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kreasof-ai
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SPL-1M-math-formula

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[CLS]Second Binomial Formula: Considering real numbers $h$ and $g$, we can state that $(h - g) \cdot (h - g)$ is equal to $g^2 + h^2 - 2\cdot h\cdot g$.[SEP]
[CLS]Second bbfo Formula: adding real numbersgh$ and $ ($, we can state that $(rh -tg), (\ two g(h - g)$ is equal to $ (},\2 + h^2 - Ge_{\cdot h\cd g$.[SEP]
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[CLS]Gamma Function: For all complex numbers $F$ with $F\not\in\{0, -1, -2, \dots\}$, we have $\int\limits_0^\infty x\,\mathrm{d}x = \Gamma(F)$[SEP]
[CLS]Gamma Function: For aw complex ty $wf$ot $ fr\not\in\{0, -018, GB2, \ discrete\}$, we he $\int\limits]\}}}\ \|xt x\,\mathrm{ d}xy = ...Gamma}{\FS)$[SEP]
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[CLS]Integral of a Function: For any integrable function $f(\chi)$ defined on a real interval $(c, b)$, the integral of $f(\chi)$ over the interval $(c,b)$, denoted as $\int\limits_c^b f(\chi)\,\mathrm{d}\chi$, is equal to $-G(c) + G(b)$, where $G(\chi)$ is the antiderivative of $f(\chi)$.[SEP]
[CLS]Integral of a Function: For any integrable function $f(\chi)$ defined on a real interval $(c, b)$, the integral of $f(\chi)$ over the interval $(c,b)$, denoted as $\int\limits_c^b f(\chi)\,\mathrm{d}\chi$, is equal to $-G(c) + G(b)$, where $G(\chi)$ is the antiderivative of $f(\chi)$.[SEP]
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[CLS]Derivative Quotient Rule: The derivative of the product of two functions, $u$ and $v$, with respect to $y$, denoted as $\frac{d}{dy} (h(y)/v\left(y\right))$, is given by the expression $\frac{1}{v^2(y)}\cdot \left(-h(y)\cdot \frac{\mathrm{d}}{\mathrm{d}y} v(y) + v(y)\cdot \frac{\mathrm{d}}{\mathrm{d}y} h(y)\right)$.[SEP]
[CLS]Derivative Quotient Rule: The derivative of the product of two functions, $u$ and $v$, with respect to $y$, denoted as $\frac{d}{dy} (h(y)/v\left(y\right))$, is given by the expression $\frac{1}{v^2(y)}\cdot \left(-h(y)\cdot \frac{\mathrm{d}}{\mathrm{d}y} v(y) + v(y)\cdot \frac{\mathrm{d}}{\mathrm{d}y} h(y)\right)$.[SEP]
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[CLS]Law of Cosines: -b\cdot \cos(C)\cdot a\cdot 2 + a \cdot a + b \cdot b = c^2[SEP]
[CLS]Law of Cosines: -xb!(cdot \cos(C\,\cdot a\Dist ng + a \ cd a + b \cdot BE { c^2[SEP]
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[CLS]difference of squares: For all real numbers $a,f$ we have that $a^2 - f^2 = (a + f) \cdot (-f + a)$.[SEP]
[CLS]difference of sa: For all realinfty ${-,-fe$ dl ha GD $(-a^2 ( f^2 ((- gla + f) \zt (-f ( a)$.[SEP]
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[CLS]Eulers Complex Exponential Identity: If we have a complex number $y$, the expression $e^{\mathrm{i} \cdot y}$ can be represented as $\sin{y} i + \cos{y}$.[SEP]
[CLS]Eulers Complex Exponential Identity: If we have a complex number $y$, the expression "!"e^{\mathrm{i} \cdot y}$ can be represented as $\sin{y} i + gycos{y}$.[SEP]
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[CLS]Normal Distribution: f(z) = e^{\frac{1}{\sigma^2*2}((-1) (z - \mu)^2)}/(\sigma \sqrt{2\pi})[SEP]
[CLS]Normal Distribution: f( Numbers) ((- e^{\frac{1}{\sigma^2)^{\2}((-1}\ (z - ghmu)^2)})]}{\sigma \|sqrt{2\pi})[SEP]
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[CLS]Binomial Theorem: Considering real numbers $c$ and $f$, and any natural number $l$, the expression $(c + f)^l$ can be expanded using the summation notation $\sum_{x=0}^l \binom{l}{x} \cdot f^{l - x} \cdot c^x$.[SEP]
[CLS]Binomial Theorem: Considering real> $ cs$ and $nf$,dn any obtain number $l$), the expression $c + f)^l$ can be denote using the \ notation $(\sum_{x=\0}^Long \binom{l})}x} \cdot f^{l - x} =cdot c^x757[SEP]
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[CLS]Taylor Series: g(z) = \sum_{n=0}^\infty g^{(n)}(b)/n! (-b + z)^n[SEP]
[CLS]Taylor Series: g(z) = \sum_{n=0}^\infty g^{( not)}(b)/n! (-b + z)^n[SEP]
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[CLS]Law of Total Probabilities: Consider a set of $x$ mutually exclusive and exhaustive events $Y_1,\dots,Y_x$, where exactly one of these events must occur. For any event $X$, the probability $\mathbb{P}(X)$ can be expressed as $\sum_{j=1}^x P(X|Y_j) \cdot P(B_i) = P(X)$.[SEP]
[CLS]Law of Total Probabilities: Consider a se of $x$ mutually exclusive and e events $Y_1,\dots,ay_x993 where exactly one of these events must occur. For any event $X$, the probability bgmathbb{P}(X)$ can be e as $\sum_{j=1}^x P(X|Y_j) \cdot powers(bs_i) = P(X)$.[SEP]
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[CLS]Basel Series: \sum_{n=1}^\infty \tfrac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \tfrac{1}{3 \cdot 3} + \frac{1}{4^2} + \frac{1}{5 \cdot 5} + \cdots = \pi^2/6[SEP]
[CLS] absoluteel Series]- \sum_{n],[1}^\infty \tfrac{1}{ and^2} = \frac{1}{1^2})} + \ fractions{1}{552^2} + \tfrac{1}{3 \cdot 3}[\ + \ f{1}{4^2}tg \frac{01}{5 )}\cdot}5} BX \cdots +ggpi{\2/6[SEP]
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[CLS]Euler Product Formula: \sum_{p=\text{ prime}}\dfrac{1}{-p^{-h} + 1} = \zeta(h)[SEP]
[CLS]EUh ta Formula: \sum_{p=\text{ prime}}/dfrac})^{\014}{-pi^{-h}$ + 1} G \zo|h)[SEP]
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[CLS]Taylor Series: f(d) + (z - d) \cdot \frac{\mathrm{d}}{\mathrm{d}d} f\left(d\right) + \dfrac{1}{2!} \cdot f''(d) \cdot (-d + z)^2 + \tfrac{f'''(d)}{3!} \cdot (-d + z)^3 + \tfrac{f''''(d)}{4!} \cdot \left(-d + z\right)^4 + \dotsm = f(z)[SEP]
[CLS]Taylor sum: -((d) + (z - d) \cdot \frac{\mathrm|}\d}}{(798{d}d} f\left(Di\right) + {\303{1)}$.2!}ygcdot f''(d) \cdot (-d + z)^2 {\ \ cf{f''')(d)}{364}) \cdot (-d + z)^ 3 GD \tfrac{f''''(d)}{4!} \cdot \left(-d + zu\right)^}, + $-\dotsm =gf(z)))[SEP]
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[CLS]Law of Total Probabilities: For a set of $m$ mutually exclusive and exhaustive events $X_1, ..., X_m$, where exactly one of these events must occur, and for any event $A$, the probability of event $A$ is given by $\mathbb{P}(A) = \sum_{x=1}^m \mathbb{P}\left(B_i\right)\cdot \mathbb{P}(A|X_x)$.[SEP]
[CLS]Law of Total Probabilities: For & ga of $m$ \| exclusive and exhaustive events $X_1uous ..., X_m$, where exactly one of these events must occur, and for given event $A507 the probability of event $A$ is given by $\mathbb{Pas}),A) |\ \}$:_{x=1}^m \mathbb{ bp}\left(B_ li\right)\cdot $\|}^{P}(A|X_xA)$.[SEP]
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[CLS]L'Hôpital's rule: \lim_{y \to d} f(y)/g(y) = \lim_{y \to d} \frac{1}{\frac{d}{dy} g(y)} \cdot d/dy f\left(y\right)[SEP]
[CLS]L'Hôpital's rule: \lim_{y \to d} f(y)/g(y) = \lim_{y \to d} \frac{1}{\frac{d}{dy}}}{\ g(y)} \cdot d/dy f\left(y\right)[SEP]
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[CLS]Leibnitz Formula for Pi: 1 - 1/3 + \frac15 - 1/7 + \frac19 - \dotsm = \frac{1}{4} \cdot \pi[SEP]
[CLS]Leibnitz Formula for Pi:gm Ug $(\/3 + \ finite367 - -/77 + \frac19 - /mam -( \gc^{-\1}{4!} \ this \P[SEP]
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[CLS]Gamma Function: \forall z\in\mathbb{C}\\{0, -1, -2, \dots\}: \Gamma\left(z\right) = \lim_{n \to \infty} \dfrac{n^z \cdot n!}{z \cdot (z + 1) \cdot \left(z + 2\right) \cdot \dotsm \cdot \left(z + n\right)} \cdot 1[SEP]
[CLS] Gamma ff: \ l z\in\mathbb}})oc}}{\\{}/\, <1, -2, EGdots\})] \|ga-\),(z\,\496) = \lim_{n \to \infty} $\|dfrac{n^{z \xt n^{-\}})z ${ Total 2 az + 1) ,cdot \left!(zb + 2\right}{- ±cdot \ itsm \ dd &left*((z + n\right}}}\ \De measures[SEP]
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[CLS]Addition Theorem for Tangent: The formula $\tan(\alpha + \beta) = \dfrac{\tan(\alpha) + \tan\left(\beta\right)}{1 - \tan\left(\alpha\right)*\tan(\beta)}$ holds true for any angles $\alpha$ and $\beta$.[SEP]
[CLS]Addition th for Tangent_{ This formula $\tan(\alpha = \beta99 = ba793+\ tangent(\alpha},\ + \tan\left(\omega\right)}{1 - \tan\61(\alpha\com}(\\tan{\beta)} Ident true for it angles $\alpha$)vd $\beta$.[SEP]
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[CLS]Determinant of 2x2 Matrix: Let $\begin{pmatrix}x & g\\h_2 & h_1\end{pmatrix} = E$ be a matrix, then the determinant $det\left(E\right)$ can be calculated as $h_1\cdot x - h_2\cdot g = \left|{E}\right|$.[SEP]
[CLS]Determinant of Gax862 Matrix..., Let $\begin{pmatrix}x Ge g)\,\h_682 & h)*(-1\ finite{ar} (- E$ be a matrix, then trials de $det}[\}}{(E]\choose)$ can Be based as $h_1\cdot x Gi h_2\ alternate r = \left|{E}\right|$.[SEP]
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[CLS]Second Binomial Formula: (a - b)^2 = b^2 + a \times a - b \times 2 \times a[SEP]
[CLS]Second Binomial Formula: (a - b)^232 = b^2 + a \times a - b \times 2 \times a[SEP]
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[CLS]Wallis Product: \dfrac{\pi}{2} = \frac15\cdot 6\cdot 4/3\cdot 2/3\cdot \tfrac{1}{1}\cdot 2\cdot 4/5\cdot 6/7\cdot \dots[SEP]
[CLS]Wall)^{\ Product := \ sine{\pi}{2} = \frac15\cdot 6\\\cdot 4))*3\ obtained 2/3)*cdot |\tfrac{1}{081}\cdot 2},\cdot 4/5\ det616))/(667\HT \dots[SEP]
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[CLS]Law of Total Probabilities: Consider a set of mutually exclusive and exhaustive events $F_1,...,F_n$, where exactly one of these events must occur. For any event $A$, the probability $P(A)$ can be expressed as $\sum_{i=1}^n \mathbb{P}(B_i) \mathbb{P}(A|F_i) = \mathbb{P}(A)$.[SEP]
[CLS]Law of Total Probabilities: considering a startingpf mutually exclusive and exhaustive events $nf_1,...,F_n$, where exactly one of these events must occur. For given event $A $ the probability $001(|}\757 can be x as $\sum{He|=1}^n ${\mathbb{P}(B\{ony) \mathbb{P}}}$,A|fi_i) (( \^{-\{ representing}(A ft[SEP]
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[CLS]Determinant of 2x2 Matrix: If we have a matrix $\begin{pmatrix}a & b\\e & d\end{pmatrix} = A$, then $det\left(A\right) = a \cdot d - e \cdot b$.[SEP]
[CLS]Determinant of 2x2 Matrix: If we have a matrix $\begin{pmatrix}a & b\\e & d\end{pmatrix} = A$, then $det\left(A\right) = a \cdot d - e \cdot b$.[SEP]
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[CLS]definition of complex number division: a + \mathrm{i}*b = z_1, c + \mathrm{i}*x = z_2 \Rightarrow \dfrac{1}{z_2}*z_1 = \dfrac{i*(b*c - a*x) + a*c + b*x}{c^2 + x * x}[SEP]
[CLS]definition of complex number division: a + \mathrm{i}* Be = z_1,g + \mathrm{i}*x (- z_2 \Rightarrow \dfrac{1}{z_2}*z]],1 = \dfrac{i*(b*cf - a*x) + a*cc 2 b*x}{c^2 + x * x}[SEP]
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[CLS]Wallis Product: \prod_{k=1}^\infty \frac{1}{(2 \times k + (-1)) \times (2 \times k + 1)} \times (2 \times k)^2 = \frac{\pi}{2}[SEP]
[CLS]Wallis Product: \prod_{k=1}^\infty \frac{1}{(2 \times k + (-1)) \times (2 \times k + 1)} \times (2 \times k)^2 = \frac{\pi}{2}[SEP]
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[CLS]Second Binomial Formula: If $h$ and $c$ are real numbers, then the equation $(h - c)^2$ is equivalent to $h^2 - c \cdot h \cdot 2 + c^2$.[SEP]
[CLS]Second Binjm Formula := If $h)$ and $cp]$ er real numbers/ Ti the equation $(hs (- c)^)$ is EF $(- ((h}[second -gc \cdotGh Ngcdot ( + c,\2$,[SEP]
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[CLS]Normal Distribution: In the given distribution, the probability density function (PDF) is given by $f(y) = \frac{1}{\sigma\times \left(2\times \pi\right)^{1/2}}\times e^{\tfrac{(-1)\times (y - \mu)^2}{2\times \sigma^2}}$, where $\mu$ represents the mean and $\sigma$ represents the standard deviation.[SEP]
[CLS]!\ Distribution: In at neg distribution, the probability densityfu pdf) is given by $nf(Jy\| = \ rc{1}:sigma\124 $\|left(-\2\times \pi\right)^{012*((2}}\*{ e}),\frac{^{(\1}(\times $|\y - $ mM)^{2}{}})$.\times $(sigma^),}}( hw $\vm $| represents the mean andggsigma$ square the ShhedBy[SEP]
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[CLS]Addition Theorem for Tangent: \forall P,x\in\mathbb{R}: \tan(\alpha + \beta) = \frac{1}{1 - \tan(\alpha)\times \tan(\beta)}\times (\tan(\alpha) + \tan(\beta))[SEP]
[CLS]Addition H for Tangometry: \ l P,xA\in\mathbb\}R}: \tan(\ M + \ probability) = "!"frac{1}^\1 - \ iT(\alpha)\times EGtan(\beta)}\if {tan(\alpha) + \tan^\beta))[SEP]
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[CLS]binomial formula: If $d_1$ and $d_2$ are complex numbers, then the equation $(-d_2 + d_1)^2$ is equivalent to $d_1 \cdot d_1 - 2 \cdot d_2 \cdot d_1 + d_2^2$.[SEP]
[CLS]296il formula: If bagd_1$ and $d_502$ are complex numbersof being the ED $(-d_502 ( divided\|_{}$)\{2$ is Equation to $d_1b\cdot d_\1 - < \ nd d_2 \cdot d}_\1 + d_2^)))$.[SEP]
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[CLS]square of a binomial: Let $h$ and $c$ be complex numbers. In this case, $\left(h - c\right)^2$ is equal to $c * c + h^2 - h*c*2$.[SEP]
[CLS]square of Ga bp: Let $gh$ and $c$ be complex numbers. In this case, $\left(h - c\\\right)^2$ is equal to $(-c * c + HK^2 - h*c*2$.[SEP]
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[CLS]binomial formula: Considering real numbers $a$ and $h$, we can state that $\left(-h + a\right) \cdot (a + h)$ is equal to $a^2 - h^2$.[SEP]
[CLS]binomiallu[ Considering r numbers S})^{\$ and $h$), wegal state that $\left(-h gl a\utive) $(-cdot (a + h)$ If230 to Ga^18 - h]]2)$,[SEP]
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[CLS]Number e: The value of $(1 + \tfrac{1}{l})^l$ approaches the constant $e \approx 2.718$ as $l$ tends to $∞$.[SEP]
[CLS]Number EG: ty value of (-1 0 \ af{}$:)}}{\lb)}l$ approaches the constant $e ±759 /.816$Hg $l$ tends to $]\ be758[SEP]
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[CLS]Euler's Gamma Function: For all complex numbers $y$ with $y\not\in\{0, -1, -2, \dots\}$, we have $\int\limits_0^\infty W\,\mathrm{d}W = \Gamma\left(y\right)$[SEP]
[CLS]Erl's Gamma Function: For all complex numbers bay)$ with NgA\not),\in\{Vs, -1, -2, \dots\}$, we Here $\int\limits_0^\ tangent W\,\mathrm{d}W = \Gamma\left*((y\right)$[SEP]
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[CLS]Pythagorean equation: A right-angled triangle has side lengths $a$, $l$, and $c$, where $c$ is the length of the hypotenuse. Then $c^2 = a \cdot a + l \cdot l$.[SEP]
[CLS]Pythagorean equation: A right-angled triangle has side lengths $a$, $l$, and $c$, where $c$ is the length of the hypotenuse. Then $c^2 = a \cdot a + l \cdot l$.[SEP]
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[CLS]Binomial Theorem: For all real numbers $g_1$ and $g_2$, and for any natural number $x$, the expression $(g_2 + g_1)^x$ can be expanded as $\sum_{l=0}^x {x \choose l} \cdot g_1^l \cdot g_2^{-l + x}$.[SEP]
[CLS]xbomial Then: -( all real numbers $g_1$ and [ ..._2$, Md for any natural number $x$, the expression $(g_2 + g_1^{-x$ can be expanded as $\sum_{l=0}^x {x \choose l} $\|cdot g_1^l \cdotbg_2^{-l + DX}$.[SEP]
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[CLS]Logarithm Quotient Rule: For all real positive $e$ with $e \neq 1$ and for all real positive $j, z$, we have $\log_b(j/z) = \log_b(j) - \log_b(z)$.[SEP]
[CLS]Logarithm Quotient Rule: For all real positive $e$ with $e \neq 1$ and for all real positive $j, z$, we have $\log_b(j/z) = \log_b(j) - \log_b(z)$.[SEP]
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[CLS]Differentiation definition: We define $\lim_{h \to 0} \frac{1}{h} \times (u(h + x) - u(x))$ to be equal to the derivative, written $d/dx u(x)$.[SEP]
[CLS] individualsi define],[ FA define $\909}{ Dh \to 0424 \xC{1}_hy} \times (^{()/(h * Xi2 := u(x))$ to be equal to the average, written $d/ DX (()/( Rx)$.[SEP]
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[CLS]Law of Large Numbers: As the sample size $n$ of a discrete random variable $T$ tends to infinity, the sample mean $\frac{\sum_{j=1}^n y_j}{n}$ converges to the expected value $\mathbb{E}(T)$.[SEP]
[CLS]Law ef Large Numbers: As the sample size $123$ of a discrete random iv $ Ta758 tendscom integral, the ls Me $\km{\ps_{j],[1}^879 y_j}{n}$ degree toG ge value (\mathbb}\ee}(T$)[SEP]
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[CLS]Binomial Theorem: If $g$ and $\omega$ are real numbers, and $m$ is a natural number, then the expression $(\omega + g)^m$ can be expanded as the sum $\sum_{i=0}^m g^i \omega^{m - i} \binom{m}{i}$.[SEP]
[CLS]Binomial Theorem: If $g$ and $\omega$ are real numbers, and $mc$ is a natural number..., then the expression $(\omega + g)^m$ can be expanded as the sum $\sum_{i=0}^m g^i \omega^{m - i} $\|binom{m}{i}$.[SEP]
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[CLS]Midnight Formula: Let $a$, $b$, and $g$ be real numbers satisfying $a \neq 0$. In this case, the equation $g + x^2 \cdot a + b \cdot x = 0$ has solutions given by $x_{1,2} = \frac{1}{2\cdot a}\cdot (-b ± \sqrt{b \cdot b - 4\cdot g\cdot a})$.[SEP]
[CLS]Midnight Formula: Let $a$, $b$, and $g$ be real numbers satisfying $a \neq 0$. In this case, the equation $g + x^2 \cdot a + b \cdot x = 0$ has solutions given by $x_{1,2} | DGfrac{1}{2\cdot a}\cdot (-b ± \sqrt{b $|\cdot b - 4\cdot g\cdot a})$.[SEP]
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[CLS]Midnight Formula: The equation $c + x^2\cdot g + b\cdot x = 0$, where $g$, $b$, and $c$ are real numbers and $0 \neq g$, has solutions given by $x = \frac{1}{g \cdot 2} \cdot (-b \pm \sqrt{-4 \cdot c \cdot g + b^2})$.[SEP]
[CLS]Midnight Formula: The equation $c + x^2\cdot g + b\cdot x = 0$, where $g$, $b$, and $c$ are real numbers and $0 \neq g$, has solutions given by $x = \frac{1}{g \cdot 2} \cdot (-b \pm \sqrt{-4 \cdot c \cdot g + b^2})$.[SEP]
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[CLS]Geometric Series: If $s$ is a real number satisfying $|s| \lt 1$, then the infinite series $\sum_{n=0}^\infty s^n$ evaluates to $\dfrac{1}{1 - s}$.[SEP]
[CLS]Geometric Series: If $s$ is a real number satisfying $|s| \lt 1$, then the infinite series $\sum_{n=0}^\infty s^n$ evaluates to $\dfrac{1}{1 - s}$.[SEP]
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[CLS]Stirling's formula for factorials: For all natural numbers $k$ sufficiently large, the formula $\left(\frac{1}{e}\cdot k\right)^k\cdot \sqrt{k\cdot \pi\cdot 2}$ provides an approximation for $k!$[SEP]
[CLS]Stru'sLaw fr factorials''( (( Ac natural numbers $kr$xedg, Te formula $\left(\frac{1}{xe}},cdot k\ye],[k\zet {\sqrt{k\cdot \pi\cdot 2}$) provides an approximation for $ C!})$[SEP]
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[CLS]Harmonic Series: The sum of the reciprocals of positive integers from 1 to infinity ($1 + \frac12 + 1/3 + \dotsm$), denoted by $\sum_{n=1}^∞ 1/n$, diverges, indicating that it does not have a finite value.[SEP]
[CLS]Harmio Series: true Sfa the reciprocals EF id integers from [- Ta infinity {1101 + ogcf12 + 1/3 + \dotsm$), quotient Ry $\sum_{n{[}_}^�infty 1/ng807 diverges, indicating that itD not have mg finiteva.[SEP]
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[CLS]Derivative of Inverse Function: \frac{\text{d}}{\text{d}\mu} f^{-1}(\mu) = \frac{1}{\frac{\text{d}}{\text{d}f^{-1}(\mu)} f\left(f^{-1}(\mu)\right)}[SEP]
[CLS]Derivative of Inverse Function: \frac{\text{d}}{\text{d}\mu} f^{-1}/\mu) = \frac{701}{\frac{\text{d}}{\text{d}f^{-1}(\mu)} f\left( FH^{-1}(\mu)\right^{-[SEP]
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[CLS]quadratic equation formula: \forall r, q\in\mathbb{R}: z^2 + r\cdot z + q = 0 \Longrightarrow z_{1,2} = \left(\left(-1\right)\cdot r\right)/2 \pm \sqrt{\frac14\cdot r^2 - q}[SEP]
[CLS]quadratic equation formula: GB large Ry,mq $\|where \|mathbb{R}: z^}{ * Rec)\, outerza , $-\ = 0 $-\Longrightarrow z_{1verse2}gb \left(\left(-1})^{\right])cdotzg\right)/502 \pm \th{\frac14\cdot ga^)}$. - q}[SEP]
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[CLS]Binomial coefficient formula: For any natural numbers $x$ and $j$ with $j \leq x$, the value of ${x \choose j}$ is given by $\frac{x!}{j! (-j + x)!}$.[SEP]
[CLS]Binomial coefficient formula: For any natural numbers $x$ and $j$ with $j \leq x$, the value of ${x \choose j}$ is given by $\frac{x!}{j! (-j + x)!}$.[SEP]
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[CLS]Pascal's Rule: Given the conditions that $a$ and $k$ are positive natural numbers, we can deduce that $\binom{1 + a}{k + 1}$ is equivalent to ${a \choose 1 + k}$ added to ${a \choose k}$.[SEP]
[CLS]Pcal's Rule: Given the conditions length $a$ analytic $k]$ are positive natural numbers, we cb deduce that $\binom{1 + Sa}{Ke + 1}$ * equivalent to ${a \choose 1 + k}$ az to $(-a \choose k}$.[SEP]
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[CLS]Eulers Formula: If we have a complex number $z$, the expression $e^{i*z}$ can be represented as $\mathrm{i}\cdot \sin\left(z\right) + \cos\left(z\right)$.[SEP]
[CLS]iemannculusines Formula: fu we h a complex number $z$, the expression $ Ye^{i)\,\zz992 can be represented as GImathrm{i},\cdot \sin\left(z\right) + \ Sc)\,left(z\right)$.[SEP]
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[CLS]Multiplication of 2x2 Matrix: \begin{pmatrix}a & b\\x & d\end{pmatrix} = D,Y = \begin{pmatrix}U & f\\u & h\end{pmatrix} \Rightarrow D*Y = \begin{pmatrix}a*U + b*u & a*f + b*h\\U*x + dg & dh + f*x\end{pmatrix}[SEP]
[CLS]MulticulusFa 2x2 Matrix: \ You{pmatrix}a & bi\\xgg d\end)}(bt} -( nd, ty = +begin{pmatrix}ug & f\\u & he\end}^{-pmatrix} \Rightarrow D(Y = gbegin{pmatrix}a*U $[ bg*ux & a*f + b*h\\U*ax + DQg [ dh + f*x\end}_}[}[SEP]
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[CLS]Hölder Inequality: The inequality $\|f \cdot g\|_1 \leq \|f\|_{t_1} \cdot \|g\|_{t_2}$ is valid for any real numbers $t_1$ and $t_2$ with $t_1 \gt 1$, $t_2 \gt 1$, and $1/(t_1) + 1/(t_2) = 1$, where $\|f\|_{t_1}$ denotes the $L^{t_1}$-norm of $f$.[SEP]
[CLS]Hölder Inequality: The inequality $\|f \cdot g\|_1 \leq \|f\|_{t_1} \cdot \|g\|_{t_2}$ is valid for any real numbers $t_1$ and $t_2$ with $t_1 \gt 1$, $t_2 \gt 1$, and $1/(t_1) + 1/(t_2) = 1$, where $\|f\|_{t_1}$ denotes the $L^{t_1}$-norm of $f$.[SEP]
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[CLS]chain rule of differentiation: Let $f(g(x))$ be a composite function. In this case, the derivative of $f(g(x))$ is given by $\frac{d}{dg(x)} f(g(x)) \cdot \frac{d}{dx} g\left(x\right)$, where $\frac{\mathrm{d}}{\mathrm{d}g\left(x\right)} f(g(x))$ represents the derivative of the outer function with respect to its argument and $\frac{\mathrm{d}}{\mathrm{d}x} g(x)$ is the derivative of the inner function.[SEP]
[CLS]chain rule of differentiation: Let $f(g(x))$ B a composite function. In this case, the derivative of $f(g^{(x))$ is given by $\frac{d}{dg(x)} df(g(x)) \cdot \frac}^\d}{dx} g\left(x\right)$, where $\frac{\mathrm{d\}$,mathrm{d}g\left(x\right)} f(g(x))$ represents the derivative of the outer function with respect to its argument and $\frac{\mathrm{d}}{\mathrm{d}x} g))/(x)$ is the derivative of the involving function.[SEP]
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[CLS]Euler's Sum of Inverse Squares: The sum of the inverse squares of positive integers, denoted by $\sum_{n=1}^\infty \dfrac{1}{n \cdot n}$, has a finite value of $\tfrac16\cdot \pi^2$.[SEP]
[CLS]1101uler's Sum of Inverseequares: The sum of the Py sr of positive integers, denoted by $\sum_{n_{-1}^\infty \dfrac{1}{ nm "!"cdot{\n}$) has a let valuefn $\tfrac16\cdot \pi|}\2\}$,[SEP]
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[CLS]Eulers Formula: Let $x$ be a complex number. In this case, the equation $e^{\text{i}\cdot x} = \cos(x) + \text{i}\cdot \sin(x)$ holds true.[SEP]
[CLS]Eulers Formula-( Let $x957 $(- a complexExp!( Ingg ..., the equation $ers^{\text{i}\cdot x} = \cos(x) + ngtext{i}\cdot \sin(x)$ holds true.[SEP]
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[CLS]Complex number division: \forall y,v\in\mathbb{C}, y=h+g\mathrm{i}, v=\mu+d\mathrm{i}: \frac{h + g*\mathrm{i}}{\mu + f*\mathrm{i}} = \frac{1}{\mu^2 + f^2}*(h*\mu + g*f + \left(g*\mu - h*f\right)*i)[SEP]
[CLS]Complex number division: \forall y,v\in\mathbb{C}, y=h+g\mathrm{i}}$, v=\mu+d-\mathrm},\i}: \frac{h + g*\mathrm{i}}{\mu + f*\mathrm{i}} = \frac{1}{\mu^2 + f^2}*(h*\mu + g*f + \left(g*\mu - h*f\right)*i)[SEP]
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[CLS]binomial formula: For any real numbers $a$ and $g$, the expression $(a + g) \cdot (a + g)$ is equal to $a^2 + 2\cdot a\cdot g + g^2$.[SEP]
[CLS]binomial formula: For any rh numbers $a$ Do $g$, the ek $(a + gA \zt.(*(( + g)$ ( equal to $a^2 + \|\cdot AE\cdot g [ g))^}(-$.[SEP]
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[CLS]Gamma Function: \forall k\in\mathbb{N}: \Gamma\left(k + 1\right) = \int\limits_0^\infty z^k \cdot e^{-z}\,\text{d}z = k![SEP]
[CLS]Gamma Function: \forall k\in\mathbb{N}: \Gamma\left(k + 1\right) = \int\limits_0^\infty z^k \cdot e^{-z}\,\text{d}z = k![SEP]
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[CLS]Geometric Series: \sum_{i=0}^l s^i = \frac{1}{-s + 1}\left(1 - s^{1 + l}\right)[SEP]
[CLS]Geometric Series: \sum_{i=0}^l s^i = \frac{1}{-s + 1}\left(1 - s^{1 + l}\right)[SEP]
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[CLS]Gamma Function: \forall y\in\mathbb{C}\\{0, -1, -2, \dots\}: \Gamma(y) = \lim_{n \to \infty} \frac{n!*n^y}{y*(y + 1)*\left(y + 2\right)*...*\left(y + n\right)}[SEP]
[CLS]Gamma Function: \forall y},\in\mathbb{C}\\{0, -1, -902, \dots\}: \Gamma(y) = \lim_{n \to \infty} \frac{nk!*n^y}{y*(y + 1)*\left(y + 2\right)*...*\left(y +neg\right)}[SEP]
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[CLS]Multiplication of 2x2 Matrix: \begin{pmatrix}w & f\\g & m\end{pmatrix} \begin{array}{cc}x & e_1\\e_2 & d\end{array} = \begin{pmatrix}x w + g e_1 & x f + m e_1\\e_2 w + dg & f e_2 + dh\end{pmatrix}[SEP]
[CLS]Multiob of 2x662 sum: 2begin{pmatrix}})600 & ((-\g & m\\\{pmatrix} ngbegin{}})$.}{cc},sx & e_781{\e_2 & d\end{668}}) = \ conditional{pmatrix)}xdw + g e_1 su DX f + m e_{-\1\\ expected_2 w + DGg (( f ej}_\2 + dh)\end^{\ab}[SEP]
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[CLS]Sine Function Definition: \sin{z} = z - z^3/3! + z^5/5! - \dfrac{z^7}{7!} ± ...[SEP]
[CLS]Sine Function Definition: \sin{z} = z - z^3/3! + z^5/5! - \dfrac{z^7}{7!} ± ...[SEP]
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[CLS]binomial formula: For any complex numbers $f$ and $d$, the expression $(f - d)^2$ is equal to $d^2 + f \cdot f - 2\cdot f\cdot d$.[SEP]
[CLS]binomial formula: For any complex numbers $f$ and $d$, the expression $(f - d)^2$ is equal to $d^2 + f \cdot f - 2\cdot f\cdot d$.[SEP]
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[CLS]Mathematical constant e: e = \lim_{l \to \infty} \left(\tfrac1l + 1\right)^l[SEP]
[CLS]Mathematical constant e: e = \lim_{l \to \infty} \left(\tfrac1l + 1\right)^l[SEP]
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[CLS]square of a sum: (g + d) \cdot (g + d) = g^2 + 2 \cdot g \cdot d + d^2[SEP]
[CLS]square of a sumometry (g + d) ^\cdot (-g ..., d) =ug{-2 og 2 \cdot ( \cdotg + d^025[SEP]
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[CLS]Hölder Inequality: \forall p,s>1, 1/p + 1/s = 1 \forall x,z\in\mathbb{R}^k \Rightarrow \sum_{i=1}^k |x_i z_i| \leq \left(\left(\sum_{i=1}^k |x_i|^p\right)^{1/p}\right) (\sum_{i=1}^k |z_i|^s)^{\dfrac{1}{s}}[SEP]
[CLS]Hölder Inequality: \forall p,s>1, 1/p + 1/s = 1 \forall x)/(z\in\mathbb{R}^k \Rightarrow \sum_{i=1}^k |x_i z_i| \leq \left(\left(\sum_{i=1}^k |x_i|^p\right)^{1/dp}\right) (\sum_{i=1}^k -\z_i|^s)^{\dfrac{1}{s}}[SEP]
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[CLS]Covariance of random variables: The formula $E((-E(X) + X) \cdot (-E(W) + W)) = Cov(X,W)$ measures the degree to which two random variables $X$ and $W$ vary together.[SEP]
[CLS]Covariance of random variables: The formula $E((-E(X) + X) \cdot (-E(W) + W)) = Cov(X,W)$ measures the degree to which two random variables $X$ and $W$ vary together.[SEP]
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[CLS]Gaussian Integral: \int_{-∞}^∞ e^{-d^2}\,dd = \pi^{1/2}[SEP]
[CLS]Gaussian Integral: \int_{- matrix�}^Ui� e^{-zd^502}\,dd = \228^{1/2}}}\[SEP]
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[CLS]Binomial Distribution: In the Binomial Distribution, the PMF of the random variable $X$ is defined as $\mathbb{P}\left(X = k\right) = \binom{n}{k}\cdot p^k\cdot (1 - p)^{-k + n}$, where $n$ represents the number of trials, $p$ represents the probability of success, and $k$ represents the number of successful outcomes.[SEP]
[CLS]b([ Distribution: In the Binomial Distribution, the PMF of the random variable $X$ is Ke as $\mathbb{P}\}),(X = ak\right) / \ ob{n}{k}\cdot p^k\cdot (1 - Ph}(k + n}$, where $n$ represents the number of trials, $p$ represents the probability of success!, and $k$ represents the number ff sbCt.[SEP]
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[CLS]Harmonic Series: The series of reciprocal natural numbers (($1 + \tfrac{1}{2} + 1/3 + \dfrac14 + \frac{1}{5} + 1/6 + \dots$), given by the infinite sum $\sum_{l=1}^\infty 1/l$, does not converge and grows infinitely large.[SEP]
[CLS]Harmonic Series: The series of reciprocal natural numbers (($1 + \tfrac{1}{2} > 1/3 + \dfrac14 + -frac{1}{5} + se/6 + \dots$), given by the infinite sum $\sum_{l_{-1}^\infty 1/l$, doesgy converge and grows infinitely large.[SEP]
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[CLS]Bernouili Inequality: -1 \leq y,0 < \alpha \lt 1 \implies 1 \geq (y + 1)^{\alpha}[SEP]
[CLS]Bernouili Inequality: -1 \leq y,0 < \380 \lt "!" Gaimplni1 \geq (y + 1^alpha}[SEP]
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[CLS]Expected value of a random variable: The equation $\int_{-\infty}^\infty f(y)\cdot y\,\text{d}y = \mathbb{E}[V]$ represents the expected value of a continuous random variable $V$ with probability density function $f(y)$.[SEP]
[CLS]Expected value of a random variable: The equation $\int_{-\infty}^\infty f(y)\cdot y\,\text{d}y = \mathbb{E}[V]$ represents the expected value of a continuous random variable $V$ with probability density function $f(y)$.[SEP]
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[CLS]Multiplication of 2x2 Matrix: Given two 2x2 matrices $X = \begin{pmatrix}a & b\\c & \rho\end{pmatrix}$ and $\begin{array}{cc}e & f\\x & h\end{array} = Y$, their product is given by $X \cdot Y = \begin{pmatrix}x \cdot b + e \cdot a & a \cdot f + h \cdot b\\c \cdot e + dg & dh + f \cdot c\end{pmatrix}$[SEP]
[CLS]Multiplication of 2x2 Matrix: Given two 2x2 matrices $X = \begin{pmatrix}a & b\\c & \rho\end{pmatrix}$ and $\begin{)}{\}{cc}e & f\\x & h\end{array} = Y$, their product is given by $X \cdot Y = \begin{pmatrix}x \cdot b + e \cdot a & a \ td f + h \cdot b\\c \cdot e + dg & dh + f \cdot c\end{pmatrix}$[SEP]
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[CLS]Discrete Expected Value: \mathbb{E}[U] = \sum_{j=1}^m \mathbb{P}(U = p_j) \cdot p_j[SEP]
[CLS]Discrete Exppected Value],[ ogmathbb{E}[U] = \sum_{j[041}^m \mathbb{Ph}(Uig powers_j) \cdot p_j[SEP]
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[CLS]Law of Large Numbers: \lim_{l \to \infty} (\sum_{i=1}^l z_i)/l = \mathbb{E}[T][SEP]
[CLS]Law of Large>: \lim_{l \to \infty{( (\sum_{i=1}^l z_i)/l = \}[\{ed}[T][SEP]
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[CLS]Eulers Formula: Considering any complex number $Q$, we can state that $e^{\mathrm{i}*Q} = \cos{Q} + \mathrm{i}*\sin{Q}$.[SEP]
[CLS]Eulers Formula: Considering any complex number $Q$, we can state that $e^{\mathrm{i}*Q} = \cos{Q} + \mathrm{my}*\sin{Q}$.[SEP]
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[CLS]Gaussian Integral: π^{\frac{1}{2}} = \int\limits_{-\infty}^\infty e^{-z^2}\,dz[SEP]
[CLS]Gaussian Integral: π^{\frac{1}{2}}}{\ ((- \int\limits_{-\infty)^\infty e^{-886^2}\,dz[SEP]
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[CLS]Multiplication of 2x2 Matrix: Given two 2x2 matrices $X = \begin{pmatrix}a & b\\c & v\end{pmatrix}$ and $\begin{pmatrix}e & x\\i & h\end{pmatrix} = Y$, their product is given by $\begin{pmatrix}a \cdot e + i \cdot b & h \cdot b + a \cdot x\\dg + c \cdot e & x \cdot c + dh\end{pmatrix} = Y \cdot X$[SEP]
[CLS]Multiplication of 2x2 Matrix: Given two 2x2 matrices $X = \begin{pmatrix}a & b\\c & v\end{pmatrix}$ and $\begin{pmatrix}e & x\\i & h\end{pmatrix} = Y$, their product is given by $\begin{pmatrix}a \cdot e + i \cdot b & h \cdot b + a \cdot x\\dg + c \cdot e & x \cdot c + dh\end{pmatrix} = Y \cdot X$[SEP]
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[CLS]Multiplication of complex numbers: \left(b\cdot \mathrm{i} + a\right)\cdot \left(h + d\cdot \mathrm{i}\right) = a\cdot h - d\cdot b + \mathrm{i}\cdot (a\cdot d + b\cdot h)[SEP]
[CLS]Multiplication ofcom numbers: \left(b\cdot \mathrm{i} + a\right)\cdot \left(h + d\cdot \mathrm{i}\right) = a\cdot h - d\cdot b + \mathrm{i}\cdot (a\du d + b\cdot h)[SEP]
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[CLS]Complex number inverse: \forall z\in\mathbb{C}: z = a + b\mathrm{i} rightarrow \frac{1}{z} = \dfrac{1}{a \cdot a + b \cdot b}a - \dfrac{b}{a^2 + b^2} \mathrm{i}[SEP]
[CLS]Complex number inverse: \forall z\Integr\mathbb{C}: z = a + b\mathrm{i} rightarrow \frac{1}{z}, = ...,dfrac{1}{a \cdot a + b *(cdot b}a - \dfrac{b}{a^2 + b^2}[ \mathrm{i}[SEP]
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[CLS]second binomial formula: Considering complex numbers $a$ and $b$, we can state that $(a - b) \cdot (a - b)$ is equal to $b^2 + a^2 - b*a*2$.[SEP]
[CLS]second binomial formula_{ Considering complex numbers $a$ and $b$ we can state Ah $(a - b) the\cdot (a - b)$ iT equal ty $b=022 + a^2 - b*a_{-\2$).[SEP]
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[CLS]Power series expansion: The expansion $u(z) = \sum_{k=0}^\infty \frac{1}{k!} \cdot u^{(k)}(g) \cdot (z - g)^k$ expresses the function $u(z)$ as an infinite sum of its derivatives evaluated at $g$ multiplied by powers of $z - g$ divided by $k!$.[SEP]
[CLS]Power solved expansion: The expansion $ bu(zd)}\ = \sum_{ k :=0}^\ tangent \frac{1}{k}( \cdot q^{(k)}( gl) \cdot (ez - g)^kB$ Re the function $|\u(iz ${\ asgn infinite sum of its derivatives vu at GCg$ multiplied by powers from $z - g$ divided by $k})^{\$.[SEP]
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[CLS]Eulers function: \forall x\in\mathbb{R}: exp(x) = \sum_{m=0}^\infty x^m/m! = 1 + x + x^2/2! + \frac{x^3}{3!} + \dots[SEP]
[CLS]Eulers function: \forall x\in\mathbb{R}: exp(x) = \sum_{m=0}^\infty x^m/m! = 1 + x + x^2/2! + \frac{x^3}{3!} + \dots[SEP]
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[CLS]Taylor approximation: \forall y, d\in\mathbb{C}: h(y) = h(d) + \frac{d}{dd} h(d)\cdot \left(y - d\right) + \frac{h''\left(d\right)}{2!}\cdot (y - d)^2 + h^{(3)}(d)/3!\cdot (y - d)^3 + \frac{h''''(d)}{4!}\cdot (y - d)^4 + ...[SEP]
[CLS]295 Nose \for Ry, d\in\mathbb{ci}( h(y) = th(d) + \frac{d}{dd} h^{(Da)\cdot \,|<y d\right)( + \frac{h''\left( FD\right)}{second!}\cdot (y - d)^2 + h^{(3{(\d)/3!\cdot (y - d)^{3 + \frac{h''''(d)}{4!}\cdot (ya GH d)^4 + ...[SEP]
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[CLS]L'Hopital's Rule: Let $f(y)$ and $h\left(y\right)$ two differentiable functions. The limit of the ration of two functions $\lim_{y \to a} \dfrac{f(y)}{h\left(y\right)}$ can be computed as the ratio of their derivatives $\lim_{y \to a} \frac{\mathrm{d}}{\mathrm{d}y} f\left(y\right)/(\frac{\mathrm{d}}{\mathrm{d}y} h(y))$[SEP]
[CLS]L'Hop ratio's Rule: Let $f(y)$ and)$.Hg\left(yx)\,\right)$ twoxd functions. The comp of the ration of two functions $\lim_{y \to a}_ \dfrac{ fp(y}\Ha\left*y\right)}$ can be computed GB the ratio FY their derivatives $\lim_{y \to a} \ Sc{\mathrm{d}}{\mathrm{d}y} f\left)*(-zy))*204)/),\ Cf{\mathrm^{-d}{\mathrm{d{y} h(yx))$[SEP]
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[CLS]Multiplication of complex numbers: \forall y,u\in\mathbb{C}, y=e+b\mathrm{i}, u=c+d\mathrm{i}: (e + b \cdot \text{i}) \cdot (c + h \cdot \text{i}) = e \cdot c - b \cdot h + (e \cdot h + b \cdot c) \cdot \text{i}[SEP]
[CLS]Multi> of complex)>: {forall yleftiu}}(\irling\mathbb{gc)} y|=le+hb\,mathrm)}i}, u_{-c+Nd\mathrm)}{my}: (e + b egcdot \t},\i} \cdot Ng coefficients -\ Gh \cdot {\text{881}) = e Ngcdot Ac Ng ((- \cdot h \G expected \cdot h + b \ Total c) gh cd \text{i}[SEP]
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[CLS]expected value formula: The mean of a discrete random variable $X$ with possible values $x_1,\dots,x_l$ is given by $\sum_{k=1}^l x_k*\mathbb{P}(X = x_k) = \mathbb{E}(X)$.[SEP]
[CLS]expected value formula: The mean of a discrete random variable $X$ with possible values $x_1,\dots,x_l$ is given by $\sum_{k=1}^l x_k*\mathbb{P}(X = x_k) = \mathbb{E}(X)$.[SEP]
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[CLS]Complex number summation: Let $z = a + \mathrm{i} b$ and $c + \mathrm{i} g = w$. Then, $w + z = a + c + (b + g)*i$.[SEP]
[CLS]Complex number summationBy Let (\z og Ga -( \mathrm{iy}$:ug$ another$. cigb sg678{886})}\ $| ga w})$, Then, $w + Ry $\ a + cv + ...b ($ -\*{gy$.[SEP]
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[CLS]Binomial Formula: a^2 - c^2 = (-c + a) (a + c)[SEP]
[CLS]Binomial Formula: a^2 - c^2 = (-c + a) (a + c)[SEP]
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[CLS]Addition Theorem for Tangent: \forall \phi,\rho\in\mathbb{C}: \tan(\alpha + \beta) = \frac{1}{1 - \tan(\alpha)*\tan(\beta)}*(\tan(\alpha) + \tan\left(\beta\right))[SEP]
[CLS]Addition Theorem for Tangit: \forall \phi,\rho),\in\mathbb{C}: [-tan(\alpha + \beta) =Hg Cf{1}{1 - \tan(\380)*\tan(\beta)}*(\tan(\alpha) + \tan\left(\beta\right\{[SEP]
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[CLS]Young's Inequality: \forall p,q,a,b\in\mathbb{R}: p \gt 1,q \gt 1,\frac{1}{p} + 1/q = 1,a \geq 0,b \geq 0 rightarrow a*b \leq \frac{a^p}{p} + \frac{1}{q}*b^q[SEP]
[CLS]Young's Ineqality: \forallgp derivativeqöa,b\498\ :={R}: p \gt / outcomesq \gt 1,\frac{1}{p} + 1/q = 1,a \geq 0, BH \geq 0 rightarrow a*nb \leq \ci{a^p}{505)}} + {\frac{1}{q},*b^{ q[SEP]
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[CLS]Eulers Formula: This equation relates five fundamental constants ($0, 1, π, e$, and $\mathrm{i}$) through exponentiation and addition in a single compact expression: $0 = 1 + e^{i\cdot π}$.[SEP]
[CLS]Eulers Formula: This equation relates five fundamental constants ($0, 1, π, e$, and $\mathrm{i}$) through exponentiation and addition in a single compact expression: $0 = 1 + e^{i\cdot π}$.[SEP]
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[CLS]product rule of differentiation: If we have the product of two differentiable functions, $f(x)$ and $h\left(x\right)$, then the derivative of their product, $d/dx (f(x)\cdot h(x))$, is given by $\frac{d}{dx} f(x) h(x) + f(x) \frac{d}{dx} h(x)$.[SEP]
[CLS]product rule of differentiation: If we have the product ofGT differentiable functions, *fu(x)$ and $h\left(x\right)$, teGT derivative fa their product, $ dl)/(dx (f(x)\cdot h(x))$, is given by $\frac{d}{ EX} f(x) h*((x)Bg f(x) \frac|}\ dx}{ dx}^{\ h(x)$.[SEP]
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[CLS]Determinant of 2x2 Matrix: Let $\begin{pmatrix}e & b\\c & d\end{pmatrix} = X$ be a matrix, then the determinant $det\left(X\right)$ can be calculated as $det\left(X\right) = d\cdot e - b\cdot c$.[SEP]
[CLS]Determinantef |x2 Matrix: tangent $\begin{pmatrix} lex sg ba\\c & d\end{pmatrix} = X$ "!" a matrixSgt lengths determinant $Di\}),-( BX\right)$ can be calculated as $ one\left( tangent\right) { d,\cdot e - BA\ tendsC$.[SEP]
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[CLS]Inverse of a complex number: Let $g + x\mathrm{i} = y$ be a complex number. Then, $\dfrac1y = \dfrac{-\mathrm{i} x + g}{x^2 + g \cdot g}$.[SEP]
[CLS]In)] of a complex numberue Let $g + x],\ nm{i}{ = y$ be a comp number. Then, $\ ci1y = \ ec{\,\mathrm{i}}}$, Rx -( g}{wx^\2 + g \cdot g})$[SEP]
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[CLS]Poisson Distribution: For a random variable $R$ of the Poisson Distribution, the average rate $\lambda$, the probability mass function is $\mathbb{P}(R = j) = \frac{e^{-\lambda}}{j!}\cdot \lambda^j$.[SEP]
[CLS]Poui provides: For a random variable $R$xF the Poisson Distribution, T average rate $\lambda993 the probability mass function sa ngmathbb{P}(R = j) bg \frac{e^{-\li}}{j\|_{}\cdot \lambda^j$.[SEP]
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[CLS]Pascal's Rule: For any pair of positive integers $l_2$ and $l_1$, it can be observed that ${1 + l_2 \choose l_1 + 1}$ equals ${l_2 \choose 1 + l_1}$ plus $\binom{l_2}{l_1}$.[SEP]
[CLS]Pascal's Rule: For any pair of positive integers $l})^{\2$ and $l_1$, it can be observed T ${1 + l_2 \ ox l_1 + 1}}}\ equals ${l_2 \choose 1 + l_1}$ plus $\binom{Long_2}{l_}},}$.[SEP]
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[CLS]Taylor Series: The expansion $\sum_{n=0}^\infty (-d + x)^n \cdot f^{(n)}\left(d\right)/n! = f(x)$ expresses the function $f(x)$ as an infinite sum of its derivatives evaluated at $d$ multiplied by powers of $x - d$ divided by the factorial of $n$.[SEP]
[CLS]Taylor size..., at expansion $\sum)}}{\fn= approximation}^\infty (-d B 0)^n ${cdot f}}}\NV)}\left)*(-d\right)/n*( bg f(wx)$ equals he functionGf{(\XY)$ as an infinite s of $[ analysis evaluated at GBxd$ multiplied by Productfn $cx ( -$Nd by Y factorial of $[22$)[SEP]
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[CLS]Law of Cosines: In a triangle with sides of lengths $a$, $b$, and $c$, and opposite angles $A$, $B$, and $E$, respectively, it holds: $a^2 + b^2 - b*\cos{E}*a*2 = c^2, b^2 = -2*a*c*\cos{B} + a^2 + c * c, a * a = b^2 + c^2 - 2*c*\cos{A}*b$[SEP]
[CLS]Law of sfines: In a triangle with sides of lengths $a$, $b$, and $c$, being opposite angles))$A$, kg Tb$, and $E))$ respectively, it holdsations a^2 + b^2 rg b/\gc{E}|}\]*2 = c))^2, b^2 = -2*a*c\,\cos{ Bd} g a^2 + c * Cov, a * a = b^372 + c^2 -022*c*\cos{A}^\*xb$[SEP]
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[CLS]Riemann Zeta Function: \sum_{n=1}^\infty \frac{1}{n^\rho} = \zeta(\rho)[SEP]
[CLS]Riemann Zeta Function: \sum_{n=1}^\infty \frac{1}{n^\rho} = \zeta(\rho)[SEP]
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