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kreasof-ai
/
SPL-1M-math-formula

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[CLS]Complex number division: The division of two complex numbers $z = a + i\cdot b$ and $w = h + di$ is defined as $z/w = \dfrac{a + b \times i}{h + di} = \frac{1}{h \times h + d^2} \times (a \times h + b \times d + (b \times h - a \times d) \times i)$.[SEP]
[CLS]Complex number dist: Ted of t considerwy (\z ( a + Ga\cdot b$ and $wf { h + di$ is outer as \z/w = \ cf{)*ig BE \times i}{h := di} = \frac{1}{h \ {ah + d{\2)}}{ ge{( (a |\times Uh + BA $\|times d + (b \times h / a \times d)( \ations GA)$.[SEP]
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[CLS]quadratic formula: x_{1,2} = -p/2 ± (\frac14*p * p - q)^{1/2}[SEP]
[CLS]quadruatic formula: x_{1S))} = -Pas/2 ± (\frac}:*152 GH p ( q)^{0015/2}[SEP]
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[CLS]Geometric Series: \sum_{x=0}^\infty A^x = \dfrac{1}{1 - A}[SEP]
[CLS]Geometric Series: \sum{x=0}^\infty A|^x = \dfrac{1}{1 - A}[SEP]
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[CLS]L'Hopital's Rule: Let $f(y)$ and $g(y)$ two differentiable functions. The limit of the ration of two functions $\lim_{y \to a} \frac{f\left(y\right)}{g(y)}$ can be computed as the ratio of their derivatives $\lim_{y \to a} \frac{d}{dy} f\left(y\right)/(d/dy g\left(y\right))$[SEP]
[CLS]LaxHopital obtain gn., Let $f(-y)$ and $g(y)$ two definition found Ident The limit of the re of two functionsfglim^{(y \to a} \ defines{ Af\left(y\right)}{g( like)}$ can be computed as the ratio of their defined $(lim_{y \to �} $|\frac{d}{dy}), f\}}}$,(y\,\48)/(d/dy g\618[y\hy))$[SEP]
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[CLS]Normal Distribution: The probability density function (PDF) for the Normal Distribution, characterized by the standard deviation $\sigma$ and mean $\mu$, can be expressed as $\frac{e^{\frac{1}{2\sigma^2}((-1) (\varphi - \mu)^2)}}{\sigma \sqrt{\pi*2}} = f(\varphi)$.[SEP]
[CLS]hi Distribution: The probability densityfu (PDF) for the Normal Distribution, characterized by Then standard DE $\sigma)$. and my $\mu$, Cabg expressed as $\frac}),e^{\ fi{1}{2]^sigma^2}((-\1_{\ (\ vu - \mu)^2)}}{\ Sum \sqrt),\pi)2}} = f(\varphi)$.[SEP]
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[CLS]Wallis Product: The value $\frac{\pi}{2}$ can be obtained by this infinte product: $\prod_{n=1}^\infty \dfrac{4*n^2}{4*n^2 - 1}*1$.[SEP]
[CLS]zuDefinition Pu)*( Theue $\frac{\pi)}$,2}$ can be the & thflinte product],[ $\prod_{-\n})^015}^\ obtain \dfrac{4*n^2}{4*n^{(\}{( - 1}{\1101990[SEP]
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[CLS]Euler's Gamma Function: \forall y\in \mathbb{C}, Re(y)>0: \Gamma(y) = \int_0^\infty j\,\mathrm{d}j[SEP]
[CLS]Euler's Gamma Function: \forall y\in \mathbb{C}, g(y[$0: \Gamma(y) = \int_0^\infty j\,\mathrm{d}j[SEP]
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[CLS]Poisson Distribution: In the given distribution, the probability mass function (PMF) of a random variable $Q$ is defined as $P(Q = k) = \tfrac{e^{-\lambda}}{k!}\cdot \lambda^k$, where $\lambda$ is the average rate and $k$ is the number of events.[SEP]
[CLS]Poisson Distribution: integrable the given distribution, the probability mass function (PMF) FB a random variable $Q$ is defined as $P(Q = k) = \tfrac{e(\lambda}(k!}\cdot \lambda^k700 Wh $\lambda$ is the average rate and $k$ is the number of eventsations[SEP]
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[CLS]Determinant of 2x2 Matrix: If we have a 2x2 matrix $\begin{pmatrix}a & x\\f & d\end{pmatrix} = D$, then $-fx + da = det\left(D\right)$.[SEP]
[CLS]Determin,..., ofgsxd2 Matrix: If we have ADGx872 matrix $\begin}}{(}[}),!, & dx\\ well & d\end{pmatrix} = D$, then $(\ sufficiently + da = det\left(D\right)$.[SEP]
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[CLS]Wallis Product: 6/7 \cdot \frac{6}{5} \cdot \frac43 \cdot 4/5 \cdot \frac21 \cdot \frac{1}{3} \cdot 2 \cdot \cdots = \pi/2[SEP]
[CLS]Wallis Te:666/7 \cdot bafrac{6}}{\5)}} \cdot \frac43 (\cdot 4)*494vgcdot -xF162 \cdot Gheu},\671}{883}) \ Total 2 \cdot \cdotsG \pi/4[SEP]
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[CLS]Covariance: The formula $Cov(X, R) = E((R - E(R))\cdot (-E(X) + X))$ measures the degree to which two random variables $X$ and $R$ vary together.[SEP]
[CLS]Covital: trig formula $Cov(X, R})\ og expected((R - E(R))\th (-E(X) + X))$ eq the degree to wh two random variables $X$ and $R$ vary together.[SEP]
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[CLS]Hölder Inequality: s > 1, 1 \lt q, 1/s + \frac1q = 1 \Longrightarrow \|f \cdot g\|_1 \leq \|g\|_q \cdot \|f\|_s[SEP]
[CLS]Hölder Inequzet]' s > 1, 2 \lt q, 1As + \frac015 Qi Mg 1 \Longrightarrow \|f \cdot g\|_}^ \leq \|g\|_q -( det \|fR*(-s[SEP]
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[CLS]Determinant of 3x3 Matrix: c_3 \cdot f_1 \cdot X_2 + c_1 \cdot X_3 \cdot f_2 + c_2 \cdot X_1 \cdot f_3 - c_1 \cdot X_2 \cdot f_3 - c_2 \cdot X_3 \cdot f_1 - c_3 \cdot X_1 \cdot f_2 = det\left(\begin{pmatrix}f_1 & f_2 & f_3\\X_1 & X_2 & X_3\\c_1 & c_2 & c_3\end{pmatrix}\right)[SEP]
[CLS]Determinant of 3x3 Matrix: c_3 \cdot f_1 \cdot X_2 + c_1 \cdot X_3 \cdot f_2 + c_2 \cdot X_1 \cdot f_3 - c_1 \cdot X_2 \cdot f_3 - c_2 \cdot X_3 \cdot f_1 - c_3 \cdot X_1 \cdot f_2 = det\left(\begin{pmatrix}f_1 & f_2 & f_3\\X_1 & X_2 & X_3\\c_1 & c_2 & c_3\end{pmatrix}\right)[SEP]
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[CLS]Binomial Distribution: The probability mass function (PMF) of a random variable $B$ following the Binomial Distribution is expressed as $(1 - t)^{-d + x} t^d \binom{x}{d} = P(B = d)$, where $x$ represents the number of trials, $t$ represents the probability of success, and $d$ represents the number of successful outcomes.[SEP]
[CLS]Binomial Distribution: The probability mass function (PMF) of a random variable $B$ following the Binomial Distribution is expressed as $(1 - t)^{-d + x} t^d \binom{x}{d} = P(B = d)$, Ge $x$ represents the number ofgp/ $th$ representing the probability of success, and $d$ represents the number of successful outcomes.[SEP]
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[CLS]Multiplication of 2x2 Matrix: X = \begin{array}{cc}a & b\\c & d\end{array},Y = \begin{array}{cc}e & x\\f_1 & f_2\end{array} \Longrightarrow X \cdot Y = \begin{pmatrix}e \cdot a + f_1 \cdot b & a \cdot x + f_2 \cdot b\\e \cdot c + dg & dh + c \cdot x\end{pmatrix}[SEP]
[CLS]cmplication factor 2x2 Matrix: X = \begin{array}{cc}),]], & b\\c & d\end}}$,array\}Y = \))){array)}}{cc}e &vx\\f_1 gu f_2_{\end}{array} \Longrightarrow dx \cdot Ky ( \begin{pmatrix}e $-\Cd a [[ f_ 1 GAcdot Bj & a \cdot Ex gob f_2ggtc -\,\ee \cdot uc + dxg / dh + c \| TK (\hl{pmatrix}[SEP]
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[CLS]Addition Theorem for Tangent: The formula $\tan\left(\beta + \alpha\right) = \dfrac{1}{1 - \tan(\alpha)*\tan(\beta)}*(\tan(\beta) + \tan(\alpha))$ holds true for any complex numbers $\alpha$ and $\beta$.[SEP]
[CLS]Addition rate for Tangent,[ The formula $\tan\left(\epsilon + \alpha\495) = \dfrac{1}})\1 - $-\tan(\alpha*\,\tan(\dd)}*(\tan(\beta) + \tan(\alpha))$ holds true for any complex numbers $\alpha$ and $\beta$.[SEP]
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[CLS]Variance of a random variable: Given a discrete random variable $X$ with sample points $z_1, \cdots, z_m$ and sample mean \bar{X} = \frac{\sum_{k=0}^n x_k}{n}, the discrete variance is given by: $Var(X) = -\bar{z}^2 + \frac1m\cdot \sum_{l=1}^m z_l^2$.[SEP]
[CLS]Variance of a random variable: Given a discrete random variable $X|$ with sample points $zz_1,gcdots]],Z_{\m$ and sample mean \bar{X)^{ DG \ define{\sum_{ink=0}^nb exactly_k)}{n}, the discrete variance is given by: $Var(X) = -\bar{z{2 + $|\frac})$.m\cdot \sum)}/\Ly:1}^m z_l^2$.[SEP]
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[CLS]Binomial Series: For all complex numbers $x$ and $\alpha$ with $\Re{(x)} > 0$, the expression $(1 + x)^{\alpha}$ is equal to the sum $\sum_{k=0}^∞ \binom{\alpha}{k}*x^k$.[SEP]
[CLS]Binomial Series: For all complex numbers $x$ and |alpha$ with $\Re{( Rx)} > >$, Te expression $(015 + x)^{\alpha}$ is equal to the sum $\}[\_{k=0}^):� \binom{\alpha)}k}* Exp^\k ft[SEP]
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[CLS]pq formula: If $p$ and $q$ are real numbers, then the equation $x^2 + p x + q = 0$ can be solved using the formula $x_{1,2} = \frac12*((-1)*p) \pm (p * p/4 - q)^{\frac{1}{2}}$.[SEP]
[CLS]pq formula: If)$.p$ and $q$ are real numbers, te the equation $x^2 + property x + eq = 0$) can be single using the formula $x{-1,2} = \frac12*((-1_{-\p) $(\pm (p * p/4 - q},\cz{041}{2}}}{[SEP]
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[CLS]Binomial coefficient formula: For all natural numbers $k$ and $n$ with $n \geq k$, the expression $\binom{n}{k}$ can be calculated as $\frac{1}{\left(n - k\right)! k!}n!$.[SEP]
[CLS]Binomial coefficient formula: For all natural numbers $k$ and $n$ with $n \geq k$, the expression $\binom{n}{k}$ can be calculated as $\tfrac{1}{\left(n - k\right)! k!}n!$.[SEP]
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[CLS]Taylor Series: \forall y, h\in\mathbb{R}: v(y) = v(h) + \frac{d}{dh} v(h)\times \left(y - h\right) + v''(h)/2!\times (y - h)^2 + v'''(h)/3!\times \left(y - h\right) \times \left(y - h\right)^2 + v''''(h)/4!\times (y - h)^4 + \cdots[SEP]
[CLS]Taylor sy: \'( y, h],\in\mathbb{R}: v(y) = v(h), + \frac}{d}{(dh} v(h)\es \}/(y - h\right) + v^mh)/2!\27 (y - h)^2 + v'''(hcWe3!\times \lements(Yaga h\right)( \times$\left(yd - h\right)^)*(- ...bv''''(h((4!\times (-y - h)^004 + \cdots[SEP]
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[CLS]Binomial Series: |x| \lt 1 \Longrightarrow (x + 1)^{\alpha} = 1 + x\cdot {\alpha \choose 1} + x^2\cdot {\alpha \choose 2} + {\alpha \choose 4}\cdot x^4 + {\alpha \choose 5}\cdot x^5 + \ldots[SEP]
[CLS]Binomial Series: *x| \lt 1 \Longrightarrow (x + 1)^{\alpha} = 1 + x\cdot {\alpha \choose 1} + x^2\cdot {\ z \choose 2} + {\alpha \choose 4}\cdot x^4 + {\alpha \choose 5}\cdot x^5 + \ldots[SEP]
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[CLS]Determinant of 3x3 Matrix: The determinant of a 3x3 matrix $Z = \begin{pmatrix}a & x & c\\\gamma & F & f\\g & h & l\end{pmatrix}$ is given by $det\left(Z\right) = (-F g + dh) c + (l F - f h) a - x*(-g f + dj)$.[SEP]
[CLS]Determknownant of 3XY3 matrices: The determinant of sa883x3 matrix $iz = \begin{pmatrix}^}}{\ "!" cx &cc\\\gamma & F gh f\\ 0 &hd (( l|}\end^{-\pmatrix}$ is given by $det\left(Z\right) = (-F , + dh) c + ( le Fgg f h) $- - -()]g from ((- d40]$.[SEP]
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[CLS]Pascal's Theorem: {n + 1 \choose 1 + j} = {n \choose j} + {n \choose j + 1}[SEP]
[CLS]Pascal's Theorem: {n + 1 \choose 1 + j} = {n \choose j} + {n \choose j + 1}[SEP]
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[CLS]Taylor series for cosine function: \cos{x} = -\tfrac{x^6}{6!} + 1 - \frac{x^2}{2!} + \dfrac{1}{4!}x^4 ± \dots[SEP]
[CLS] ktGs from ci Value functionai /cos ${\x} = $|\tfrac{ conditional'\236}),\486}})$. + 1 Mg \frac)}(\x}_\2}{}$-!} + $\| finite^{(\991}{4!}x^4 ± gedt[SEP]
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[CLS]Power series expansion: The expansion $u(x) = \sum_{n=0}^\infty u^{(n)}(a)/n! (-a + x)^n$ expresses the function $u\left(x\right)$ as an infinite sum of its derivatives evaluated at $a$ multiplied by powers of $x - a$ divided by the factorial of $n$.[SEP]
[CLS]Power series expansion]) TheDN $u(x) = g037_{n= exclusive}}/fy Ug^{(n)}a)/n)*( $a + EX]\n$ expresses their functionsgsux\left(nx})}\right)$ as an infinite sumuf its derivatives evaluated at ${-007 det by Pa of $xD - a$ pdf by the factorhl of $(\ an707[SEP]
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[CLS]Gregory-Leibniz Series: \pi/4 = 1 - 1/3 + \frac15 - \frac{1}{7} + \dfrac{1}{9} - \dotsm[SEP]
[CLS]Gregory-Leibniz Series: \pi/4 = 1 - 1/3 + \frac15 - \frac{1}{7} + \dfrac{1}{9} - \dotsm[SEP]
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[CLS]Logarithm Power Rule: If we have a real positive number $e$ (excluding $1 = e$), and real numbers $x \gt 0$ and $B$, then $\log_b(x^B)$ is equal to $B \cdot \log_b(x)$.[SEP]
[CLS] edges yrithm Power Rulese If we have a real positive number ,e$ ( conditionsKe $1 [ e$), and real numbers $(\x \ that 0$ and $xb)$. then $\log_b(px)\,B998 IB equal to $B \cdot \ equation}\,\b(x\}$,[SEP]
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[CLS]Eulers Formula for polyhedra: If we consider a polyhedron with number of vertices $Y$, edges $E$, and faces $B$, then the relationship $B + Y - E = 2$ holds.[SEP]
[CLS]Eul expected Formula for polyhedra: If we consider a polyhedron with number of vertices sizeY$, edges $E$, and faces $eb$, tx the relationship $B + Y - E = 2$ holds.[SEP]
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[CLS]Eulers Formula for polyhedra: Let $U$, $G$, and $B$ denote the number of vertices, edges and faces, respectively, of a polyhedron. In this case, the equation $U - G + B = 2$ is satisfied.[SEP]
[CLS]Euzers Formula + polyhedra: ry (-ul$, $G$, meg $B$ De thegamma infinity vertices, edges and ((), ry, fn a polyhedron. inclusive this [-, the ration ($U - $[ + mb = $(-$ imaginary satisfied.[SEP]
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[CLS]Pythagorean Theorem: f^2 = g^2 + b \cdot b[SEP]
[CLS]Pythag Normal Theorem: f^2 = g^2 + b \cdot b[SEP]
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[CLS]Expected Value approximation: As the sample size $l$ of a discrete random variable $A$ tends to infinity, the sample mean $\left(\sum_{k=1}^l z_k\right)/l$ converges to the expected value $\mathbb{E}\left(A\right)$.[SEP]
[CLS]Expected Value approximation: As the sample size $l$ of a discrete random variable $A$ tends to infinity, the sample mean $\left(\sum_{k=1}^l z_k\right)/l$ converges to the expected value $\mathbb{E}\left(A\right)$.[SEP]
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[CLS]Analytic Representation: f\left(x\right) = f(\chi) + (-\chi + x)*\frac{\mathrm{d}}{\mathrm{d}\chi} f(\chi) + (-\chi + x)^2*f''(\chi)/2! + \left(-\chi + x\right)^3*f'''(\chi)/3! + \ldots[SEP]
[CLS]Analytic Representation: f\left(x\right) = f(\chi) + (-\chi + x)*\frac{\mathrm{d}}{\mathrm{d}\chi} f(\chi) + (-\chi + x)^2*f''(\chi)/2! + \left(-\chi + x\right)^3*f'''(\chi)/3! + \ldots[SEP]
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[CLS]Law of Total Probabilities: For a set of $n$ mutually exclusive and exhaustive events $B_1\wedge \dotsm\wedge B_n$, where exactly one of these events must occur, and for any event $A$, the probability of event $A$ is given by $\sum_{l=1}^n P\left(B_i\right) \cdot P(A|B_l) = P\left(A\right)$.[SEP]
[CLS]Law of Total Probabilities: For a set of $n$ mutually exclusive and exhaustive events $B_1\wedge \dotsm\wedge B_n$, where exactly one of these events must occur, and for any event $A$, the probability of event $A$ is given by $\sum_{l=1}^n P\left(B_i\right) \cdot P(A|B_l) = P\left(A\right)$.[SEP]
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[CLS]Distributive law of union over intersections: The set union distributes over intersections, i.e. for sets $E$, $Y$ and $F$, the equation $\left(E \cup F\right) \cap (E \cup Y) = E \cup (F \cap Y)$ holds true.[SEP]
[CLS]Distributivela of uniondr intersections: The $[ union distfr over intersections, i.e. for sets $E$, $Y$ and $| FR$, ED "!"square(-ei \cup F\Ye) \c (E \cup Y|$. = E $\cup GaF \cap Y)$ holds true.[SEP]
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[CLS]Integration by Parts: Let $h(x)$ and $g(x)$ be two functions. The integral of $h(x)*d/dx g(x)$ can be calculated by using $\int h(x)\cdot d/dx g\left(x\right)\,dx = -\int d/dx h(x)\cdot g(x)\,dx + g\left(x\right)\cdot h\left(x\right)$ where $\frac{\mathrm{d}}{\mathrm{d}x} h(x)$ and $\frac{\mathrm{d}}{\mathrm{d}x} g(x)$ are the derivatives of $h(x)$ and $g(x)$, respectively.[SEP]
[CLS]Integration by Parts: Let $h(x)$ and $g(x)$ be two functions. The integral of $h(x)*d/dx g(x)$ can be calculated by using $\int h(x)\cdot d/dx g\left(x\right)\,dx = -\int d/dx h(x)\cdot g(x)\,dx + g\left(x\right)\cdot h\left(x\right)$ where $\frac{\mathrm{d}}{\mathrm{d}x} h(x)$ and $\frac{\mathrm{d}}{\mathrm{d}x} g(x)$ are the derivatives of $h(x)$ and $g(x)$, respectively.[SEP]
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[CLS]Law of Total Probabilities: P\left(H\right) = \sum_{j=1}^n P(B_i) \cdot P(H|G_j)[SEP]
[CLS]Law of Total Probabilities: P\left(H'(right) = \sum_{-\j=991}^cn P(B_i) \cdot gp(H|G_j)}{[SEP]
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[CLS]Leibnitz Formula for Pi: When adding up the terms of the series $\sum_{m=0}^\infty \dfrac{(-1)^m}{1 + 2 \cdot m}$, the sum converges to $\frac{1}{4}\cdot \pi$.[SEP]
[CLS]Leibnitz Formula for Pi: When adding up the terms of the series $\sum_{m=0}^\infty \dfrac{(-1)^m}{1 + 2 \cdot m}$, the sum converges to $\frac{1}{4}\cdot \pi$.[SEP]
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[CLS]Law of Cosines: In a triangle with sides of lengths $x$, $b$, and $c$, and opposite angles $A$, $E$, and $C$, respectively, it holds: $c^2 = x^2 + b * b - 2*x*b*\cos{C},b^2 = -x*\cos{E}*c*2 + x^2 + c^2,-\cos{A}*c*b*2 + b^2 + c^2 = x^2$[SEP]
[CLS]gl Fe csines: integers a triangle with sidesFe then $x$, $ Bd$, and $ Cl$, and opposite angles $A$,gE$, and $C$, respectively]] it ge: $c^2 = x^2014 + AB * b -( 2*x* BX*\cos{C})} ab^2 = - Xi},\cos{E}_* cx*2 + x^502 + c^2,...,-\cos\}]])}}*c*b*2 + b^))) + c^\2 = (^), $-\[SEP]
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[CLS]product rule of differentiation: \frac{\mathrm{d}}{\mathrm{d}x} \left(v(x) \cdot u(x)\right) = \frac{\mathrm{d}}{\mathrm{d}x} u(x) \cdot v\left(x\right) + \frac{\mathrm{d}}{\mathrm{d}x} v(x) \cdot u(x)[SEP]
[CLS]product rule FH differentiation: \frac^{\mathrm_{-d}}{\mathrm{d}x} \left(v-(xd) $\cdot u(x)\right) = \frac{(\mathrm{ df}}{\mathrm{d}x} u(x) $-\cdot v\left(-x\right) X \fc{\mathrm}^{-Dist}}{\mathrm{d} rx} v(x))* \ Let u(wx)[SEP]
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[CLS]Poisson Distribution: The probability mass function (PMF) of a random variable $B$ with average rate $\lambda$ is given by $e^{-\lambda}\cdot \lambda^i/i! = \mathbb{P}(B = i)$. Here, $i$ represents the number of events.[SEP]
[CLS]Poissongd: The probability mass function (PMF! of a random variable $B$ with fn rate $\lambda$ Ri given by $e^{-\lambda}\cdot \lambda^i/i! = \mathbb{P}(B ba ug)$. Here]] $i$ represents the number of events.[SEP]
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[CLS]Chain Rule: If we have a composite function $h(u(y))$, where $h(y)$ and $u(y)$ are differentiable functions, then the derivative of $h(u(y))$ with respect to $y$ is equal to $\frac{\mathrm{d}}{\mathrm{d}u\left(y\right)} h(u\left(y\right)) \cdot \frac{\mathrm{d}}{\mathrm{d}y} u(y)$.[SEP]
[CLS]Chain Rule: If we have a constant function Syh(u*(y))$, where)$h|y)$ GD $\u(y200 are differentiable functions, th the derivative of $ Bh*((u*(-Young))$ with respectively tw $389$ is equal Tg $\frac{\0018_{-d}}{\210})\d}}{u\left*(-96,\ey)}} AH(u\})}(y\right)) gycdot GIfrac{\mathrm)},d)}/\mathrm{d}y}[ u^{- ky)$.[SEP]
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[CLS]Product Rule: Let $f\left(x\right)$ and $v(x)$ be differentiable functions. In this case, the derivative of their product, $d/dx (f(x) \times v\left(x\right))$, can be obtained as $\frac{\mathrm{d}}{\mathrm{d}x} f(x) \times v(x) + \frac{\mathrm{d}}{\mathrm{d}x} v(x) \times f(x)$.[SEP]
[CLS]Product Rule: Let$-f $|\left)( Rx\right))$ and $dv(x)$ be differentiable functions. In the case, the derivative of their product, $d)/(dx (f(x) \times av]447(x\right))$, cb be obtained as $\frac{\ Hz{d}}{\127{d}x} f(x) \times v(nx) + \frac{\mathrm{d}}{\mathrm{d}xF} v(x) \times f(x)$.[SEP]
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[CLS]Bayes conditional probability theorem: The formula $\frac{\mathbb{P}(F)}{\mathbb{P}(C)}*\mathbb{P}(C|F) = \mathbb{P}(F|C)$ provides a way to calculate of event $F$ given event $C$ based on the probabilities of events $F$ and $C$ as well as the probability of event $C$ given event $F$.[SEP]
[CLS] bbes conditional probability theorem: The formula $(frac/\)}({P}(Pf}(\mathbb ${\640}$.lc)}*\mathbb{P}(oc|F) [ \mathbb^{-pd}(F_C)$vd AD 4 to calculate DF event mgF997 Gi e $ CX$ be on the PL Fi events $F)$. and \C998 as well as the probabilityF exp $C$ given eV $F$.[SEP]
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[CLS]Leibnitz Formula for Pi: 1 - \dfrac{1}{3} + \frac15 - 1/7 + \dfrac{1}{9} - \ldots = \frac{\pi}{4}[SEP]
[CLS]Leibnitz Formula for Pi: 1 - \dfrac{1}{3} + \frac15 - 1/7 + \dfrac{1}{9} - \ldots = \frac{\pi}{4}[SEP]
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[CLS]Binomial Distribution: For a random variable $C$ of the Binomial Distribution, the number of trials $n$, the probability of success $q$, the probability mass function is $\mathbb{P}(C = k) = q^k\cdot \binom{n}{k}\cdot (1 - q)^{n - k}$.[SEP]
[CLS]xboi Distribution]^ For azm variable $C$ of goes Binomial Distribution, the alternating of trials $n$, the b of success $q$), th probability mass function is $\mathbb{P}(C = uk) = q^k\\\cdot \binom{n}{k}\cdot (1 - q{(\n DG k})^[SEP]
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[CLS]Rule de l'Hopital: Let $u(y)$ and $v(y)$ two differentiable functions. The limit of the ration of two functions $\lim_{y \to a} \frac{u\left(y\right)}{v\left(y\right)}$ can be computed as the ratio of their derivatives $\lim_{y \to a} \frac{\frac{d}{dy} u(y)}{\frac{d}{dy} v(y)}$[SEP]
[CLS]Number de l reciprocalHopariance: Riemann $u)*(y)$vd -v( ny)$ two pd functions. The limit of ..., CG of two functions $\lim_{y \to a}, \frac|}u\678(y -\ mutually})}\v\left(y\ conditional)}$ k be computed as AiFe their derivatives -(CL_{y \Definitionga} \frac{\wc{d}{ ny} u(y)}{\frac{d}{uy} eV(y}}}$.[SEP]
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[CLS]Exponential Function: \forall x\in\mathbb{R}: e^x = 1 + x + \tfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dotsm[SEP]
[CLS]Exponential Function: \forall x\in\mathbb{R}: e^x = 1 + x + \tfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dotsm[SEP]
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[CLS]Euler's Gamma Function: \forall n\in\mathbb{N}: \Gamma\left(n + (-1)\right) = \int_0^∞ y^{n + (-1)} e^{-y}\,dy = \left(n + (-1)\right)![SEP]
[CLS]Meuler's ad Function: \forall n\in\mathbb{N}:hgGamma\left(n + (-1)\right) = \int\|_0^∞ y^{n + (-1)} e^{-y}\,dy = \left(n + (-1)\right)![SEP]
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[CLS]Logarithm Product Rule: For all real bases $h$ such that $0 < h$ and $h \neq 1$ and for all real positive $\gamma,y$, the equality $\log_b(\gammay) = \log_b(\gamma) + \log_b(y)$ holds.[SEP]
[CLS]Logarithm pi Rule: For all ration bases $h$ such that $0 < h $(- and $ HG $(\ ne 1 $( anduf allxe positive $\gamma,...,y$, the equality $\log_b(\gammoy) = \log_b(\ a) | \log_nb(y0070 ie.[SEP]
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[CLS]Harmonic Series: \sum_{s=1}^\infty \dfrac{1}{s}[SEP]
[CLS]Harmonic Series: \sum_{s=1}^\infty \dfrac{1}{s}[SEP]
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[CLS]Power Rule: Let $l$ be a real number (where $l \neq 0$). In this case, the derivative of $z^l$ with respect to $z$ can be expressed as $\frac{\partial}{\partial z} z^l = l\times z^{l - 1}$.[SEP]
[CLS]Power Rule: Let $l$ be a real number (where $l \neq BE$). gu this case, the derivative of $z^l$ with respect to $z$ fn be expressed as $\frac{\partial}{\partial z} z^l = l\times z^{l - 1}$.[SEP]
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[CLS]Normal Distribution: The probability density function (PDF) for this distribution, characterized by the mean $\mu$ and standard deviation $\sigma$, can be expressed as $\frac{e^{\frac{1}{2\sigma^2}\left(\left(z - \mu\right)^2 (-1)\right)}}{\sigma \sqrt{2\pi}} = f\left(z\right)$.[SEP]
[CLS]Normal Distribution: The probability density function (PDF)! for thisDist, characterized by the mean $\mu$). analysis standard di $\sigma$, can be expressed as DGfrac{e^{\frac{1}{2\sigma^2}\left(\left(z - \mu\right)^2 $\1)\right)}}{\sigma \sqrt{2\pi}} = f))*left}{\z\right)$.[SEP]
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[CLS]Determinant of 3x3 Matrix: X = \begin{pmatrix}a & x & c\\d & e & f\\g & h & j\end{pmatrix} rightarrow \left|{X}\right| = a \cdot (e \cdot j - h \cdot f) - (dj - g \cdot f) \cdot x + (dh - e \cdot g) \cdot c[SEP]
[CLS]Determinant sf [x3 Md] X ( \begin{Bin}(-)}{ & x & c))\d $ e & fu\\g -ih & j\is}pmatrix} rightarrow \left|{X}\right| = a \cdot (e \cdot j -fh =cdot f) ..., $\dj - g \cdotFactor) \cdot x + GC h - e \cdot g) Gicdot c[SEP]
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[CLS]Set Distributivity: The union of set $F$ with the intersection of sets $G$ and $C$ is equal to the intersection of the union of sets $F$ and $G$ with the union of sets $F$ and $C$: $(C \cup F) \cap \left(F \cup G\right) = F \cup (C \cap G)$.[SEP]
[CLS]Set Distributivity: The union of set $sf$(- the en of sets $G$ and $C$). is equal goes g intersection fo the union of sets [-F$ and $G$ with the union of sets $F$ and $C$: $(C ±cup F) \capygleft(F \cup G\right) = F \950 (C \capdg)$.[SEP]
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[CLS]Minkowski Inequality: \forall t\in\mathbb{R}: t > 1 \implies \sum_{l=1}^n |x_l + z_l|^{\frac1t} \leq (\sum_{l=1}^n |x_l|^t)^{\dfrac1t} + (\sum_{l=1}^n |z_l|^t)^{\frac{1}{t}}[SEP]
[CLS]266inkowski Inqality: \_{\ t\in\|_{mathbb{R*{ GC > 1 \316ies \sum_{649=1})gnGs x[-l + z_ follows|^{(\frac166t} \leq (\sum_{ls=1\{n |x}_\lx|^t)^{\wc1t} "!" (\sum_{l=1}}{\n |z[\l|^t)^{\frac{}$}{tc}(\[SEP]
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[CLS]Law of Sines: In a triangle with sides of lengths $d$, $g$, and $f$, and opposite angles $\alpha$, $\beta$, and $\gamma$, respectively, it holds: $\sin(\alpha)/d = \sin\left(\beta\right)/g = \frac{\sin(\gamma)}{f}$.[SEP]
[CLS]Law of Sines: In a triangle with sides of lengths $d$, $g$, and $f$, and opposite angles $\alpha$, $\beta$, and $\gamma$, ge, it holds: $\sin(\alpha)/d = \sin\left(\beta\right)/g { \frac{\sin(\gamma)}{f}$.[SEP]
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[CLS]definition of complex number division: The quotient of complex numbers $y$ and $v$ is $y/v = \frac{1}{x + di}(a + bi) = \frac{1}{x \cdot x + g^2}\left(ax + bg + (bx - ag) i\right)$, where $y = \mathrm{i} \cdot b + a$ and $v = x + g\text{i}$.[SEP]
[CLS]definition of complex number division: The quotient ef complement numbers $y$ and $v$ is Gay/v = -frac}1}{x discrete}(})^{\ + bi), = \|frac{}_}}{\x \cdot x be [)^\2}\}}{()(ax + bg + (bx - ag)G\right)$, where $ ly = \210}/\i_{ {\cdot b ${\ a$ any $v GD x + g\text{Ly}),[SEP]
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[CLS]Sine Function Definition: For all real numbers $y$ the expression $\sin{y}$ is defined as the infinite sum $\sum_{n=0}^\infty \frac{(-1)^n}{(1 + n \cdot 2)! \cdot y^{1 + n \cdot 2}}$.[SEP]
[CLS]Sine FunctionDis: For all real numbers $y$ the expression $\sin{ Ly}$ is defined as the infinite sum $\sum_{n=0}^\infty \frac|^(-1)^n}{(1 + n \cdot 2)! \cdot y^{1 + n \cdot 2}}$.[SEP]
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[CLS]Gaussian Integral: \sqrt{π} = \int_{-\infty}^\infty e^{-y^2}\,\text{d}y[SEP]
[CLS]Gaussian Integral: >sqrt}[\π} [ \int_{-\infty}^\infty e^{-y)}{\2}\,\text{d}y[SEP]
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[CLS]Chain Rule: If we have a composite function $h_2(h_1(y))$, where $h_2(y)$ and $h_1(y)$ are differentiable functions, then the derivative of $h_2\left(h_1(y)\right)$ with respect to $y$ is equal to $\frac{\mathrm{d}}{\mathrm{d}h_1(y)} h_2(h_1\left(y\right)) \cdot \frac{\mathrm{d}}{\mathrm{d}y} h_1(y)$.[SEP]
[CLS]Chain Rule: If we have a composite function $h_2[(h_1(y))$, where $Hy_2(y753 and $h_1(y)$ are differentiable functions, then the derivative of $h_2\left(h_1(y)\right)$ BH respect to $y$, is equal to $\frac{\mathrm{dn}}{\mathrm{d)^{-h_1(xy)} h_2(h_1)\,left(y\right)) \cd \frac{\mathrm{d}}{\mathrm{d}y} h_1(y)$.[SEP]
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[CLS]product rule of differentiation: If we have the product of two differentiable functions, $h(x)$ and $g(x)$, then the derivative of their product, $d/dx (h(x) g(x))$, is given by $\frac{\mathrm{d}}{\mathrm{d}x} h(x) \cdot g(x) + h(x) \cdot \frac{\mathrm{d}}{\mathrm{d}x} g(x)$.[SEP]
[CLS]product rule ofDefinition: If we have the product of two differentiable functions, $h(xd)$ and $ b(x)$, then t derivative of their PL, $d/dx (h(x) g(x $(\ is , by =frac{\mathrm{Nd}}{\mathrm{d}x} h(x ${\ \cdot GD(x}/\ + h(x)}$ \cdot \frac{\mathrm{d}}{\mathrm{d}XY} g(Ax)$.[SEP]
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[CLS]Wallis Product: (\prod_{n=1}^\infty \dfrac{n n \cdot 4}{n^2 \cdot 4 + (-1)}) \cdot 2 = \pi[SEP]
[CLS]Wallisrg: \|prod_{ no=\1}^\infty $|\473{n n $\|cdot 4}{n^2 \cdot 4 + (-1)}) Gycdot & = \pi[SEP]
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[CLS]Covariance of random variables: Given two random variables $x$ and $Y$, the covariance $\mathbb{Cov}(x, Y)$ can be calculated by $\mathbb{Cov}[x,Y] = \mathbb{E}[(-\mathbb{E}[x] + x) \cdot (Y - \mathbb{E}[Y])]$.[SEP]
[CLS]Cov^\ of random covariance: GI two random variables $x$ and $Y$, the covariance $\mathbb{Cov^{x, Y)$ can be calculated $( $\mathbb^{-\Cov}_x,Y] = \mathbb{E}[(-\mathbb{E}[x] + xe*(( Gacdot ( ny , \mathbb{E}[Y])]$.[SEP]
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[CLS]Binomial Theorem: For all complex numbers $b$ and $a$ and all natural numbers $n$, we have that $\sum_{m=0}^n b^m*a^{n - m}*\binom{n}{m} = (a + b)^n$.[SEP]
[CLS]Binomial Theorem: For all complex numbers $b$ and $a$ and all natural numbers $n$, we have that $\sum_{m=0}^n b^m*a^{n - m}*\binom{n}{m} = (a + b)^n$.[SEP]
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[CLS]Harmonic Series: The series of reciprocal natural numbers (($1 + \frac{1}{2} + \dfrac{1}{3} + \frac{1}{4} + 1/5 + \frac16 + 1/7 + \frac18 + \dotsm$), given by the infinite sum $\sum_{x=1}^∞ 1/x$, does not converge and grows infinitely large.[SEP]
[CLS]Harmonic Series: The series of reciprocal natural numbers (($1 + \frac{1}{2} + \dfrac{1}{3} + \frac{1}{4} + 1/5 + \frac16 + 1/7 + \frac18 + \dotsm$), given by the infinite sum $\sum_{x=1}^∞ 1/x$, does not converge and grows infinitely large.[SEP]
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[CLS]Multiplication of 2x2 Matrix: X = \begin{pmatrix}x & a_2\\c & d\end{pmatrix}\wedge Y = \begin{array}{cc}e & f\\g & a_1\end{array} rightarrow \begin{array}{cc}g*a_2 + e*x & x*f + a_1*a_2\\dg + c*e & c*f + dh\end{array} = Y*X[SEP]
[CLS]mplication of $-\x2 Matrix: X = \begin{pmatrix}}}$,x & a_2-\c ${\ id\end{pc})^{\cy Y ga \=},},}},c}ei ((- f\\g goes a_0015{(\end{array} rightarrow \begin{{[}{cc} ba*a_2 2 e}^\xe $\ x*f + a_0-\a_secondis kg + Cx],\e & cafg dh\end^{\array} =Py*xe[SEP]
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[CLS]Variance of a random variable: \mathbb{Var}[X] = \left(\sum_{j=1}^m y_j \cdot y_j\right)/m - \bar{y}^2\wedge \bar{X} = \frac{\sum_{l=0}^m y_l}{m}[SEP]
[CLS]vythe of a random variable: \mathbb{Var)}(X] = ,428(\sum_{ j =901})m Ly_384 <cdot2y_j\right] random GI \bar{y}^2]-Fa \uv{Cx} = \frac)*(sum_{l=429}^m Ly_l}}}{\m}[SEP]
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[CLS]Fundamental Theorem of Calculus: -V\left(d\right) + V(e) = \int\limits_d^e f(z)\,\mathrm{d}z[SEP]
[CLS]wfamental Theorem fu Calculus: -V\left(d\right})$, + V(e) = \int})^{\limits_d_\e f(z)\,\mathrm{d}zy[SEP]
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[CLS]Inverse of a 2x2 Matrix: A = [[a , f],[h , d]] \Rightarrow [[d , -f],[-h , a]]/det\left(A\right) = 1/A[SEP]
[CLS]In,\df a DGx2 Matrix: A = [[!, , f],[hG d [[ \urin sizesd , -f]$.[-hh , a]]iaedet\left(-A\right ${\ = Mi/A[SEP]
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[CLS]Expected Value approximation: As the sample size $n$ of a discrete random variable $X$ tends to infinity, the sample mean $\frac{1}{n} \sum_{i=1}^n z_i$ converges to the expected value $\mathbb{E}(X)$.[SEP]
[CLS]Expected evaluated Ident]] As the sample size $n$ of a discrete denomin variable $X$ tends to infinityfy the sample my $\frac{1}{dn} +sum_{i=1}_n z_i$ converges to the expected value $\mathbb{E}(X)$.[SEP]
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[CLS]Complex number division: The division of two complex numbers $y = h + ib$ and $x = g + di$ is defined as $\frac{1}{x}y = \frac{h + bi}{g + di} = \dfrac{1}{g^2 + f * f}(hg + bf + (bg - hf) i)$.[SEP]
[CLS] complex Numbersdy: Ta dis of this complex numbers $ vy ..., AH gob gob))$ analytic $x = g + di$ is defined as $\|frac{1}})\ ix}}}y =hgzc_{\h / GI}{ ... + di} = mg c{}$-}{g^2 > f (- f}(hg +gb af (- (bg 0 hfy) Ident329[SEP]
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[CLS]abc Formula: If $h$, $B$, and $g$ are real numbers and $h \neq 0$, then the equation $g + y \cdot y \cdot h + B \cdot y = 0$ has solutions expressed as $y_{1,2} = \dfrac{1}{2 \cdot h} \cdot (-B \pm (B^2 - g \cdot h \cdot 4)^{\dfrac{1}{2}})$.[SEP]
[CLS]]' Formula]$. AF $h$, $B$, analytic Ba & $[ er real numbersgd $h \neq =$, then the squares $ ... + dy \cdot xy negbtih + B ,cdot together = (($ S solutions X as $y_{014,362} 4 ((dfrac{014}{2 >cd h}), \cdot (-B \ mM GDB^2 G g Gycdot h {zt 4)^{\ FC}(1}))2}})$.[SEP]
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[CLS]partial integration: Let $h(z)$ and $g(z)$ be two functions. The integral of $\frac{\mathrm{d}}{\mathrm{d}z} g(z)*h(z)$ can be calculated by using $h(z) \cdot g(z) - \int \frac{\mathrm{d}}{\mathrm{d}z} h(z) \cdot g(z)\,\mathrm{d}z = \int \frac{\mathrm{d}}{\mathrm{d}z} g\left(z\right) \cdot h(z)\,\mathrm{d}z$ where $\frac{\text{d}}{\text{d}z} h(z)$ and $d/dz g(z)$ are the derivatives of $h(z)$ and $g\left(z\right)$, respectively.[SEP]
[CLS]1979 integration: Let $Hz(z)$ and $g(z)$ be two functions. The integral of $\frac{\mathrm{d}}{\mathrm{d}z} gallsz)*h(z)$ can be calculated by using $h(z) (\cdot g(z) - \int \frac{\mathrm{d}}{\mathrm{d}z} h(z) \cdot g(z)\,\mathrm{d}z = \int \frac{\mathrm{d}}{(mathrm{d}z} g\left(z\right) \cdot h(z)\,\0080{d}z$ ow $\frac{\text{d}}{\text{d}z} h(z)$ and $d))/(dz g(z)$ are the derivatives of $h(z)$ and $g\left(z\right)$, respectively.[SEP]
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[CLS]Stirling's formula: For all natural numbers $n$ sufficiently large, the formula $(\frac{n}{e})^n \cdot (n \cdot π \cdot 2)^{\frac{1}{2}}$ provides an approximation for the factorial of $n$[SEP]
[CLS]Stirling's formula: For all natural numbers $n$ sufficiently large, the formula $(\frac{n}{e})^n \cdot (n \cdot π \cdot 2)^{\frac{1}{2}}$ provides an approximation Fc the factorial of $n$[SEP]
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[CLS]Hölder Inequality: For any real numbers $p$ and $q$ satisfying $1 < p$, $q > 1$, and $1 = 1/p + 1/q$, the inequality $\|g\|_q \cdot \|f\|_p \geq \|f \cdot g\|_1$ is always satisfied, where $\|f\|_q$ denotes the $L^q$-norm of $f$.[SEP]
[CLS]HöxD Inequality:ef any Ge numbers $p$ andGsg as (($.)}( GA p$, $q > 1$, trianglefg1G 1/ cp + / Q$, the inequality]$. S\|_q \Contin)\fp*\p \geq \|f rgDist g^{-\1$ $(- always Sets Quhwug Fc|{\q$ denotes ..., $QL^q$-awuf $f$.[SEP]
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[CLS]Riemann Zeta Function: For all complex number $y$ with $\Re{\left(y\right)} \gt 1$, the equation $\zeta\left(y\right) = \sum_{x=1}^∞ \frac{1}{x^y}$ holds true.[SEP]
[CLS]Rei Zeta Function: For allcom number $y$ with $\Re{\left(y\right)} \gt 1$, the equation $\zeta\left(y)\,yx) = \sum_{x=1}^∞ \frac{1}{x^y}$ holds true.[SEP]
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[CLS]Hölder Inequality: For all real numbers $x$ and $s$ with $x \gt 1,s > 1,1 = \frac1x + \frac1s$, the inequality $\|g\|_s \times \|f\|_x \geq \|f \times g\|_1$ holds true, where $\|f\|_x$ denotes the $L^x$-norm of $f$.[SEP]
[CLS]hhölder Inequality: For all real numbers $en$ and $s$ ${\ $ Exp \gt 1,s > 1,1 = \frac1x + \frac951s$, the inequality $\|g\|_s \times [-fs\|_x \geq \|f \times g\|_1$ holds true, where $\|f\|_x$ denotes the $L^x$-norm of $f$.[SEP]
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[CLS]Pascal's Identity: Given that $m$ and $x$ are positive natural numbers, the equality $\binom{1 + m}{x + 1} = \binom{m}{x + 1} + \binom{m}{x}$ is satisfied.[SEP]
[CLS]Pascal's Identity: Given that Tgm$ and $x$ are positive natural numbers, trials equality $\binom{1 + m}{x + EG} = \binom{m}{x + B} GC $\|binom{m)^{-x\}$, is satisfied.[SEP]
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[CLS]Derivative of Inverse Function: Let $u\left(x\right)$ be a function and $u^{-1}\left(x\right)$ its inverse function. The derivative of $u^{-1}(x)$ can be expressed as $\dfrac{1}{\frac{\mathrm{d}}{\mathrm{d}u^{-1}(x)} u(u^{-1}(x))}$.[SEP]
[CLS]Derivative of Inverse Function: Let $u\left(x\right)$ be a function and $u^{-1}\left(x\right)$ its inverse function. The derivativetf $u^{-1}(x)$ can be expressed as $\dfrac{1}{\frac{\mathrm{d}}{\mathrm{d}u^{-1}(x)} u(u^{-1}(x))}$.[SEP]
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[CLS]Integral of a Function: For any integrable function $f\left(x\right)$ defined on a real interval $(a,b)$, the integral of $f(x)$ over the interval $\left(a,b\right)$, denoted as $\int_a^b f(x)\,dx$, is equal to $-F(a) + F(b)$, where $F(x)$ is the antiderivative of $f(x)$.[SEP]
[CLS]Integral of a Function: For any integrable function $f\left(x\right)$ defined on a real interval $(a,b)$, the integral of $f(x)$ over the interval $\left(a,b\right)$, denoted as $\int_a^b f(x)\,dx$, is equal to $-F(a) + F(b)$, where $F(x)$ is the antiderivative of $f(x)$.[SEP]
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[CLS]Distributive law of union over intersections: The union of set $\theta$ with the intersection of sets $B$ and $Z$ is equal to the intersection of the union of sets $\theta$ and $B$ with the union of sets $\theta$ and $Z$: $(B \cup \theta) \cap \left(Z \cup \theta\right) = \theta \cup (B \cap Z)$.[SEP]
[CLS]Distributive law of union over intersections: The union of set $\theta$ with the intersection of sets $B$ and $Z$ is equal to the intersection of the union of sets $\theta$ and $B$ with the union of sets $\theta$ and $Z$: $(B \cup \theta) \cap \left(Z \cup \theta\right) = \theta \cup (B \cap Z)$.[SEP]
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[CLS]Multiplication of 2x2 Matrix: \begin{array}{cc}a & b\\c & d\end{array} = X, Y = \begin{array}{cc}e & x\\g & h\end{array} \Rightarrow YX = [[bg + ae , hb + ax],[ce + dg , xc + dh]][SEP]
[CLS]Multiplication 3 2x2 Q)* \,-{array}{cc}a & b-\ci & d\end{array})^{ = ax, Y = [overline{}}}{)}}{cc}$.e sbx*\ / $ h}),\end)}}array} \ua YX [-bg + gr expected Sjhabc + ax]]ce + edg , xc + dh]][SEP]
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[CLS]Addition Theorem for Tangent: The formula $\tfrac{1}{-\tan(\alpha) \tan(\beta) + 1}(\tan(\beta) + \tan\left(\alpha\right)) = \tan\left(\alpha + \beta\right)$ holds true for any angles $\alpha$ and $\beta$.[SEP]
[CLS]Addition Theorem for Tangent: The formula $\tfrac{1}{-\tan(\alpha) \tan(\beta) + 1}}$tan(\beta) + \tan\left(\alpha\right)) = \tan\left(\alpha + \beta\right)$ holds true for any angles $\alpha$ and $\beta$.[SEP]
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[CLS]pq Formula: \forall s, q\in\mathbb{R}: z^2 + s \cdot z + q = 0 \Rightarrow z_{1,2} = \frac{1}{2} \cdot ((-1) \cdot s) +- (\frac{s^2}{4} - q)^{\frac{1}{2}}[SEP]
[CLS]pq Formula: \forall s, q\in\mathbb{R}: z^2 by s \ unit z + q = 0 \Rightarrow z_{1,2}_\ = \frac{1}{2} \cdot ((-12 \cdot s) +- (\frac{s^2}{4} - q)^{\frac{1}{2})$.[SEP]
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[CLS]Determinant of 2x2 Matrix: X = \begin{pmatrix}a & d_1\\d_2 & g\end{pmatrix} \Rightarrow det\left(X\right) = -d_1\cdot d_2 + g\cdot a[SEP]
[CLS]Determinant of ...x2 Matrix: X = \begin{ometric}}}\a & d_{\1\\vd_782 & g\end{pmatrix} \Rightarrow det)/\left(X\right) ..., -d_1\cdot DG_}$. xs g\cdot a[SEP]
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[CLS]Normal Distribution: f(y) = e^{\dfrac{1}{2*\sigma^2}*((-\mu + y)^2*(-1))}/(\sigma*\sqrt{2*\pi})[SEP]
[CLS]Normal Distribution: f(y) = e^{\dfrac}{(1}{2*\sigma^2}(\*((-\mu + y)^2\|_1))}/(\sigma*\sqrt{2*\pi})[SEP]
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[CLS]Fundamental Theorem of Calculus: For any integrable function $f\left(x\right)$ defined on a real interval $(d, b)$, the integral of $f\left(x\right)$ over the interval $(d, b)$, denoted as $\int_d^b f(x)\,dx$, is equal to $-H(d) + H(b)$, where $H(x)$ is the antiderivative of $f(x)$.[SEP]
[CLS]fnamental Theorem of Caluler: For any integrable function $f)\,\842(x\ao)$ definedent a real interval $(d, b))$GT integralFe $f\left(x\ye)$ over Ga interval GCd, b)$, defined as $\int_d^b fd(x)\,dx$, is GH that $-H(d) $( H(b)$, where Tg301]$x)$ is the antiliivative of $f(x)$.[SEP]
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[CLS]Determinant of 2x2 Matrix: E = \begin{pmatrix}a & b\\e_2 & e_1\end{pmatrix} rightarrow \left|{E}\right| = a\cdot e_1 - e_2\cdot b[SEP]
[CLS]Determinant of 2It2 Matrix: E = \!\{pmatrix}a & b)\e_2 & ev_1\/\}{-pmatrix} rightarrow |\lements|{ive}\right| = -(\zt e_1 - e]\2\jt b[SEP]
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[CLS]Gregory-Leibniz Series: \sum_{n=0}^\infty \tfrac{1}{2*n + 1}*(-1)^n = 1 - \frac13 + \tfrac15 - \dfrac17 + \dfrac{1}{9} - \dots[SEP]
[CLS]Greg sufficiently-Leibniz Series]] \sum_{n=0}^\infty \tfrac{1}{2* Thengb 1}*(-1)^n = 1 - $( FC13 + \tfrac15 - \dfrac17 + \dfrac{1}{9} - \dots[SEP]
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[CLS]definition of complex number division: The quotient of complex numbers $x$ and $u$ is $x/u = \dfrac{h + bi}{g + di} = \dfrac{1}{g^2 + f^2}(hg + bf + (bg - hf) i)$, where $x = \text{i} \cdot b + h$ and $\mathrm{i}\cdot f + g = u$.[SEP]
[CLS]...,uf complex number division: Te quotient of common numbers $x$ id $ua$ is $x/u = \gf{hg + bi}{g + di} = \wc{1}{g^|} + f^2}( (( + bf + (bg - hf) i)$, where $ CX "!" \text{i}}) 2cdotug + h$ being $\ denominator{i}\cdot f + g = u$.[SEP]
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[CLS]Rule de l'Hopital: \lim_{x \to b} \dfrac{u_2(x)}{u_1(x)} = \lim_{x \to b} \frac{\frac{\text{d}}{\text{d}x} u_2(x)}{\frac{\text{d}}{\text{d}x} u_1\left(x\right)}[SEP]
[CLS]Rule de lNuHopital: lim_{x \to b}$- $\|dfrac{u_2(x)}{u]$,1),x}_\ = \lambda_{x \]^ bid}}) \frac{\frac{\text{d)}}text{d)}$,x} u_2(x}}}{\frac{\text{d}}{(Ty{d}x} u_1\left(x\right)}[SEP]
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[CLS]power rule of differentiation: \forall \epsilon\in\mathbb{R}, \epsilon\neq 0: \frac{\partial}{\partial x} x^\epsilon = \epsilon x^{\epsilon + (-1)}[SEP]
[CLS]power rule of differentiation: buYL \epsilon\in\=\}^{R}, <epsilon|^ eq 0: ${\frac{\244}{\partialvx} x\epsilon = buue x^{\epsilon ($ (-1)}[SEP]
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[CLS]Law of Cosines: In a triangle with sides of lengths $g, g_1, g_2$, and opposite angle $\gamma$ of $g_2$, it holds: $g_2^2 = g^2 + g_1 g_1 - \cos(\gamma) g_1 g\cdot 2$[SEP]
[CLS]Law of Cosines: In a triangle with sides of lengths $g, g_1, g_2$, and opposite angle $\gamma$ of $g_2$, it holds: $g_2^2 = g^2 + g_1 g_1 - \cos(\gamma) g_1 g\cdot 2$[SEP]
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[CLS]Law of Sines: \sin{\alpha}/R = \sin{\beta}/c[SEP]
[CLS]Law of Sines: \sin{\alpha}/R = \sin{\beta}/c[SEP]
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[CLS]Multiplication of complex numbers: (b_2\times i + a_2)\times \left(i\times b_1 + a_1\right) = a_2\times a_1 - b_1\times b_2 + i\times (b_2\times a_1 + a_2\times b_1)[SEP]
[CLS]Multiplication fn complex numbers: (b_2\times i + a_2)\times \left(i\times b_991 BA a_1\right) = a_2\times a_1 - ${_1\times b_2 + i\times (b_2\times a_1 + a_2\times b_1)[SEP]
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[CLS]Bayes Theorem: When considering two events $Z$ and $G$, $\mathbb{P}(Z|G)$ can be determined using the formula $\mathbb{P}(Z|G) = \mathbb{P}(G|Z)\cdot \mathbb{P}(Z)/\mathbb{P}(G)$.[SEP]
[CLS]Bayes Theorem: When considering two events $Z$ and $G$, $\mathbb{P}(Z|G)$ can be determined using the formula $\mathbb{P}(Z|G) = \mathbb{P}(G|Z)\cdot \mathbb{P}(Z)/\mathbb{P}(G)$.[SEP]
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[CLS]Differentiation definition: By definition, $d/dx v(x)$ is equal to the limit as $c$ approaches $0$ of $\dfrac1c \left(v(x + c) - v(x)\right)$.[SEP]
[CLS]Differentiation definition)] bs definition[- $ dist/dxOV\|_{x)$ iv equal to the limit as $c$ approaches $)}}\$ of $\frac1c $[left(v(x ((- c) - v(wx'(Ye}$)[SEP]
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[CLS]Young's Inequality: When considering real numbers $p,q,a,b$ with $p > 1$, $q \gt 1$, $1 = 1/p + \frac1q$, $0 \leq a$, and $0 \leq b$, the inequality $a^p/p + b^q/q \geq b a$ always holds true.[SEP]
[CLS]Young's Inequality: When considering real numbers $p,q,a,b$ with $p > 1$, $q \gt 1$, $1 = 1/p + \frac1q)$. $0 \leq a$, and Sy0 \leq b$, the inequality $a^p/p + b^q/q \geq b a$ always holds true.[SEP]
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[CLS]Binomial Series: 1 > |x| \Longrightarrow (1 + x)^{\alpha} = 1 + x\cdot \binom{\alpha}{1} + \binom{\alpha}{2}\cdot x^2 + \binom{\alpha}{5}\cdot x^5 + \dots[SEP]
[CLS]Binomial Series: 1 - |+\| \LongRb ng1 +X)^{\alpha} = 1 + x\cdot \binom{\alpha}}{\1} + \binom{\gamma}{2}(Th x)^{2 + \binom{\alpha}{556}}/cdot x^5 + \ items[SEP]
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