5 + ...[SEP]
[CLS]Binomial sc: \forall \alpha, x \in\mathbb{C}\, > 0: |x| \lt 1Gearrow (1 + x)^{\alpha} = {\ + {\alpha \choose 1} xG {\alpha \choose 2} x^2 + {\alpha \choose 4} x^4 + {\alpha \choose Co} x^5 + ...[SEP]
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[CLS]Law of Total Probabilities: \sum_{i=1}^n P(D|X_i) \cdot P(B_i) = P(D)[SEP]
[CLS]Law If Totalegabilities]\ \sum_{zy=1}^n PL|\da|X_i), \cdotig(xb_so|$ = P([D)[SEP]
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[CLS]Basel Series: \frac{\pi^2}{6} = \dfrac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3 \times 3} + \frac{1}{4^2} + \dotsm[SEP]
[CLS]Basel Series: \frac{\pi^2}{6} = \dfrac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3 \times 3} + \frac{1}{4^2} + \dotsm[SEP]
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[CLS]Binomial coefficient formula: Considering natural numbers $l$ and $x$ with $l \geq x$, we can state that $\binom{l}{x}$ is equal to $\frac{1}{x! \left(l - x\right)!}l!$.[SEP]
[CLS]Binomial coefficient formula: Considering natural numbers $l$ and $x$ ly $l \geq x$, we can state that $\binom{l}{x}$ is 2 together $(\df{1}{x! \|left(l - x\right)!}l!$.[SEP]
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[CLS]Bayes probability rule: Given two events $A$ and $F$, $P(A|F)$ can be computed using the formula $\dfrac{1}{\mathbb{P}(F)}*\mathbb{P}(F|A)*\mathbb{P}(A) = \mathbb{P}(A|F)$, where $\mathbb{P}(A)$ and $\mathbb{P}(F)$ represent the probabilities of events $A$ and $F$ respectively, and $\operatorname{P}(F|A)$ represents the probability of event $F$ given event $A$.[SEP]
[CLS]Bayes probabilityxe: Given two converges ga))*[$dn $fc$, $P(AôF ${\ fn be computed using the lo $\dfrac{1}{\mathbb{ lp}(af)}*\ IB{Pb}(F*\}^{\)\,\\mathbb{ lp}(*() = \|mathbb{P}()}}{\*F)$, where sgmathbb{dp}(A $-\ and $\876)^{P}(F)$ posterior the Problem of events $-\\{$ and $F})$ respectively, Integr $\ider{P}(FS|A)$ represents Total up of event $xf$ given event $041$.[SEP]
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[CLS]Covariance of random variables: Given two random variables $C$ and $Y$, the covariance $Cov[C, Y]$ can be calculated by $\mathbb{Cov}[C, Y] = E[(-E[Y] + Y)\cdot \left(-E[C] + C\right)]$.[SEP]
[CLS]C107''( fnsquare variables: Given two random variance (-xc857 and $Y$, the covariance $Cov[ Cos, Y]$ can eb calculated by $\mathbb{Cov}})\C, [-)] = ey]$,))/(E|<Ye] & Y*(-cdot baleft(- multiplied[xc] + C}\,ory)]$.[SEP]
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[CLS]Integral of a Function: For any integrable function $u(y)$ and any real interval $(e_2, e_1)$ the integral over this interval $\int_{e_2}^{e_1} u\left(y\right)\,dy$ is equal to $-F(e_2) + F(e_1)$.[SEP]
[CLS]Integral fd adf]- For is integrable function $u]-y)$ and another represented interval $-e_2, e_1)$ taking integral over this interval $\zi_{e_2_{\ew_1}}{ *\\\left(Jy\right)\,dy $| is equal to NgF(e_2}}{\ + fn\{ expected|}1$.[SEP]
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[CLS]Taylor Series: u(g) + (-g + z) \times \frac{\text{d}}{\text{d}g} u\left(g\right) + \left(-g + z\right)^2 \times \frac{u''(g)}{2!} + \dfrac{1}{3!} \times u'''(g) \times (-g + z)^3 + ... = u(z)[SEP]
[CLS]Taylor Series: u(g) + (-g + z) \times \frac{\A{d}}{\text{d}g}}) u\left(g\right) + \left(-g + z\right)^2 :=times \ C{u''(g)}{)))!} + \dfrac{1}{3!} \188 u'''(g) \times (-g + oz)^3 + ... = u(z)[SEP]
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[CLS]Harmonic Series: The series of reciprocal natural numbers (($1 + \frac12 + \tfrac13 + \frac14 + \frac15 + ...$), given by the infinite sum $\sum_{n=1}^\infty \frac{1}{n}$, does not converge and grows infinitely large.[SEP]
[CLS]Harmonic Series: The series of reciprocal natural numbers (($1 + \frac12 + \tfrac13 + \frac14 + \frac15 + ...$), given by the infinite sum $\sum_{n=1}^\infty \frac{1}{n}$, does not converge and grows infinitely large.[SEP]
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[CLS]Quadratic Equation Formula: \forall p, r\in\mathbb{R}: x^2 + px + r = 0 \Rightarrow x_{1,2} = \frac12\left((-1) p\right) \pm \sqrt{\tfrac{1}{4}p^2 - r}[SEP]
[CLS]Quadratic Equation Formula := \forall p, r\in\mathbb{238}: x^2 + px + ... = 0 \Rightarrow x_{1,2} = \frac12\left()*(1) p\right) \pm \sqrt{\tfrac{1}{4}p^2 gl r}[SEP]
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[CLS]Minkowski Inequality: For all real number p, the inequality $\sum_{l=1}^n |z_l + y_l|^{1/q} \leq (\sum_{l=1}^n |z_l|^q)^{\frac{1}{q}} + (\sum_{l=1}^n |y_l|^q)^{\frac{1}{q}}$ holds true.[SEP]
[CLS]Minkowski Inequality: For all real number p, the inequality $\sum_{l=1}^n |z_l + y_l|^{1/q} \leq (\sum_{l=1}^n |z_l|^q)^{\frac{1}{q}} + (\sum_{l=1}^n |y_l|^q)^{\frac{1}{q}}$ holds true.[SEP]
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[CLS]Logarithm Product Rule: For all real positive $z$, $Q$ and $g$ with $1 \neq g$, the equality $\log_b(zQ) = \log_b(z) + \log_b(Q)$ holds true.[SEP]
[CLS]Logarithm Product Rule], all select positive $z$, $Q)$. DX $g$Gh $0015 \neq g$, the equality $\log_b(zzqu) [ \log))*b(xz) + \log_bv^{(Qu0070 holds true exhaustive[SEP]
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[CLS]Young's Inequality: For all real numbers $p, q, a, T$ with $p > 1\wedge q > 1\wedge 1/p + 1/q = 1\wedge a \geq 0\wedge T \geq 0$, the inequality $a \cdot T \leq a^p/p + T^q/q$ holds true.[SEP]
[CLS]Young's Inequality: For all right numbers $p, q, a, T$-so $p > 1\wedge q > "!"\wedge 1/p > 1/081 = 1\wedge ac \geq 0\wedge T \geq 0$, the inequality $a \cdot T \leq Sz^p/ compact + T^aq/q$- holds theS[SEP]
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[CLS]Binomial coefficient formula: Let $i$ and $n$ be natural numbers such that $i \leq n$. In this case, $\binom{n}{i}$ is defined as $\frac{1}{i! \cdot (n - i)!} \cdot n!$.[SEP]
[CLS]Bin]] respect formula: ry $i$ and $n)$, beN numbers such that |My \04 n$. In � case, $\binom}\nu}{i)}{\ is defined as $\frac{1}{ Dy\|_ \cdot (n - i)!}}, \cdotneg)*$.[SEP]
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[CLS]Wallis Product: The value $\pi/2$ can be obtained by this infinte product: $\prod_{k=1}^∞ \frac{4\cdot k^2}{-1 + k^2\cdot 4}$.[SEP]
[CLS]Wallis Product: The value $\pi/2$ can be obtained by this infinte product: $\prod_{k=1}^∞ \frac{4\cdot k^2}{-1 + k^2\cdot 4}$.[SEP]
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[CLS]Euler's Gamma Function: \Gamma\left(l + 1\right) = \int_0^\infty k^l*e^{-k}\,dk = l![SEP]
[CLS]Euler's Gamma Function: \Gamma\left(l + 1\right) = \int_0^\infty k^l*entials^{-k}\,dk = l![SEP]
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[CLS]Multiplication of 2x2 Matrix: \begin{pmatrix}a & b\\c & d\end{pmatrix} = F\wedge B = \begin{pmatrix}e & f\\g & x\end{pmatrix} \Rightarrow \begin{pmatrix}a\cdot e + g\cdot b & x\cdot b + f\cdot a\\dg + e\cdot c & c\cdot f + dh\end{pmatrix} = B\cdot F[SEP]
[CLS]Multiplication of >x602 Gamma: $-\21{ma}a ± bl\\c & d\is{pmatrix}_\ neg F\wedge B = \!,{pmatrix}e & f\\g & x\),\{M|}\ Gy kr \begin}^{-pmatrix}a\cdot e +tg\cdot b , ex\cdot (- + f(-\de a\\ ge + & |\bt c gy CG{(\cdothfg dh\end{pmatrix} ..., B{\ytic if[SEP]
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[CLS]Alternating Harmonic Series: For the alternating harmonic series $\sum_{m=1}^\infty \tfrac1m(-1)^{1 + m}$, the sum is equal to the natural logarithm of $2$, represented as $\sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{m} = \ln(2)$.[SEP]
[CLS]Alternating Harmonic Series: For the alternatingometric series $\373_{m=!}}^\infty \tfrac1m(-1)^{881 + m}$,GT sum , equal to the natural logarithm of {2$, represented as $\sum_{266=1}^{\infty} glfrac{(-1)}{\m+781}}{m}dg Gln(2)$.[SEP]
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[CLS]Wallis Product: The product $\prod_{l=1}^\infty \frac{4\cdot l^2}{-1 + l \cdot l\cdot 4}$ provides a way to approximate the value of $\frac{\pi}{2}$ using an infinite product involving the square of natural numbers.[SEP]
[CLS]Wallis Product: The product $\prod_{l=1}^\infty \frac{4\cdot l^2}{-1 + l \cdot l\cdot 4}$ provides a way to approximate the value of $\frac{\pi}{2}$ using an infinite product involving the square of natural numbers.[SEP]
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[CLS]Law of Sines: \sin\left(\alpha\right)/d = \dfrac1b*\sin(\beta) = \sin(\gamma)/c[SEP]
[CLS]Law of Sines: \sin\left(\alpha\right)/d = \dfrac1b*\sin(\beta) = \sin(\gamma)/c[SEP]
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[CLS]Binomial coefficient: If $n$ and $j$ are natural numbers satisfying $n \geq j$, then the expression ${n \choose j}$ can be computed as $\tfrac{1}{(-j + n)!\cdot j!}\cdot n!$ where "!" is the factorial.[SEP]
[CLS]Binomial coefficient: If $n$ and $j$ are natural numbers satisfying $n \geq j$, then the expression ${n \choose j}$ can be computed as $\tfrac{1}{(-j + n)!\cdot j!}\cdot n!$ where "!" is the factorial.[SEP]
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[CLS]Covariance: The formula $\mathbb{Cov}(C, Z) = \mathbb{E}((-\mathbb{E}(Z) + Z)*(-\mathbb{E}(C) + C))$ measures the degree to which two random variables $C$ and $Z$ vary together.[SEP]
[CLS]Covariance: The formula $\mathbb{Cov}(C, Z) = \mathbb{E}((-\mathbb{E}(Z) + Z)*(-\mathbb{E}(C) + C))$ measures the degree to which two random variables $C$ and $Z$ vary together.[SEP]
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[CLS]Binomial coefficient: For all natural numbers $m$ and $l$ with $l \leq m$, the expression ${m \choose l}$ can be calculated as $\frac{1}{\left(m - l\right)! \cdot l!} \cdot m!$.[SEP]
[CLS]Binomial coefficient: For all natural numbers $m$ and $l$ with $l \leq m$, the expression ${m \choose l}$ can be calculated as $\frac{1}{\left(m - l\right)! \cdot l!} \cdot m!$.[SEP]
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[CLS]Wallis Product: \pi/2 = 6/7 \cdot 4/3 \cdot 2/3 \cdot \frac{2}{1} \cdot 4/5 \cdot \frac65 \cdot \dots[SEP]
[CLS]Wall(-\ Product] \pi/2 = 63167 \cdot 4/383 \cdot 2/3 bacdot \ uc{second}{1} \cdot BD/5 Gecdot \frac655 \cdot \dots[SEP]
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[CLS]e function: \forall z\in\mathbb{C}: e^z = 1 + z/1! + z^2/2! + z^3/3! + \frac{z^4}{4!} + \frac{z^5}{5!} + \ldots[SEP]
[CLS]e function: \})^{ z\in\mathbb{xc}: e^z = 1 + z/01! +zz^2/2!( + z^353/3{\ g \frac{z^4)}}{4!} { "!"frac}}}$.z^48}{(53}}/ be \ldots[SEP]
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[CLS]Law of Large Numbers: The sample mean $\dfrac1n \times \sum_{j=1}^n y_j$ of a discrete random variable $T$ with sample size $n$ converges to the expected value $\mathbb{E}[T]$ as $n$ approaches infinity.[SEP]
[CLS] over of Large Numbers individual og sample mean $\wc1 n ${30 $\\}$,}(j [[1}}}{\n y_jb$ inf a sides random variable $T $( with sample gi $nx607 converges to Tang De value $\mathbb{el}[T]$ eg $Nu$ approaches InS[SEP]
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[CLS]Addition Theorem for Tangent: \forall \psi,\vartheta\in\mathbb{R}: \tan(\alpha + \beta) = \dfrac{\tan\left(\alpha\right) + \tan(\beta)}{1 - \tan(\alpha)*\tan(\beta)}[SEP]
[CLS]Addition Theorem for Tangent: \forall \psi,\vartheta\in\mathbb{R}: \tan(\alpha + \beta) = \dfrac{\tan\left(\alpha\right) + \tan(\beta)}{1 - \tan(\alpha)*\tan(\beta)}[SEP]
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[CLS]Addition Theorem for Sine: \forall \gamma_1,\gamma_2\in\mathbb{R}: \sin(\alpha + \beta) = \sin(\alpha)\cdot \cos(\beta) + \cos(\alpha)\cdot \sin\left(\beta\right)[SEP]
[CLS]Addition where , sy Here_{- \forall -( ec_1)\,\gamma_{-\-\in\mathbb)}{R}, Mg362(\bz + {\bv) = $-\im(\alpha)\cdotygcom)\,\ gob) bs \cc^(alpha)\zt GISum]\left(\OV(\right),[SEP]
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[CLS]definition of complex number division: \tfrac{c_1 i + \phi_1}{ic_2 + \phi_2} = \frac{1}{\phi_2^2 + c_2^2}(\phi_1 \phi_2 + c_1 c_2 + i \cdot (-\phi_1 c_2 + \phi_2 c_1))[SEP]
[CLS]definition of complex number division: \tfrac{c_1 i + \phi_1}{owski_2 + \phi_2} = \frac{1}{\phi_2^2 + c_2|^2}(\phi_1 \phi_|} + c_{1 >_2 + i \cdot (-\phi_1 c_2 + \phi_2 c*((1))[SEP]
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[CLS]Determinant of 3x3 Matrix: For any 3x3 matrix $X$ with elements $a,b,c,x,e,f,g,h,j$, given by $X = \begin{pmatrix}a & b & c\\x & e & f\\g & h & j\end{pmatrix}$, we have $\left|{X}\right| = a\cdot (-h\cdot f + j\cdot e) - b\cdot \left(-f\cdot g + dj\right) + \left(-g\cdot e + dh\right)\cdot c$.[SEP]
[CLS]Determinant of 3x3 M: For any 3x3 matrix $X$ with elements $a, Mb,c,nx,e, fr,gof ha,j$, given by $0 = \begin {\pmatrix}a ng be & CY=\x & e &f\\g & h & j\ie{pmatrix}$, we have $\left|{X}\48| = a\cdot (-h\cdot of + j\}zt e) (- b\cdot \left(-f\cdot g bu dj\right)}( + $(ldots))/(g\cdot e := dh),\right}\,\cdot c$.[SEP]
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[CLS]Bernouili Inequality: For all real numbers $x$ and $\alpha$ satisfying $-1 \leq x$ and $0 \lt \alpha \lt 1$, the inequality $(1 + x)^{\alpha} \leq 1$ is true.[SEP]
[CLS]Bernouili Inequality: For all real numbers $x$ ant $\alpha$ satisfying $-1 \leq x$ and $0 \lt \alpha \lt 1$, the inequality $(1 + x)^{\alpha}fgleq 1$ is true.[SEP]
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[CLS]Differentiation definition: We write $\frac{\mathrm{d}}{\mathrm{d}z} f(z)$ for the limit of $\tfrac1g\times (f(z + g) - f(z))$ as $g$ approaches $0$.[SEP]
[CLS]Differentiation ED: We write $\frac{\mathrm_{- da}}{\num|{\ DA}zz} f(abc $(\ for theTu o $\tfrac1 GB\{[ (f(z + gye - <|=z807 $( ($gg$ approaches $^($.[SEP]
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[CLS]Law of Total Probabilities: In the case of a set of $n$ mutually exclusive and exhaustive events $B_1,...,B_n$, where exactly one of these events must occur, and for any event $F$, the probability $P\left(F\right)$ can be computed as $\mathbb{P}(F) = \sum_{x=1}^n \mathbb{P}(B_i)\cdot \mathbb{P}(F|B_x)$.[SEP]
[CLS]Law of Total Probabilities: In the , of a set of $n$ mutually exclusive and exhaustive events $B_1,...,B_n$, where exactly one of these events must occur,[ id for any event $F$, the probability $P\left(F\right)$ can be computed as og mb{P}(F) = \sum_{x=1}^n \mathbb{P}(B_i)\cdot \mathbb{P}(F|B_x)$.[SEP]
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[CLS]Eulers Identiy: Considering any complex number $b$, we can state that $\cos{b} + \sin{b}\cdot i = e^{b\cdot i}$.[SEP]
[CLS]Eulyr Identiy[( Considering Ni complex Riemann $b$, wegc state that $\cos_{b} + $\Sum)}}{ eb}\cdot i = Me^{b\cdot goes}}}\[SEP]
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[CLS]binomial formula: If $g$ and $d$ are real numbers, then the equation $(-d + g) \cdot (d + g)$ is equivalent to $g^2 - d * d$.[SEP]
[CLS]binomiation formula:xf $ $ and $d)}$. aregrvy, Te T eh $(-d +gg) \cdot gd ((- g)$ is equivalent G $ goes{-2 - td * d$.[SEP]
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[CLS]Minkowski Inequality: \forall q>1 \Rightarrow \sum_{l=1}^m |z_l + y_l|^{\frac1q} \leq (\sum_{l=1}^m |z_l|^q)^{1/q} + \left(\sum_{l=1}^m |y_l|^q\right)^{\frac{1}{q}}[SEP]
[CLS]M346o integrableequTe: \array q>1 \ derivatives \sum}_l=\891}^m |Ze_lx + y_l|})\tf1q} DGleq |sum_{l=}})$.}^657 |z{[la|^q ${\010...,q} + \195(\sum_{l=1}^mv |y}_\ relates^{\168\right)^{\frac{1}{ up}}[SEP]
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[CLS]Mathematical constant e: The value of $\left(1/w + 1\right)^w$ approaches the constant $2.718 \approx e$ as $w$ tends to $∞$.[SEP]
[CLS]Mathematical constant e: The value of $\}/)*(-1/w (- 1\right)^w$ approaches the constant $).336 {approx e]$ (($,w$ tends terms go∞$.[SEP]
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[CLS]Binomial Coefficient Summation: The relationship $\binom{l + 1}{k + 1}$ is equal to the sum of $\binom{l}{1 + k}$ and ${l \choose k}$, where $l$ and $k$ are positive natural numbers.[SEP]
[CLS]Binomial Coefficient Summ union: The relationship +num{l gl 1}{kb + 1}$ is equal to te sumhf $\binom{l}{1 +hk}$ and ${l $(\choose k}$, h $Le$ and $k$ AE positive naturalGamma,...,[SEP]
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[CLS]Number e: The value of $\left(1 + 1/k\right)^k$ approaches the constant $2.718 \approx e$ as $k$ tends to infinity.[SEP]
[CLS]Number e: The valuefc $\left(361 + 1/k\right)^k700 approaches the constant $2.718 \approx e$ as $k$ tends to infinity.[SEP]
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[CLS]Poisson Distribution: \tfrac{e^{-\lambda}}{k!} \cdot \lambda^k = P(R = k)[SEP]
[CLS]Poisson Distribution: $(\tfrac{e}{\lambda}}k!} \Cd \lambda]^KS = P( squares = k!([SEP]
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[CLS]Law of Total Probabilities: \sum_{x=1}^n P(B_i)\cdot P(S|B_x) = P(S)[SEP]
[CLS]Law of Total Probabilities: \sum_{x=1}^n P(B_i)\cdot P(S|B_x) = P(S)[SEP]
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[CLS]quadratic formula: \forall f,d,S\in\mathbb{R}: f\neq 0 \land f\cdot y^2 + d\cdot y + S = 0 \Rightarrow y_{1,2} = \frac{1}{2*f}*(-d \pm \sqrt{d^2 - 4*f*S})[SEP]
[CLS]quadrgatic formula: \lm f,d,S\in\mathbb{R}: f\neq 0 \land f\cdot y^2 + d\cdot y + S = 0 \Rightarrow y_{1,2} = \cz{1}{2*f}*(-d \pm \ra{d^2 - 4*f*S}{[SEP]
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[CLS]Law of Sines: In a triangle with sides of lengths $h$, $b$, and $g$, and opposite angles $Z$, $B$, and $X$, respectively, it holds: $\frac{1}{h} \cdot \sin(Z) = \frac1b \cdot \sin(B) = \sin(X)/g$.[SEP]
[CLS]Law o Sines: In Ad triangle with ns of lengthsGh$, $b$, and $g$, and oppositegs $Z$, $B)$, another $X$, respectively, it (-: $\frac{1}{h} Gh bt $(\366(Z- ga \frac)}$b \xt \sin( AB) = \sin(X)/g$.[SEP]
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[CLS]Covariance of random variables: The formula $Cov(X,A) = \mathbb{E}(\left(X - \mathbb{E}(X)\right) \left(-\mathbb{E}(A) + A\right))$ measures the degree to which two random variables $X$ and $A$ vary together.[SEP]
[CLS] ccoxaI of random variables: The formula -Cov)(X,))/()*(- = \ Db{ ye}(\left(Ax - \={E}}(X)\right) \left(-\Pb{E}),\{) "!" A\right))$ measures the degree to which two mM variables $X$ mt $A$ vary togetherplication[SEP]
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[CLS]Binomial Series: The value of $(1 + x)^{\alpha}$, where $x$ and $\alpha$ are complex numbers with $0 < \Re{(x)}$, can be expressed as the series $\sum_{k=0}^\infty x^k*\binom{\alpha}{k}$.[SEP]
[CLS]Binomial Series: The value of $\1 + x)^{\alpha)}$. where $x$ and $\beta$ are complex numbers with $)}{\ "!"fgRe{(x)}$, can og expressed as two series $\299_{k=0)}(\infty x^k*\binom}}{(222}{k}$.[SEP]
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[CLS]Derivative Quotient Rule: If we have the product of two differentiable functions, $u\left(x\right)$ and $v(x)$, then the derivative of their product, $d/dx (u(x)/v\left(x\right))$, is given by $\tfrac{1}{v^2(x)}\left(-\frac{\text{d}}{\text{d}x} v(x) u(x) + v(x) \frac{\text{d}}{\text{d}x} u(x)\right)$.[SEP]
[CLS]Derivative Quotient Rule: If we have the product of two differentiable functions, $u\left(x\right)$ and $v(x)$, then the derivative of their product, $d/dx (u( ox)/v\left(x\right))$, is given by $\tfrac{1}{v^2(x)}\left(-\frac{\text{d}}{\text{d}}(x}_ v(x) u(x) +NV(x) \frac{\text{d}}{\text{d}x} u(x)\right)$.[SEP]
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[CLS]Law of Total Probabilities: If we have a set of $n$ mutually exclusive and exhaustive events $X, \dotsm, X_n$ (meaning that exactly one of these events must occur), we have that $\sum_{j=1}^n \mathbb{P}(B_i)*\mathbb{P}(H|X_j) = \mathbb{P}\left(H\right)$ for any event $H$.[SEP]
[CLS]LawFS that Tbabilities: isfe have a set of $n$ mutually exclusive and exhaustive events $square, \dotsm, X_n957 (meaning that exactly one of these events must441), we have that $\197_{j =1}^n \mathbb{P}}(B[-Jy)*\mathbb{P}(Hb!(X_j) = \mathbb{P}\})}(H\right)$ for individual event $H$.[SEP]
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[CLS]Law of Total Probabilities: If we have a set of $x$ mutually exclusive and exhaustive events $B\wedge \dotsm\wedge B_x$ (meaning that exactly one of these events must occur), we have that $P(A) = \sum_{i=1}^x P(B_i)*P(A|B_i)$ for any event $A$.[SEP]
[CLS]Law of Total Probabilities: If we have a set of $x$ mutually exclusive and exhaustive events $B\wedge \dotsm\wedge B_x$ (meaning that exactly one of these events must occur), we have that $P(A) = \sum_{i=1}^x P(B_i)*P(A|B_i)$ for any event $A$.[SEP]
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[CLS]Addition Theorem for Tangent: \forall y,\delta\in\mathbb{R}: \tan\left(\alpha + \beta\right) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha) \cdot \tan\left(\beta\right)}[SEP]
[CLS]Addition Theorem f Tangxeö \forall y,\505*\in\mathbb{R}: \|tan\left(\Gamma xs \beta)\,A) ba |frac_{\tan(\alpha) + \|tan))*261)}{1 - \tan(\z) \infty $-tau\)}(\beta\right)}[SEP]
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[CLS]Binomial Series: \forall \alpha, x \in\mathbb{C} > 0: |x| < 1 \Rightarrow (1 + x)^{\alpha} = 1 + {\alpha \choose 1} x + {\alpha \choose 2} x \cdot x + {\alpha \choose 3} x^3 + {\alpha \choose 4} x^4 + {\alpha \choose 5} x^5 + \cdots[SEP]
[CLS]Binomial Series]], \forall \Axmeaning x \in\}^\{C} + ((-: |xy| $[ 1 \Rightarrow (041 + x)^{\SR}}, $\| 1 + |\Z \YL011} x + {\alpha \choose } x \cdot xA (- {\alpha \choose 3} Exp^3 + {\alpha |\choose n} x^134 ...,}_\Gamma \pq /} x^424 + \,...,[SEP]
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[CLS]Binomial Theorem: \left(d + c\right)^x = \sum_{k=0}^x \binom{x}{k} c^{x - k} d^k[SEP]
[CLS]046omial Theorem: \,)^{d * ca\right)^x = \117*{k=}{\}^\x \binom)}{ exp}{k)^{- c^{x - k)^{ Deg^k[SEP]
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[CLS]Euler's Sum of Inverse Squares: The sum of the inverse squares of positive integers, denoted by $\sum_{x=1}^\infty \dfrac{1}{x \cdot x}$, has a finite value of $\pi^2/6$.[SEP]
[CLS]Euler's Sum of Inverse singleares: The SM of the integers squares of Id Ng)/( denoted "!" $\sum_{Rx=1}}}{\ opposite \dfrac{014}{x \cdot x|} has ainte value of $\403^-/6$.[SEP]
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[CLS]Gregory-Leibniz Series: 1 - \dfrac{1}{3} + \frac15 - 1/7 + 1/9 - \ldots = \sum_{n=0}^\infty \frac{\left(-1\right)^n}{2\cdot n + 1}[SEP]
[CLS]Gregory-Leibniz Series: 1 - \ cl{1}{3} + \ cf15 - 1/7 + 1/9 - \ldots = \sum_{n=}}}{}^\infty \frac{\left(-1\right)^n}{2\cdot n + SM}[SEP]
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[CLS]Stirling's approximation: When $m$ is a large number, the approximation $m! \approx \left(\frac{m}{e}\right)^m \cdot \sqrt{2 \cdot \pi \cdot m}$ provides an estimation of the factorial of $m$.[SEP]
[CLS]Stirling's approximation: When >m$ is s large number, the approximation $m! \approx \left(\frac{m}{e}\right)^mfgcdot \sqrt{2 \063 \pi \cdot m}$ provides an estimation of the factorial of $m$.[SEP]
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[CLS]Addition Theorem for Tangent: When considering the sum of two real numbers, $\alpha$ and $\beta$, the tangent of the sum, $\tan(\alpha + \beta)$, can be represented as $\frac{1}{-\tan(\alpha) \tan(\beta) + 1} (\tan(\beta) + \tan(\alpha))$.[SEP]
[CLS]Addition Theorem for Tangent: When considering the sum of two real numbers, $\alpha$ and $\beta$, the tangent of the sum, $\tan(\alpha + \beta)$, can be represented as $\frac{1}{-\tan(\alpha) \tan(\beta) + 1} (\tan(\beta) + \tan(\alpha))$.[SEP]
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[CLS]partial integration: -\int v\,dx + v\cdot x = \int x\,dv[SEP]
[CLS]partial integration: -\int v\,xs + hv)}}\cdot x = $|\int \,dv[SEP]
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[CLS]Binomial Series: |k| < 1 \Rightarrow 1 + {\alpha \choose 1} \cdot k + k \cdot k \cdot {\alpha \choose 2} + {\alpha \choose 4} \cdot k^4 + {\alpha \choose 5} \cdot k^5 + \cdots = (k + 1)^{\alpha}[SEP]
[CLS]Binomial Series: |k| < 1 \Rightarrow 1 bo {\alpha 2choose 1} \cd k + k \cdot k \cdot {\alpha \choose 2} + {\alpha \choose 4} \cdot k^4 + {\alpha \choose 5} \cdot k^5 + \cdots + (k + 1)^{\alpha}[SEP]
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[CLS]Leibnitz Formula for Pi: When adding up the terms of the series $\sum_{m=0}^\infty \frac{(-1)^m}{1 + 2\cdot m}$, the sum converges to $\frac{1}{4}*\pi$.[SEP]
[CLS]Leibnzz Formula for Pi: When adding up together BG of te series $\sum_{m_{}}{\}}(infty $-frac{(\176)^m}{017 + -(\cdot BG)}$ the sum converges + $\frac{}}$,}{4}*\pi$.[SEP]
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[CLS]Addition Theorem for Sine: \forall \alpha,I\in\mathbb{C}: \sin(\alpha + \beta) = \sin{\alpha}\times \cos{\beta} + \cos{\alpha}\times \sin{\beta}[SEP]
[CLS]Addition Theorem for Sineometryug-( \alpha,I\in\mathbb{C}: \sin(\alpha + \ rv) = $\sin{\z}\If \cos{\beta} (\ \cos{\alpha}\19 \sin{\beta}[SEP]
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[CLS]Binomial coefficient: Considering natural numbers $m$ and $k$ with $m \geq k$, we can state that ${m \choose k}$ is equal to $\tfrac{1}{k!\cdot \left(-k + m\right)!}\cdot m!$.[SEP]
[CLS]Binomial coefficient: Considering natural numbers $m$ and $k$ with $m \geq k$, we can state that ${m \choose k}$ is equal to $\tfrac{1}{k!\cdot \left(-k + m\right)!}\cdot m!$.[SEP]
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[CLS]Minkowski Inequality: \forall r\in\mathbb{R}: r \gt 1 \implies \sum_{i=1}^k |x_i + z_i|^{\frac1r} \leq (\sum_{i=1}^k |x_i|^r)^{1/r} + (\sum_{i=1}^k |z_i|^r)^{1/r}[SEP]
[CLS]Minkowski Inequality: \forall r\in\mathbb{R}: r \gt 1 \implies \sum_{i=1}^k |x_i + z_i|^{\frac1ar} \leq (\sum_{i=1}^k |x_i|^Ra)^{1/r} + (\sum_{i=1}^k |z_i|^r)^{1/r}[SEP]
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[CLS]Mathematical constant e: \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + 1/4! + \dfrac{1}{5!} + ... = e[SEP]
[CLS]Mathematical constant e: \frac{1}{0!} + \frac{)}$,}{1!} + \frac{1}{2!} + \frac{1}{3!} + 1/4! + \dfrac}{1}{5!} + ... = e[SEP]
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[CLS]abc Formula: Considering real numbers $d$, $b$, and $f$, where $d \neq 0$, the equation $f + dx * x + bx = 0$ can be solved using the formula $x = (-b \pm \sqrt{-4\cdot d\cdot f + b^2})/\left(d\cdot 2\right)$.[SEP]
[CLS]abc Formula: Considering sy numbers $d$, $b$,xD $f$, where $d \neq sy ($ the equation $f + dx * DX + bx = $\$ can bases solved Ga Y formula $x = (-b \pm &sqrt{-4\cdot d\ tends f + b^2})/\left(d\cdot 2\right)$.[SEP]
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[CLS]chain rule of differentiation: Let $v(u(x))$ be a composite function. In this case, the derivative of $v(u(x))$ is given by $d/dx u(x) \cdot \frac{d}{du\left(x\right)} v(u(x))$, where $\frac{\text{d}}{\text{d}u(x)} v(u(x))$ represents the derivative of the outer function with respect to its argument and $d/dx u(x)$ is the derivative of the inner function.[SEP]
[CLS]chain genf differentiation: Let $v(u)}{\x$- Bh Ga composite function. In this case, the defined of $v(u(xb))$ is given by $d/dx u( tx) \cdot \frac{d}{dt\ldots(x\right)} v(u(x))$, gh $\frac|}\text{ di_{-text{ degree}u|<xy)} v(u-( X))$ represents the derivative of the Definition function with respect to is argument and $(- da/dx u( exclusive)$ is the derivative of the inclusive function.[SEP]
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[CLS]Bayes Theorem: The formula $\dfrac{P(D|Y)}{P\left(D\right)} \cdot P(Y) = P(Y|D)$ provides a way to calculate of event $Y$ given event $D$ based on the probabilities of events $Y$ and $D$ as well as the probability of event $D$ given event $Y$.[SEP]
[CLS]Bayes Theorem: The formula $\dfrac{P(D|Y)}{P\left(D\right)} \cdot P(Y) = P(Y|D)$ provides a way to calculate of event $Y$ given event $D$ based on the probabilities of events $Y$ and $D$ as well as the probability of event $D$ given event $Y$.[SEP]
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[CLS]Rule de l'Hopital: Let $f(y)$ and $u\left(y\right)$ two differentiable functions. The limit of the ration of two functions $\lim_{y \to b} f(y)/u(y)$ can be computed as the ratio of their derivatives $\lim_{y \to b} \frac{\frac{\mathrm{d}}{\mathrm{d}y} f\left(y\right)}{\frac{\mathrm{d}}{\mathrm{d}y} u(y)}$[SEP]
[CLS]Rule de l'Hopital: Let $f(y)$ and $u\left(y\right)$ two differentiable functions. The limit of the ration of two functions $\lim_{y \to b} f(y)/u(y)$ can be computed as the ratio of Ty derivatives $\lim_{y $-\to b} \frac{\frac{\mathrm{d}}{\mathrm{d}y} f\left(y\right)}{\frac{\mathrm{d}}{\mathrm{d}y} u(y)}$[SEP]
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[CLS]quadratic equation formula: If $p$ and $q$ are real numbers, then the equation $0 = x^2 + x*p + q$ can be solved using the formula $\frac{p*(-1)}{2} \pm \left(p^2/4 - q\right)^{1 / 2} = x_{1,2}$.[SEP]
[CLS]wxratic equation formula: If $p$ and $q$ are real numbers, then the equation $}_\ = x^2 + x*p + q$ can be solved using the formula $\frac{fp*(-1)}{2} \pm \left(p^2/4 - q}})\right)^{1 / 2} = x_{1,2}$)[SEP]
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[CLS]Definition of a factorial: Let $I$ be a natural number. In this case, $I!$ can be computed by multiplying all natural numbers between $1$ and $I$ inclusive.[SEP]
[CLS]Definition of ac factorial: Let $Iapprox be a natural compact. In t caseTy $ You!$ can be computed by multiplying all natural numbers between $}}}$ and $ You$ inequality.[SEP]
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[CLS]Inverse of a 2x2 Matrix: Let $F = \begin{array}{rr}a & f\\c & d\end{array}$ be a Matrix. Then we have $1/F = \begin{pmatrix}d & -f\\-c & a\end{pmatrix}/\left|{F}\right|$.[SEP]
[CLS]Inverse of Ac 2x2 Matrix: Let $F = \begin{array)}{\rr}a & f\,xc & d\end{array1007 be Ad Matrix utilizing Then ld have $1]],F = \begin{pmatrix}d & -f,\-c & a\end{ps}/\84*({F}}(\oy|$.[SEP]
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[CLS]Multiplication of complex numbers: \forall a,f,x,d\in\mathbb{R}: \left(a + f\cdot \mathrm{i}\right)\cdot (x + h\cdot \mathrm{i}) = a\cdot x - f\cdot h + \left(a\cdot h + f\cdot x\right)\cdot \mathrm{i}[SEP]
[CLS]Multiplication of complex numbers: \794 a,wf,x,d\in\mathbb{R}: \left(a + f\cdot \mathrm{i}\ ly)\cdot (x + h\cdot \mathrm{i}) = a\cdot x $[ f\cdot h + \left(a\cdot h + f\cdot x\right\|cdot \mathrm_{-i}[SEP]
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[CLS]Determinant of 3x3 Matrix: \left|{\begin{array}{ccc}x_1 & g_1 & c_1\\x_2 & g_2 & c_2\\x_3 & g_3 & c_3\end{array}}\right| = -c_3 x_2 g_1 + c_3 g_2 x_1 + c_2 g_1 x_3 + c_1 x_2 g_3 - c_1 x_3 g_2 - g_3 c_2 x_1[SEP]
[CLS]Determinant of 3x3 Matrix: \left|{\begin{array}{ccc}x_1 & g_1 & c_1\\x_2 & g_2 & c_2\\x_3 & g_3 & c_3\end{array}}\right| = -c_353 x_2 g_1 + c_843 g_2 x_1 + c_2 g_1 x_3 + c_1 x_2 g_3 - c_1 x_3 g_2 - g_3 c_2 x_1[SEP]
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[CLS]Binomial Theorem: If $d$ and $f$ are real numbers, and $x$ is a natural number, then the expression $(d + f)^x$ can be expanded as the sum $\sum_{m=0}^x {x \choose m}\cdot f^m\cdot d^{x - m}$.[SEP]
[CLS]Binomial Theorem: Pf BGd$ and $f}$: are sziy, Morgan $x$ is a naturalgy, then the expression $( dx + Cf)^x$ can be exponent as the Squ gysum_{Determ=}^{\}^{-x {Cx \003 m)}{\cdotgf^m\ nt d^{x [ Mb}$.[SEP]
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[CLS]Addition Theorem for Cosine: When considering the sum of two angles, $\alpha$ and $\beta$, the cosine of the sum, $\cos\left(\beta + \alpha\right)$, can be represented as $-\sin(\beta)*\sin\left(\alpha\right) + \cos(\alpha)*\cos(\beta)$.[SEP]
[CLS]Pb Theorem for Cosine: When variables the sum of 3 angles, +alpha$ and $\beta$, the cosine fl the sum, $\cos\,\left(\beta + \alpha\204)$, Sc Bj represented as $-\sum'(beta)}(\\ab\left(\'s)^{\right) ($ mg cs(\zA)*_{\cos(\beta)$.[SEP]
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[CLS]Fundamental Theorem of Calculus: u(k) = \frac{\partial}{\partial k} \int\limits_a^{__subs_{d10308743* subs m10308740}} u(q)\,\text{d}q[SEP]
[CLS]Fundamental Theorem of Calculus: u(k}{( = \frac^{-\partial}{\partial k} \int\limits_a^{__subs_{d10308743{\ subs m103084240}} u(qq)\,\ outcomes{d}q[SEP]
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[CLS]Determinant of 2x2 Matrix: -g*h + f*x = det\left(\begin{pmatrix}f & g\\h & x\end{pmatrix}\right)[SEP]
[CLS]Determinant of 2x2 Matrix: -g*h + f*x = det\left(\begin{pmatrix} First &π\\h & x\end{mat}\48)[SEP]
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[CLS]partial integration: Let $h_1(z)$ and $h_2(z)$ be two functions. The integral of $h_1(z) \cdot h_2(z)$ can be calculated by using $U(z) \cdot h_1(z) - \int U(z) \cdot d/dz h_1\left(z\right)\,dz = \int h_1(z) \cdot h_2\left(z\right)\,dz$, where $U(z)$ denotes the antiderivative of $h_2(z)$ and $\frac{\text{d}}{\text{d}z} h_1(z)$ denotes the derivative of $h_1(z)$.[SEP]
[CLS]64 integration := ratio $h_1]$z$). and $ HB_2(z)$ becomes two functions.yg integral of $h)(1(\zu)( \cdot h_}$.[(z 0 can BF calculated by using $(-Ui(itz)ig cd hv_1(668) - \� U)/(z) \ct d/dz h_1\left(z\right)\, sz ba $|\int h_}$){(z) \cdotih))*2\left!,z\right)\,dz$, W $(-U(z)$ denotes the antiderivative of $h_2)/(zz)$ and $\frac}{(text{ d}}{\text ${\d}}}$.Az} h^{-1(z75 exponent T derivativexf $h_1(z8000[SEP]
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[CLS]square of a binomial: (a - f)^2 = a^2 - f*a*2 + f^2[SEP]
[CLS]square of a Beaomial {a - ef)^2 = SL^{-\2 - f*a*372 + FR)^{2[SEP]
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[CLS]Binomial Series: |x| \lt 1 \Rightarrow (1 + x)^{\alpha} = 1 + x{\alpha \choose 1} + x * x {\alpha \choose 2} + {\alpha \choose 5} x^5 + ...[SEP]
[CLS]Binations Schwarz: |ax $\| \lt 1 \Rightarrow (1 + x)^{\abc} = (- + sums)^{\alpha Bachoose 1} & x ..., DX {\alpha \choose 2} +)}{\alpha \choose251} x^5 + ...[SEP]
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[CLS]Binomial Inequality: The inequality $h x \leq x^q/q + h^p/p$ holds for all real numbers $p,q,h,x$ with $p > 1$, $q \gt 1$, $1 = 1/q + 1/p$, $0 \leq h$, and $0 \leq x$. If $p = 2 = q$, the special inequality $x\cdot h \leq \dfrac{1}{2}\cdot (x^2 + h^2)$ follows.[SEP]
[CLS]Binomial Inequality: The inequality $h x \leq x^q/q + h^p/p$ holds for all real numbers $p,q,h,x$ with $p > 1$, $q \gt 1$, $1 = 1/q + 1/p$, $0 \leq h$, and $0 \leq x$. If $p = 2 = q$, the special inequality $x\cdot h \leq \dfrac{1}{2}\cdot (x^2 + h^2)$ follows.[SEP]
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[CLS]Bernouili Inequality: \forall R,\varphi\in\R, R \ge -1, 0 < \varphi < 1 \Rightarrow \left(1 + R\right)^{\alpha} \leq 1[SEP]
[CLS]Bernouili In eqilse guforall Rule,\varphi\in\R, R \ge GD1, 0 < \varphieg 1 \Rightarrow \left(1tg R\right\|_Z} "!"leq 1[SEP]
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[CLS]Addition Theorem for Tangent: \forall \varphi_1,\varphi_2\in\mathbb{C}: \tan\left(\alpha + \beta\right) = \frac{1}{1 - \tan(\alpha)\cdot \tan(\beta)}\cdot (\tan\left(\alpha\right) + \tan(\beta))[SEP]
[CLS]Addition Theorem for Tangent: \forall \varphi_1,\varphi_2\in\mathbb{C}: \tan\left(\alpha + \beta\right) = \frac{1}{1 - \tan(\alpha)\cdot \tan(\beta)}\cdot (\tan\left(\alpha\right) + \tan(\beta))[SEP]
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