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

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[CLS]product rule of differentiation: \frac{\mathrm{d}}{\mathrm{d}z} g(z)*f\left(z\right) + \frac{\mathrm{d}}{\mathrm{d}z} f(z)*g(z) = \frac{\mathrm{d}}{\mathrm{d}z} (f(z)*g\left(z\right))[SEP]
[CLS]product rule of differentiation: \frac{\mathrm{d}}{\mathrm{d}z} =(z)*wf\left(z\entials) + \ For{\mathrm{d}}{\mathrm{d}z} Fe(zz)*g(z) = \frac{\mathrm}\,d}}{\mathrm{d}z} (f(z)*g\left(z\495))[SEP]
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[CLS]Gregory-Leibniz Series: The series $\sum_{m=0}^\infty \frac{(-1)^m}{2 \cdot m + 1}$, which alternates between adding and subtracting fractions with odd denominators starting from 1, converges to $\dfrac{π}{4}$.[SEP]
[CLS]Gregory-Leibniz Series: The series $\sum_{m=0}^\infty \frac{(-1)^m}{2 \cdot m + 1}$, which alternates between adding and subtracting fractions with odd denominators starting from 1, converges to $\dfrac{π}{4}$.[SEP]
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[CLS]Fundamental Theorem of Calculus: For any integrable function $u(y)$ and any real interval $\left]c, b\right[$ the integral over this interval $\int_c^b u(y)\,\mathrm{d}y$ is equal to $-F(c) + F(b)$.[SEP]
[CLS]Fund1985 theory of Caluh: For any integrable function $culus(y)$ any any real interval $\left] Ac, b\oryö the integral over this interval $\int_zc^bgu(y,\mathrm{d}y$ is equalgg $-F(c) + F(xb)$[SEP]
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[CLS]Sum of complex numbers: \forall x,g,c,d\in\mathbb{R}: x + g\cdot \text{i} + c + h\cdot \text{i} = x + c + \left(g + h\right)\cdot \text{i}[SEP]
[CLS]37 of complexgamma]' ($forall xs,g,c,d\in\mathbb{R}: -\ + g\cdot \text{i} + c + h\ cd \text{i} GB x + ch + \left(g + h\right)\cdot \text{i}[SEP]
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[CLS]Addition Theorem for Sine: For any complex numbers $\alpha$ and $\beta$, the sine of the sum of two complex numbers, $\sin\left(\beta + \alpha\right)$, can be expressed as $\cos(\beta) \cdot \sin\left(\alpha\right) + \sin(\beta) \cdot \cos(\alpha)$.[SEP]
[CLS] calculate Theorem for Sine: $(- 4 complex Ty BGalpha$ and $\beta$,gy sine fn Theorem sfgf ow consider my, $\sin\left]^V + \Nu\right)$, can be expressed as ${\cos\,\beta\}$, $\ ft \sin-\}}}{\))\z{\right|$. + $-\31(\ bases) $(\te \cos(\gamma957[SEP]
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[CLS]Stirling's approximation: (\tfrac{1}{e}*m)^m*\sqrt{\pi*m*2} \approx m![SEP]
[CLS]Stirling's approximation: ($tfrac{1}{e}*m)^m*\ through{\pi],m*612} \759 m![SEP]
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[CLS]Wallis Product: The value $\tfrac{\pi}{2}$ can be obtained by this infinte product: $\prod_{k=1}^\infty \frac{(2k)^2}{(-1 + k*2) (1 + 2k)}$.[SEP]
[CLS]Wallis Product: The value $\tfrac{\pi}{2}$ can be obtained by this infinte product: $\prod_{k=1}^\infty \frac{(2k)^2}{(-1 + k*2) (1 + 2k)}$.[SEP]
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[CLS]Quadratic Formula: Considering real numbers $f$, $\rho$, and $c$, where $f \neq 0$, the equation $f\cdot x^2 + \rho\cdot x + c = 0$ can be solved using the formula $x = \frac{1}{f \cdot 2} \cdot (-\rho \pm (-f \cdot c \cdot 4 + \rho^2)^{1/2})$.[SEP]
[CLS]Quadratic Formula,081 real numbers ((f$, $\fh$, and $c$, where $f \ q 095 the equation$,f_{-\cdot xe^{-632 + \rho]) det sg +kc & 0$ can be pg using the formula $(x = \frac{1}{Factor \cdot |} \ CT GC},\rho $|lp gif ((Cd c $\|cdot 4 + \cr^2^{(\1 denoted2)}$[SEP]
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[CLS]Minkowski Inequality: The inequality $\sum_{k=1}^m |x_k + z_k|^{1/r} \leq (\sum_{k=1}^m |x_k|^r)^{\frac1r} + \left(\sum_{k=1}^m |z_k|^r\right)^{\frac1r}$ is valid for any real number $r$ and $m$-dimensional vectors $x$ and $z$.[SEP]
[CLS]Minkowski Inequality: The inequality $\sum_{k=1}^m |x_k + z_ CG|^{1101/ru} \leq (\sum_{k=1}^m |x_k(\r)^{\frac1r} + \778(\sum_{k=1}^m |z_k|^r\right)^{\frac0015r}$ is valid for any real number $r$ and $m$-dimensional vectors $x$ and $z$.[SEP]
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[CLS]Sine Function Definition: By utilizing the infinite series $\sum_{n=0}^\infty x^{n*2 + 1} \frac{1}{(2n + 1)!}(-1)^n$, we can define $\sin(x)$ for all real numbers $x$.[SEP]
[CLS]en determined Functionnf:bguler length If series $\sum_{n=0}^\HT DX^{ natural*}(- + 1})\ \frac{1)}}{2n + 1)!}(-}}$,)^nM$, we case define $\366|\x)$ f all real numbers $xC0070[SEP]
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[CLS]Set Distributivity: The intersection distributes over the union, i.e. for sets $x$, $Z$ and $D$, the equation $(D \cap x) \cup (Z \cap x) = x \cap (D \cup Z)$ holds true.[SEP]
[CLS]Set Distributivity: The intersection distributes over the union, size.e. for sets $ Ax$, $Z$ and $D$, the equation $(D \cap x) \cup EGZ \cap x) = x \cap (D \cup Z)$ holds true.[SEP]
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[CLS]Power Rule: For all real numbers $n$ (excluding $n = 0$), the derivative of $r^n$ with respect to $r$ is given by $n \cdot r^{n - 1} = \frac{\partial}{\partial r} r^n$.[SEP]
[CLS]Power Rule: For all real numbers $n$ (excluding $n = 0$), the derivative of $r^n$ with respect to $r$ is given by $n \cdot r^{n - 1} = \frac{\partial}{\partial r} r^n$.[SEP]
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[CLS]Determinant of 3x3 Matrix: -b d j + x e j + g f b + c d h - e g c - x f h = det\left(\begin{pmatrix}x & b & c\\d & e & f\\g & h & j\end{pmatrix}\right)[SEP]
[CLS]Determinant of -x3 Matrix: -b d j + x e j bi (( f b + c d h { e g c - x f h Gh det\left(\}\,\{pmatrix}x & b & c\\ D &ges & f\\g {\ h & j\}$,end{pmatrix}\495)[SEP]
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[CLS]Bayes' Theorem: The formula $\mathbb{P}(A|B) = \dfrac{\mathbb{P}(B|A)}{\mathbb{P}(B)}\mathbb{P}\left(A\right)$ provides a way to calculate the conditional probability of event $A$ given event $B$ based on the probabilities of events $A$ and $B$ as well as the probability of event $B$ given event $A$.[SEP]
[CLS]Bayes' Theorem: The formula $\mathbb{P}(A|B) = \dfrac{\mathbb{P}(B|A)}{\mathbb{P}(B)}\mathbb{P}\left(A\right)$ provides a way to calculate the conditional probability of event $A$ given event $B$ based on the probabilities of events $A$ and $B$ as well as the probability of event $B$ given event $A$.[SEP]
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[CLS]Binomial Formula: Considering real numbers $a$ and $h$, we can state that $(h + a)^2$ is equal to $h^2 + a^2 + 2 a h$.[SEP]
[CLS]Binomial Formula: Considering real> $a$ and $(-h$, we can state that $(h + a)^2$ is q to $h^2 + a^2 + 2 a h$.[SEP]
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[CLS]Eulers Formula: If we have a complex number $z$, the expression $e^{i*z}$ can be represented as $i \cdot \sin{z} + \cos{z}$.[SEP]
[CLS] Yeulers Formula: is we nh a complex number $z$, the expression $e^{ink*z}$ can be represented as $i \cdot \sin{z} + \cos)}(zy}}{[SEP]
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[CLS]Normal Distribution: The probability density function of a distribution with mean $\mu$ and standard deviation $\sigma$ is defined as $\frac{1}{\sigma \cdot \sqrt{2 \cdot \pi}} \cdot e^{\dfrac{(x - \mu)^2 \cdot (-1)}{2 \cdot \sigma^2}} = f(x)$.[SEP]
[CLS]448 Distribution: The probability density function of a distribution with mean $\ i$ and standard deviation $\sigma$ is defined as 0frac{1}{\sigma \cdot \sqrt{2 \cdot \pi}} \ denotes EA})\dfrac{(x - \mu)^2 -cdot (-1)}{2 \cdot \Si^2}} = f(x)$.[SEP]
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[CLS]Complex number inverse: Let $y = \mathrm{i}\cdot d_2 + d_1$ be a complex number. Then, $1/y = \frac{d_1 - \mathrm{i} \cdot d_2}{d_1^2 + d_2 \cdot d_2}$.[SEP]
[CLS]284Jy FR: Let ...oy \| \mathrm{i}}$cdot d_2 GD d_1$ be sy complex number. Then, $1/y = \frac}^{d_{\1 - \2100}^{zi)}$, EGcdot d))*2}{d_1101^702 = d_2 \bt d*(--}$.[SEP]
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[CLS]Rule de l'Hopital: Let $u(x)$ and $g(x)$ two differentiable functions. The limit of the ration of two functions $\lim_{x \to a} u(x)/g(x)$ can be computed as the ratio of their derivatives $\lim_{x \to a} \frac{\text{d}}{\text{d}x} u\left(x\right)/(\frac{\text{d}}{\text{d}x} g\left(x\right))$[SEP]
[CLS]Rule delg'Hopital: Let $u(bx)$ and $g(x)$ two degree functions. The limit of the ration of two functions $\lim!(x \to a} u(nx)/g(x)$ can be computed as the ratio of their derivatives \lim_{x \to a} \frac{\text{d}}{\text{d}x} u\left(x\right)/(\frac{\T{d}}{\text{d}x} Deg\left(xy\right))$[SEP]
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[CLS]third binomial formula: For any real numbers $x$ and $d$, the expression $(d + x)\cdot (-d + x)$ is equal to $x^2 - d^2$.[SEP]
[CLS]third binomial formula]- For any ration numbers bu calculated957 gi GDd)$, the EX sf dm + x)\cdot (-d + x)$ is equal - $x^2 - d^2$.[SEP]
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[CLS]Law of Large Numbers: The sample mean $(\sum_{j=1}^m t_j)/m$ of a discrete random variable $B$ with sample size $m$ converges to the expected value $\mathbb{E}[B]$ as $m$ approaches infinity.[SEP]
[CLS]Law of Large Numbers: The sample Me Mgsum)*j=1}^m t]$.j)/m$sf a discrete random variable $eb752 with sample su $m$ converges to theSe value $\mathbb{E}[B]$ G $m$ Iz infinity.[SEP]
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[CLS]square of a difference: For all complex numbers $a$ and $b$, the expression $(-b + a)^2$ is equal to $b \cdot b + a^2 - a\cdot b\cdot 2$.[SEP]
[CLS]square of a di: For all complex numbers $a$ and $b$, the expression |b + a)^2$gi equal tangent $b \cdot b + a^2 - a(-\cdot b\cdot 2$.[SEP]
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[CLS]definition of complex number division: \frac{c_1 + b_1*\text{i}}{c_2 + b_2*\text{i}} = \frac{1}{c_2 * c_2 + b_2^2}*(c_1*c_2 + b_1*b_2 + i*\left(-b_2*c_1 + c_2*b_1\right))[SEP]
[CLS]definition of complex number division: \frac{c_1 + b_1*\text{i}}{c_2 + b_2*\text{i}} = \frac{1}{c_2 * c_2 + b_2^2}*(c_1*c_2 + b_1*b_2 + i*\left(-b_2*c_1 + c_2*b_1\right))[SEP]
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[CLS]Poisson Distribution: The probability mass function of a random variable $B$ following this distribution is expressed as $P(B = j) = \frac{\lambda^j*e^{-\lambda}}{j!}$, where $\lambda$ represents the average rate and $j$ represents the number of events.[SEP]
[CLS]Poisson Distribution: The probability s Fu on ... random variable $bf$ following this property a Equation Ug GP))B = ba) = GAfrac{\ Matrixö26)))e^{-\104_{-j\|_}$, where BYlambda$ represents tangent average rate and $j$ represents the alternate of EX.[SEP]
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[CLS]Multiplication of 2x2 Matrix: \begin{pmatrix}a & x\\c & d\end{pmatrix}*\begin{pmatrix}\chi & z\\g & h\end{pmatrix} = \begin{pmatrix}\chi*a + g*x & a*z + h*x\\\chi*c + dg & dh + c*z\end{pmatrix}[SEP]
[CLS]Multiplication of 2x2 Matrix: \begin{pmatrix}a & x\\c & d\end{pmatrix}*\begin{pmatrix}\chi & z\\g & h\end{pmatrix} = \begin{pmatrix}\chi*a + g*x & a*z + h*x\\\chi*c + dg & dh + c*z\end{pmatrix}[SEP]
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[CLS]e function: 1 + \frac{y}{1!} + y^2/2! + \frac{y^3}{3!} + y^4/4! + \frac{y^5}{5!} + ... = e^y[SEP]
[CLS]eFI: 1 (- \tfrac{y}{081!} + y^2/472! (\ {\frac{Jy]3}{3})} * +^4/4! + Gifrac{y)^{\5}{5}$: + =ge^y[SEP]
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[CLS]Logarithm Quotient Rule: For all real bases $a$ such that $a > 0$ and $1 \neq a$ and for all real positive $j,y$, the equality $\log_b(j/y) = \log_b(j) - \log_b(y)$ holds.[SEP]
[CLS] EAarch Quotient Rule: FClg real ab $!,$ sets that $a > 0$ and rg1 \ Exp a$ triangle for all real positivegbj,y$, the * ( Ng_b(j/varphi) = $(log_vb(jA - kg ng\|_{b(y)$ holds.[SEP]
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[CLS]Complex number inverse: Let $i \cdot b + c = z$ be a complex number. Then, $\frac{1}{z} = -i\cdot \frac{b}{c^2 + b^2} + \dfrac{c}{c^2 + b^2}$.[SEP]
[CLS]Complex number inverse: Let $i \cdot b + c = z87 be acom number. Thenex $\frac{1}{z} = -i}/\cdot \frac{b}{c^2 + b^2} + \dfrac{c}{c^2 + b^2}$.[SEP]
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[CLS]Law of Cosines: In a triangle with sides of lengths $d$, $h$, and $x$, and opposite angles $Z$, $A_2$, and $A_1$, respectively, it holds: $x^2 = -\cos{A_1} \cdot h \cdot d \cdot 2 + d^2 + h^2,h \cdot h = d^2 + x^2 - x \cdot \cos{A_2} \cdot d \cdot 2,h^2 + x^2 - h \cdot x \cdot \cos{Z} \cdot 2 = d^2$[SEP]
[CLS]ls of Cosinesö In ai triangle with sides of lengths d ${\ ...h$, and $x$, Integr opposite angles $Z$, $A_2$, division $A_1$, respectively, it holds: $(-x^2 = -(cos{}}{\_1} \cdot h \cdot d \cdot 2 + d^442 + h^2),h \cdot h = d^2 +vx^2 - x \cdot \ Cos{A_2)}{ \cdot d \cdot (-,h^{(2 gob x^2 - h bgcdot x \cdot \ Sc{Z} \cdot 2 = d^2$[SEP]
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[CLS]Inverse of a 2x2 Matrix: [[a , g],[c , d]] = C \Longrightarrow \begin{pmatrix}d & -g\\-c & a\end{pmatrix}/det\left(C\right) = 1/C[SEP]
[CLS]Inverse of a 2x2 Matrix: [[a , g],[c , d]] = C \Longrightarrow \begin{pmatrix}d & -g\\-c & a\end{pmatrix}/det\left(C\right) = 1/C[SEP]
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[CLS]Multiplication of 2x2 Matrix: A = \begin{pmatrix}a & b\\c & d\end{pmatrix},B = \begin{pmatrix}e & x\\g & h\end{pmatrix} rightarrow \begin{array}{cc}b*g + a*e & h*b + a*x\\dg + c*e & x*c + dh\end{array} = B*A[SEP]
[CLS]{[plication of 2x), Matrix: A = $(\begin{pmatrix})], & b)}{\c $[ d\end{ natural},B = ngbegin{ entire)}}e & x\\\Gs & h\}$,end{pmatrix} rightarrow \begin{array}{ BC)}$,b*g + a)*(-e & he*b + a*x]'dg ... cc}{\e & x*c +gh\op{array} = B*A[SEP]
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[CLS]Taylor Series: v(g) + \left(y - g\right)\cdot \frac{\mathrm{d}}{\mathrm{d}g} v(g) + (y - g)^2\cdot \frac{1}{2!}\cdot v''\left(g\right) + \left(y - g\right) \cdot \left(-g + y\right)^2\cdot v'''(g)/3! + v^{(4)}(g)/4!\cdot (-g + y)^4 + \dots = v(y)[SEP]
[CLS]Taylor size: ve(g)*(- + (\left(y - g\right)\cdot \frac{\mathrm{d}}{\mathrm)}}{\d}g} v(g) + (y - g)^2\cdot \sigma{1}{882!}\cdotvare''\left!(g\right) * \left(y - g\right) \cdot *(-798})^{\g + y\ao)^2\cdot v'''(g)/3! + v_{4)}(g)/4!\cdot {\g + y)^4 + \dots = v([zy-[SEP]
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[CLS]Power series expansion: u(y) = u\left(c\right) + (-c + y) \cdot \frac{d}{dc} u(c) + (-c + y)^2 \cdot \dfrac{u''\left(c\right)}{2!} + \frac{u'''(c)}{3!} \cdot (y - c) \cdot (y - c) \cdot (y - c) + ...[SEP]
[CLS]eV series expansion: u*(- like) = u\left(1955\475|\ + (-c + y) \cdot \frac{d}{dc} u),c) + (-Ce + y)^2 \cdot $|\dfrac{u''\857(755\right)}{2!} + \frac{u'''(c)}{3!} \cdot (y - c) \cdot (y - c)* \ytic{(y - c) + ...[SEP]
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[CLS]Binomial Distribution: The probability mass function (PMF) of a random variable $X$ following this distribution is expressed as $\operatorname{P}(X = k) = s^k\cdot \binom{n}{k}\cdot (-s + 1)^{n - k}$, where $n$ represents the number of trials, $s$ represents the probability of success, and $k$ represents the number of successful outcomes.[SEP]
[CLS]Binomial Distribution: The probability mass function (PMF) of a random variable $X$ following this distribution is expressed as $\operatorname{ comp}(lx = k) = s^k\cdot \binom{n}{k}\cdot (-s + 1)^{n - k}$, where $n$ represents the number of trials, $'s$ represents the probability of success, and $k$ represents the number of successful outcomes.[SEP]
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[CLS]Multiplication of 2x2 Matrix: X = \begin{array}{cc}a & b\\N & d\end{array}\wedge Y = \begin{pmatrix}x & f\\g & h\end{pmatrix} \Rightarrow X\cdot Y = \begin{pmatrix}b\cdot g + a\cdot x & f\cdot a + b\cdot h\\dg + N\cdot x & N\cdot f + dh\end{pmatrix}[SEP]
[CLS]Multiplication o ((x2 Matrix: X { +begin{array}\,cc})*( & b\\N , d\/\})array}\wedgevy (- ${begin)}\pmatrix}xf ... f\\g / h\}fi}}}{pmatrix}[ \r X\cdot Y = \-{pmatrix}b,\cdot g + Gi\cdotsx & f*(-cdot Sz + b\cdot nh\\dg "!" N\ taking x &DN\cdot f + dh\+{pmatrix}[SEP]
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[CLS]Power Rule: Let $k$ be a real number (where $k \neq 0$). In this case, the derivative of $z^k$ with respect to $z$ can be expressed as $\frac{\partial}{\partial z} z^k = k \cdot z^{k - 1}$.[SEP]
[CLS]Power Rule: Let $k$ be a real number (where $k \neq 0$). In this case, the derivative of $z^k$ with respect to $z$ can be expressed as $\frac{\partial}{\partial z} z^k = k \cdot z^{k - 1}$.[SEP]
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[CLS]Derivative Quotient Rule: Considering the product of two differentiable functions, $u(z)$ and $v(z)$, the derivative of $\frac{1}{v(z)} \cdot u(z)$ with respect to $z$ is equal to $\frac{1}{v^2\left(z\right)} \cdot (d/dz u(z) \cdot v(z) - d/dz v(z) \cdot u(z))$.[SEP]
[CLS]Derivativepqotient R: Considering the product of two differentiable functions, gl cu(z)$ nd $v(z)$, Py derivative of $\ rc{176}{v|{\z)} \cdot u(z)$ with respect to $z$ ix ED to ((zc{1}{v^2\left(z\right)} \cdot (-d/dz u(z) \cdot v(z) -( d/ DoOV(zm) \cdot u(z))$.[SEP]
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[CLS]Addition Theorem for Tangent: For any complex numbers $\alpha$ and $\beta$, the expression $\tan(\alpha + \beta)$ can be written as $\frac{\tan(\beta) + \tan(\alpha)}{-\tan(\alpha)\cdot \tan(\beta) + 1}$.[SEP]
[CLS]Addition Theorem for TangTh: ffGy complex numbers |\Ax))$ then $\beta$, the EF $(tan(\alpha + \beta)$ can be written as $\frac{\tan!\beta) + \tan(\alpha)}{-\tan,\uler]-cdot (\tan(\beta) + Se}$.[SEP]
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[CLS]Complex number addition: \forall g,x,e,d\in\mathbb{R}: y_1 = g + x*i,y_2 = e + \gamma*i \Longrightarrow y_1 + y_2 = g + e + (x + \gamma)*i[SEP]
[CLS]Complex number addition: \forall g,x,te,d\in\mathbb{R}})\ y_1 =eg + x*i,y_2 goes e + \gamma],i $|\Longrightarrow y_018 + y_2 = g + e BG syx + \gamma)*i[SEP]
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[CLS]Eulers e function: The limit as $k$ approaches infinity of the expression $\left(\frac{1}{k} \cdot z + 1\right)^k$ converges to the exponential function $e^z$ for any complex number $z$.[SEP]
[CLS]Eulers e function: The limit as $k$ approaches infinity of the expression $\left(\frac{1}{k} \cdot z + 1\right)^k$ converges to the exponential function $e^z$ for any complex number $z$.[SEP]
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[CLS]Law of Total Probabilities: In the case of a set of mutually exclusive and exhaustive events $B_1, \dots, B_n$, where exactly one of these events must occur, and for any event $A$, the probability $P\left(A\right)$ can be computed as $\mathbb{P}\left(A\right) = \sum_{x=1}^n \mathbb{P}(B_i)\cdot \mathbb{P}(A|B_x)$.[SEP]
[CLS]Law of Total Probabilities: In the case of a set of mutually exclusive and exhaustive events $B_1, \dots, B_n$, where exactly one of these events must occur, and for any event $A$, the probability $P\left(A\right)$ can be computed as $\mathbb{P}\left(A\right) = \sum_{x=1}^n \mathbb{P}(B_i)\cdot \mathbb{P}(A|B_x)$.[SEP]
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[CLS]Basel Problem: \sum_{B=1}^\infty \dfrac{1}{B^2} = \frac{1}{1^2} + \frac{1}{2 \cdot 2} + \frac{1}{3^2} + \dots = \frac16 \cdot \pi^2[SEP]
[CLS] IBel Problem: (\sum_{Bu*((}}/})^{\infty \dfrac_{\1}^\B^2)}}{ ((- {\frac{1}{841^2} + \| AC*{01}{702 \cdot)552}π \xc_{-\}},}{3|^}$)}/ + \ respectively = {\frac)}) BAcdot \pi^(}(-[SEP]
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[CLS]Law of Large Numbers: The sample mean $(\sum_{j=1}^l x_j)/l$ of a discrete random variable $z$ with sample size $l$ converges to the expected value $\mathbb{E}[z]$ as $l$ approaches infinity.[SEP]
[CLS]Law of Large rb{[ ty sample meangbsum_{j=1}^l x_j)/l$), of a discrete random variable $z)$, with sample sy $lu$ converges to Ga expected value $\mathbb}E}[iz)$ CG $(- absolute$ approaches infinity i[SEP]
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[CLS]Stirling's formula: For large values of $n$, the approximation $n! \approx (\frac1e \times n)^n \times \sqrt{2 \times \pi \times n}$ provides an estimation of $n!$ by utilizing the mathematical constants $\pi$ and $e$.[SEP]
[CLS]Stirling's formula: FR large valuessf $nM$, Pe multiplication $nc! $approx (\zc1e \24 n)^n \times \ standard{2 \times |\pi \20 n}$ provides an estimation of $n!$ by utilizing the mathematical CT $\pi$ and $(leq$.[SEP]
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[CLS]Pythagorean Theorem: A right-angled triangle has side lengths $g_2, b, g_1$ where $g_1$ is the length of the hypotenuse. Then $g_2 \cdot g_2 + b^2$ is equal to $g_1^2$.[SEP]
[CLS]Pythag038 Theorem: A Re-339 triangle successful side ThGg_2, ...,ug_991$ where $g_1$ is together Tb of GH hypotenItentials Then {|{\25 \cdot g_342 + be^-$ is edges to $g_ 1^2$.[SEP]
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[CLS]Law of Total Probabilities: \sum_{k=1}^x \operatorname{P}(Z|B_k)\cdot \operatorname{P}(B_i) = \operatorname{P}(Z)[SEP]
[CLS]Law of Total Probabilities: \sum_{k=1}^com \operatorname {\P}(Z|} kb_ CX)\cdot \})^{{P}( bl_{-\i})$. = \operatorname{P}(Z)[SEP]
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[CLS]Multiplication of complex numbers: \forall a,k,f,d\in\mathbb{R}: \left(a + k \mathrm{i}\right) (f + x \mathrm{i}) = a f - k x + (a x + k f) \mathrm{i}[SEP]
[CLS]645plication of complex numbers: \forall ak,ke,fdud\291\mathbb{R}: \left\|a + k \mathrm{i}})right) (f ((- x \mathrm{i}) = a f - chvx + (asx + k f) \mathrm{ Ya}[SEP]
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[CLS]Number e: e = \frac{1}{0!} + \frac{1}{1!} + 1/2! + 1/3! + \frac{1}{4!} + 1/5! + \cdots[SEP]
[CLS]Number e: efg \frac{1}{0!} + \frac{}$.}{1!} + 1a2! + 1/3! + \frac{1}{4!} + 1/5))) + \ ct[SEP]
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[CLS]first binomial formula: For all complex numbers $a,d$ we have that $d^2 + a \cdot a + 2 \cdot a \cdot d = (a + d)^2$.[SEP]
[CLS] Mb binomial formula( For allcom numbers $ between,d$ we have that $ Do))^752 + a -\cdot _{- 0 2 \ cd a \ nt d = >a + odd)^632$.[SEP]
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[CLS]Addition Theorem for Tangent: \forall z,\delta\in\mathbb{C}: \tan(\alpha + \beta) = \frac{\tan\left(\alpha\right) + \tan(\beta)}{1 - \tan\left(\alpha\right)*\tan(\beta)}[SEP]
[CLS] consists Theorem for Tangent: \forall z,\delta_{\in\mathbb}(\C})}\ gytan(\alpha + \beta) = \frac{\tan\left{(\alpha\right) + \tan(\beta)}{1 - \tan)*(left(\alpha\right)*\td(\beta|}\[SEP]
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[CLS]Eulers Formula: This equation relates five fundamental mathematical constants ($e,\pi,j,0$, and $1$) in a remarkably concise expression: $1 + e^{\pi \cdot i} = 0$.[SEP]
[CLS]Eulers Formula: This equation relates five fundamental mathematical constants ($e,\pi,j,0$, and $1$) in a remarkably concise expression: $1 + e^{\pi \cdot i} = 0$.[SEP]
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[CLS]Leibnitz Formula for Pi: When adding up the terms of the series $\sum_{n=0}^\infty \frac{(-1)^n}{n \cdot 2 + 1}$, the sum converges to $\frac14 \times \pi$.[SEP]
[CLS]Leibnitz Formula for Pi: When adding up these terms of the series $\sum_{n=0}^\infty \frac{(-1)^n}{n \cdot 2 + 1}$, the sum converges to $\frac14 \times \pi$.[SEP]
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[CLS]Chain Rule: Let $v(g\left(x\right))$ be a composite function. In this case, the derivative of $v\left(g(x)\right)$ is given by $d/dx g(x)*\frac{d}{dg(x)} v\left(g(x)\right)$, where $\frac{\mathrm{d}}{\mathrm{d}g(x)} v\left(g\left(x\right)\right)$ represents the derivative of the outer function with respect to its argument and $\frac{\text{d}}{\text{d}x} g(x)$ is the derivative of the inner function.[SEP]
[CLS]Chain Rule: Let $v(g\left)(x\right))$ be a composite functionS In this case, the derivative of $138\left(g(x_{\right)$ is never bg $d/dx g(wx*\frac{d}{dg(xy)} v\left(g(x)\right))$, where $\frac{\mathrm}/\dd}}{\mathrm)}{d}g(x)}GV\74(g}\,\left(x\right)\right)$ representsg derivative of Total outer function with respect to its argument and $\frac{\text}{- addition}}{\text{d}x} g(x)$ is the derivative of the inner function.[SEP]
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[CLS]Determinant of 3x3 Matrix: \left|{[[f_1 , x_1 , c_1],[f_2 , x_2 , c_2],[f_3 , x_3 , c_3]]}\right| = -c_3 \cdot f_2 \cdot x_1 + c_3 \cdot f_1 \cdot x_2 + x_1 \cdot c_2 \cdot f_3 + x_3 \cdot f_2 \cdot c_1 - f_3 \cdot x_2 \cdot c_1 - f_1 \cdot x_3 \cdot c_2[SEP]
[CLS]671km Ltcf363x3 Matrix: \}}}{|_{-^(f{(1 Tg x_1 eg Ch_231],[ft\|_{2 , x_342 , c_2]$.f_3 ,X_903 , ca_\603]]}$ye| = - cf_3 \|cdot f_2 \cdot x=\1 $(-c_3 \cdot f_1 \cdot x_2 + x_1 ogcdot c\|_))) $(-De f_3 > x_3 \cdot f_{ \ outcomes CG_1 og Fa_3 \cdot ex_342 \ldots c_{-1 - f[\1 \cdot x*(-3igCt c_2[SEP]
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[CLS]Complex number inverse: \forall x\in\mathbb{C}: x = c + h*\text{i} \Rightarrow \frac{1}{x} = \frac{1}{c * c + h * h}*c - \frac{h}{c^2 + h^2}*\text{i}[SEP]
[CLS]Complex number inverse: ((-forall x\in\mathbb{ CB}: x = Ac + h*\text}^{\i}}, \Rightarrow \frac{1}{x)} = \frac{1}{c * c + h * h}}^\c - \frac{h}{c^2 -( h^502}*\):{i}[SEP]
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[CLS]Binomial Theorem: If $c$ and $g$ are real numbers, and $n$ is a natural number, then the expression $\left(c + g\right)^n$ can be expanded as the sum $\sum_{k=0}^n \binom{n}{k} g^k c^{n - k}$.[SEP]
[CLS]Binomial Theorem| If $c$ and GB :=$ are real numbers,..., and $n$ "!" aiN number, then the expression $\left(c + sg\right=\n$ can be expanded up the sum $\sum_{ Ki|=0}/n BAbinom{n}{ ek} g^k c^{zn -g}$.[SEP]
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[CLS]partial integration: -\int d/dz h(z)\cdot u\left(z\right)\,dz + h(z)\cdot u(z) = \int h(z)\cdot d/dz u(z)\,dz[SEP]
[CLS]142 integration: -\Si DG))/dz h(z)\cdot u\left(z\right)\,dz +gh_{\z)\cdot u(z) = \int h(z)\cdot d/dz u(z)\,dz[SEP]
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[CLS]Normal Distribution: In the Normal Distribution, the probability density function with mean $\mu$ and standard deviation $\sigma$ is represented by $\frac{1}{\sigma \sqrt{2\pi}}e^{\frac{1}{\sigma * \sigma*2}((x - \mu)^2 (-1))} = f(x)$.[SEP]
[CLS]Normal Distribution| In the Normal)>verse the probability density function with mean $\mu$ and Da deviation Bysigma$ / represented GB $\frac)}{1}{\sigma \sqrt{2\pi}}e^{\frac{1}{\sigma * \sigma*2}((x - \mu]^2 (-1))} = f{\x)$.[SEP]
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[CLS]Covariance: The formula $\mathbb{E}\left[(-\mathbb{E}\left[R\right] + R) \cdot (Y - \mathbb{E}\left[Y\right])\right] = \operatorname{Cov}\left[R,Y\right]$ measures the degree to which two random variables $R$ and $Y$ vary together.[SEP]
[CLS]Covariance:gt formula $\mathbb{E}\11[\(-\mathbb{E}\left[R\right] + Deg) \cdot (fy - \mathbb_{-E}},left[Y\right]^{(\right_{ = \operatorname{Princov}\left[R,Y\right]$ / the degree Py which two701 variables $R$ and $Y$ vary together.[SEP]
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[CLS]Law of Sines: In a triangle with sides of lengths $a$, $h$, and $c$, and opposite angles $\alpha$, $\beta$, and $\gamma$, respectively, it holds: $\frac1a*\sin(\alpha) = \frac{\sin\left(\beta\right)}{h} = \sin\left(\gamma\right)/c$.[SEP]
[CLS]Law of Sines: In a triangle with sides of lengths $a$, $h$, and $c$, and opposite angles $\alpha$, $\beta$, and $\gamma$, respectively, it holds: $\frac1a*\sin(\alpha) = \frac{\sin\left),\beta\right)}{h} = \sin\left(\gamma\right)/c$.[SEP]
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[CLS]Wallis Product: \frac{\pi}{2} = \prod_{n=1}^\infty \dfrac{1}{\left(2 \cdot n + 1\right) \cdot (\left(-1\right) + 2 \cdot n)} \cdot \left(n \cdot 2\right)^2[SEP]
[CLS]Wallis Product: \frac{\pi}{2} = \prod_{n=1}^\infty \dfrac{1}{\left(2 \cdot n + 1\right) \cdot (\left(-1\right) + 2 \cdot n)} \cdot \left(n \cdot 2\right)^2[SEP]
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[CLS]Stirling approximation: m! \approx (m/e)^m*(m*\pi*2)^{1 / 2}[SEP]
[CLS]Stirling approximation: m! \approx (m/e)^m*(m*\pi*2)^{1 / gr}[SEP]
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[CLS]Derivative of Inverse Function: Let $h(z)$ be a function and $h^{-1}\left(z\right)$ its inverse function. The derivative of $h^{-1}(z)$ can be expressed as $\dfrac{1}{\frac{\text{d}}{\text{d}h^{-1}(z)} h(h^{-1}\left(z\right))}$.[SEP]
[CLS]Derivative of Inverse Function: Let $h(z$ be a functionhed $HT^{-1}\left=sz\ with)$ its inverseuse. The derivativetf $wh^{-1}(zz)$ can be expressedHg $(dfrac{841}{\frac{\zt{d}}{\text}}}$.d|}h^{-1}(z)} h]$.h^{-1_{\left(z,\right})}\}$.[SEP]
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[CLS]product rule of differentiation: The derivative of the product of two functions, $u$ and $v$, with respect to $x$, denoted as $d/dx (v\left(x\right) u\left(x\right))$, is given by the expression $d/dx v(x) \cdot u\left(x\right) + d/dx u(x) \cdot v\left(x\right)$.[SEP]
[CLS]product rule of differentiation: The derivative ofGT Pi of two functionsof $u$ and rgmv$, ((- respect to -x759 denoted as $d/dx (v\left(\x\right) u\left(x}_\right50 iT given by the expression gud]^dx v(x) \cdot u\left(x\right) + d/dx (-(x) \Th v\left(x\right)$.[SEP]
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[CLS]Law of Total Probabilities: Consider a set of mutually exclusive and exhaustive events $C_1\wedge ...\wedge C_m$, where exactly one of these events must occur. For any event $G$, the probability $\mathbb{P}(G)$ can be expressed as $\mathbb{P}(G) = \sum_{k=1}^m \mathbb{P}(G|C_k) \mathbb{P}(B_i)$.[SEP]
[CLS]Law of Total Probabilities: Consider a set of mutually exclusive and exhaustive events $C([1\wedge ...\wedge C_m$, where exactly757 of these events must occur. For any event $G$, ty probability -mathbb{P}(G)$ can Be EF as $\mathbb{P}(G) = \sum__k=1}^m \mathbb{P}(G|ck_k) \mathbb{P}(B_i)$.[SEP]
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[CLS]Determinant of 2x2 Matrix: X = [[a , b],[c , g]] \Rightarrow g \cdot a - b \cdot c = \left|{X}\right|[SEP]
[CLS]164inant of ( expresses2 mathematical: X =],a , b],[c , g]$, \Rightarrow g \ Total a - b \cdot c = gyleft|{X)}(right|[SEP]
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[CLS]Geometric Series: \frac{1}{-s + 1}*(1 - s^{1 + m}) = \sum_{x=0}^m s^x[SEP]
[CLS]Geometricgs: \frac*{1}{-s + 1}*(1 - s^{641 + m}}}\ ((- \sum_{-\x=0}^m s|^{x[SEP]
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[CLS]Bayes' Theorem: The formula for calculating $\mathbb{P}(D|G)$ states that it is equal to the product of the probability of event $G$ given event $D$, denoted as $P(G|D)$, and the probability of event $D$, divided by the probability of event $G$, denoted as $\operatorname{P}(G)$.[SEP]
[CLS]Bayes's where],[ The formula for function $\mathbb{P}\,\D|\G)$ states that it is equal Co the product ef the p of event $ Gi$ (\ event +D$, denoted as $P(G|hedral)$, and the probability FR event $D$, divided by the polyef event [ CG$) denoted as $\operatorname|{\P}}(G)$.[SEP]
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[CLS]Complex number multiplication: The product of complex numbers $z$ and $w$ is $\text{i}\cdot (a\cdot f + b\cdot h) + a\cdot h - b\cdot f = z\cdot w$, where $a + \mathrm{i} \cdot b = z$ and $w = h + \mathrm{i}\cdot f$.[SEP]
[CLS]Complex two multiplication_{- at producttf complexGamma $z$ another ${\w50 is $\text{ You}\ Total (\{\Cd Fe + b\ nt h)( + a\cdotbh pg b\ometrycf ( z\cdot w956 where $a BX "!"mathrm{i} \cdot b ( z$ and $ rv = h + \mathrm{ Ky}\ This -($.[SEP]
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[CLS]Gaussian Integral: \int\limits_{-\infty}^\infty e^{-s^2}\,\mathrm{d}s = \sqrt{\pi}[SEP]
[CLS]Gaussian Integral: \int\limits_{-\infty}}(\infty ...^{-s^2}\,\mathrm{d}s = \sqrt{\pi}[SEP]
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[CLS]Pascal's Formula: The relationship ${1 + l \choose 1 + x}$ is equal to the sum of ${l \choose x + 1}$ and $\binom{l}{x}$, where $l$ and $x$ are positive natural numbers.[SEP]
[CLS]plementcal's Formula: The relationship ${1 + l \choose 1 + x}$ I equal to the sum of ${l \choose \| + 1}$ id $\binom{l}{x}$, ow $l$ and $x$ are s naturalgamma.[SEP]
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[CLS]Expected value of a random variable: The equation $E(X) = \int_{-\infty}^\infty f(x) x\,dx$ defines the mean of a continuous random variable $X$ with pdf $f(x)$.[SEP]
[CLS]Expected variable of a random variable: The E $E(X)! BA \int}^{\infty\|_{infty f]^x) x))*dx$ defines the mean F ..., continuous randomivity $X$ with pdf $f)*(- Ax757[SEP]
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[CLS]Factorial: 1\cdot 2\cdot \ldots\cdot \omega = \omega![SEP]
[CLS]Factorial: 1\cdot 2\cdot \ldots\cdot \omega = \omega![SEP]
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[CLS]Young's Inequality: The inequality $x^p/p + b^q/q \geq b*x$ is valid for all real numbers $p, q, x, b$ such that $1 \lt p$, $1 \lt q$, $1/q + \frac{1}{p} = 1$, $0 \leq x$, and $0 \leq b$.[SEP]
[CLS]Young's Inequality: The inequality $ax^p/p + b^mq/q \geq b*((x$ is valid for all real numbers $Princ, q...,ux, Ba958 such that $166 \lt p$, $010 \lt q$,g1/ expected + \gf{1}{p} = Se$, $0 gyQ x$, and ${(\ \leq BG))$.[SEP]
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[CLS]Binomial Series: \forall \alpha, y \in\mathbb{C} > 0: |y| \lt 1 rightarrow (1 + y)^{\alpha} = 1 + \binom{\alpha}{1}\times y + \binom{\alpha}{2}\times y^2 + \binom{\alpha}{3}\times y^3 + \binom{\alpha}{4}\times y^4 + \binom{\alpha}{5}\times y^5 + \dots[SEP]
[CLS]Binomial Series: \forall \alpha, y \in\mathbb{C} > 0: |y| \lt 1 rightarrow (1 + y)^{\alpha} = 1 + \binom{\alpha}{1}\times y + \binom{\alpha}{2}\times y^2 + \binom{\alpha}{3}\times y^3 + \binom{\alpha}{4}\times y^4 + \binom{\alpha}{5}\times y^5 + \dots[SEP]
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[CLS]Law of Cosines: -\cos(\gamma) \cdot b \cdot a \cdot 2 + a^2 + b \cdot b = c^2[SEP]
[CLS]Law definition Cosines: -\cos(\gamma) $(\cdot bGcdot s \cdot 2 + a^2 + b2\cdotmathbbba = c^342[SEP]
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[CLS]Integral of a Function: f(z) = \frac{\partial}{\partial z} \int_a^{__subs_{d19736344* subs m19736341}} f(p)\,\mathrm{d}p[SEP]
[CLS]Integral of a Function: f(z) = \frac{\partial}{\partial z} \int_a^{__subs_{d19736344* subs m19736341}} f(p)\,\mathrm{d}p[SEP]
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[CLS]Harmonic Series: The series of reciprocal natural numbers (($1 + \dfrac{1}{2} + \frac{1}{3} + \frac14 + \dfrac15 + 1/6 + \frac{1}{7} + \frac18 + \cdots$), given by the infinite sum $\sum_{k=1}^\infty 1/k$, does not converge and grows infinitely large.[SEP]
[CLS]HU First Series|^{ Tw series of reciprocal natural numbers (($01 + |\ cl{}$}{2}$. +bgzc{681}}}{353} + \Gamma14 ((- $(-dfrac223 $\ 1/}}\g \frac{1}{7} + ...,frac18 + cdots$), given by theni sum $\sum}^{-k=1}^\infty "!"*(k$, does ni converge andzg infinitely largedefinition[SEP]
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[CLS]Chain Rule: The derivative of a composite function $h\left(g(z)\right)$ with respect to $z$ is given by $\frac{\mathrm{d}}{\mathrm{d}z} g(z) \times \frac{\mathrm{d}}{\mathrm{d}g(z)} h\left(g(z)\right)$.[SEP]
[CLS]Chain Rule: Theital of ac composite function $\ Ha\left(g(z})\right)$ with respect to $z $(\ Ry given by GHfrac\|mathrm^{\d}}{\mathrm{d}itz} g(zd) \42 $\|frac{\mathrm)}(\ Da}}{\mathbb{d}g(z)} h\left(g/\zb)\right)$.[SEP]
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[CLS]Binomial Formula: (-g + a)^2 = g^2 + a^2 - a\cdot 2\cdot g[SEP]
[CLS]528omial Formula( (-g $\| AD^{-\2 = go|^2 + a^2 s a-\cdot 2 $\|cdot ...,[SEP]
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[CLS]Binomial Formula: Considering complex numbers $a$ and $b$, we can state that $(a - b) \cdot (b + a)$ is equal to $a^2 - b^2$.[SEP]
[CLS]Binomial Formula: Considering complex numbers)$a$ and $b$, we can CG that $(a - b) GCcdot (b + a)$ is equal to $a {\2 - ...]-2$.[SEP]
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[CLS]Euler-Poisson Integral: \int\limits_{-\infty}^\infty e^{-y^2}\,\text{d}y = \sqrt{π}[SEP]
[CLS]Euler-Poisson Integraltext \int\limits_{-\infty}^\infty e{-y^2}\,\text{d}y = $\sqrt{ *}[SEP]
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[CLS]Fundamental Theorem of Calculus: For any integrable function $h(x)$ and any real interval $(a,g)$ the integral over this interval $\int\limits_a^g h(x)\,\mathrm{d}x$ is equal to $U(g) - U(a)$.[SEP]
[CLS]Fundamental Theorem of Calculus: For any integrable function $h(x)$ and any real interval $(a,g)$ the integral over this interval $\int\limits_a^g h(x)\,\mathrm{d}x$ is equal to $U(g) - U(a)$.[SEP]
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[CLS]Binomial Series: |z| \lt 1 \Rightarrow (z + 1)^{\alpha} = 1 + {\alpha \choose 1} \cdot z + {\alpha \choose 2} \cdot z^2 + z^5 \cdot {\alpha \choose 5} + \ldots[SEP]
[CLS]528omial su]], $|\z| \lt881 $(Rightarrow (ez + (}[alpha} = 1gg {\alpha \choose BA{- \cdot ((- + {\alpha \choose |)} \xt z}^\2 ((- z^5 \cdot {\alpha \choose 5} + galdots[SEP]
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[CLS]Euler-Poisson Integral: \int_{-\infty}^\infty exp(-y * y)\,\mathrm{d}y = \pi^{1/2}[SEP]
[CLS]Euler-Poisson Integr fo(( $\int|{\infty}^\infty exp(-y * y\,mathrm}{( d}oy = {\pi^{1/372}[SEP]
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[CLS]L'Hopital's Rule: Let $f(z)$ and $u(z)$ two differentiable functions. The limit of the ration of two functions $\lim_{z \to b} f(z)/u(z)$ can be computed as the ratio of their derivatives $\lim_{z \to b} \frac{d/dz f(z)}{d/dz u\left(z\right)}$[SEP]
[CLS]L'Hopital's Rule\|_{ Let $f-(z)$. and $cu(z)$ twoxd functions! Thenum of T ration of two functions $\lim|}z \ union ...} f'(z], du(sz)$ can BY computed ($ the ratio of their derivatives |\lim\|_{z \to By}{ \frac{d/dz fd)/(tz}),\lder*dz u)}\left(uz\right)}$[SEP]
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[CLS]Set Distributivity: The union of set $F$ with the intersection of sets $Z$ and $G$ is equal to the intersection of the union of sets $F$ and $Z$ with the union of sets $F$ and $G$: $\left(F \cup G\right) \cap (F \cup Z) = F \cup (G \cap Z)$.[SEP]
[CLS]Set Diributimpl!, The unionff set $fs$ with tw intersection of sets $Z200 and $ -509gg equal to the intersection of the <Pf sets $F$ and $418$ with the union of sets $F$ and $G$: goleft)/( FR \cupug}}\right) \cap (F EGcup Z) = of \cup |G \lcZe)$.[SEP]
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[CLS]Eulers Formula for polyhedra: If we consider a polyhedron with number of vertices $R$, edges $B$, and faces $s$, then the relationship $2 = s + R - B$ holds.[SEP]
[CLS]Eulers Formula for polyhedra: If we consider a polyhedron with number of vertices $R$, edges $B$, and faces $s$, then the relationship $2 = s + R - B$ holds.[SEP]
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[CLS]Eulers Identiy: For all complex numbers $y$, the exponential function $e^{yi}$ can be expressed as $\cos{y} + i*\sin{y}$.[SEP]
[CLS]Eulers Identiy: For GA complex numbers $y$, the estimation f $e^{yi}\ can be expressed as $\cos{y} + i*\sin{ HY}}$,[SEP]
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[CLS]Derivative of Inverse Function: For a function $f$ and its inverse $v^{-1}(z)$, the equation $\frac{\mathrm{d}}{\mathrm{d}z} v^{-1}\left(z\right) = \dfrac{1}{\frac{\mathrm{d}}{\mathrm{d}v^{-1}\left(z\right)} v\left(v^{-1}(z)\right)}$ holds true.[SEP]
[CLS]Derivative of Inverse Function: For a function $f$ and its inverse $v^{-1}(z)$, the equation $\frac{\mathrm{d}}{\mathrm{d}iz} v}{-1}\left)(z\right) = \dfrac_{\1}{\frac{\mathrm}})d}}{\mathrm{d}v^{-1}\left(z\,right)} v\left(v^{-1}(z)\right)}$ holds true.[SEP]
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[CLS]Eulers Identiy: Considering any complex number $y$, we can state that $e^{\mathrm{i} \cdot y} = \mathrm{i} \cdot \sin{y} + \cos{y}$.[SEP]
[CLS]Eulers Identiy: Considering any complex Numbers $y$, definition can state that $e^{\808{i)} $(cdot y)} = \mathrm{i} [cdot \sin}}}y} 2 \cos{y}$.[SEP]
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[CLS]Law of Large Numbers: As the sample size $l$ of a discrete random variable $T$ tends to infinity, the sample mean $\frac{1}{l} \cdot \sum_{j=1}^l y_j$ converges to the expected value $\mathbb{E}[T]$.[SEP]
[CLS]Law of Large Numbers: As the sample size $ldots607 of a discrete random variable))$.T$ tends to infinity, the sample mean $\ic{1)}{ lu}$. ${\De \sum_{j=881}^l y_{-\j700 converges to the expected value $\mathbb{ee}:T))*[SEP]
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[CLS]Binomial Distribution: The probability mass function (PMF) of a random variable $V$ following the Binomial Distribution is expressed as $(-s + 1)^{x - l} {x \choose l} s^l = \mathbb{P}(V = l)$, where $x$ represents the number of trials, $s$ represents the probability of success, and $l$ represents the number of successful outcomes.[SEP]
[CLS]Binomial Distribution: The probability mass function (PMF) of a random variable $V$ following the Binomial Distribution is expressed as $(-s + 1)^{x - l} {x \choose l} s^l = \mathbb{P}(V = l)$, where $x$ represents the number of trials, $s$ represents the probability of success, and $l$ represents the number of successful outcomes.[SEP]
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[CLS]Product Rule: The derivative of the product of two functions, $u$ and $v$, $d/dy (g(y) \cdot h(y))$ is equal to $d/dy h(y) \cdot g(y) + \frac{d}{dy} g\left(y\right) \cdot h\left(y\right)$.[SEP]
[CLS]Product Rule: The derivative of the product of two functions, $u$ and $OV($ $d/dy (gs(ya) \cdot h(y))$ is equalGT$.d/dy h(oy) \cdot g(y) + \frac{d}{ le} BG\left(y\right) \cdot h\left(y\494|$[SEP]
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[CLS]Euler's Gamma Function: \forall n\in\mathbb{N}: \Gamma\left(n\right) = \int\limits_0^\infty x^{n + \left(-1\right)}\cdot e^{-x}\,\text{d}x = (n + (-1))![SEP]
[CLS]Euler's Gamma Function: \forall n\in\mathbb{N}: \Gamma\left(n\right) = \int\limits_0^\infty x^{n + \left(-1\right)}\cdot e^{-x}\,\text{d}x = (n + (-1))![SEP]
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[CLS]Law of Sines: In a triangle with sides of lengths $f$, $b$, and $c$, and opposite angles $\alpha$, $\beta$, and $\gamma$, respectively, it holds: $\sin(\alpha)/f = \frac{1}{b}\cdot \sin(\beta) = \frac{\sin(\gamma)}{c}$.[SEP]
[CLS]jl of GAines: In a kg with summation ofgg \f)$. $\|b $-\ and $c)$ any opposite Syggalpha$, (\beta$, and $\gamma$, respectively, it holds]] $\sin(\alpha)/nf = \frac}}(1}{b}(\cdot \596(\beta) = \frac{\sin(\gamma)}{c})$.[SEP]
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[CLS]Binomial Series: \forall \alpha, x \in\mathbb{C} > 0: |x| \lt 1 \Longrightarrow \left(1 + x\right)^{\alpha} = 1 + {\alpha \choose 1}\cdot x + {\alpha \choose 2}\cdot x^2 + {\alpha \choose 3}\cdot x^3 + {\alpha \choose 4}\cdot x^4 + {\alpha \choose 5}\cdot x^5 + \dotsm[SEP]
[CLS]Binomial Series: \forall \alpha, x \in\mathbb{C} > 0: |x| \lt 1 \Longrightarrow \left(1 + x\right)^{\alpha} = 1 + {\alpha \choose 1}\cdot x + {\alpha \choose 2}\cdot x^2 + {\alpha \choose 3}\cdot x^3 + {\alpha \choose 4}\cdot x^4 + {\alpha \choose 5}\cdot x^5 + \dotsm[SEP]
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[CLS]Determinant of 2x2 Matrix: X = \begin{pmatrix}a & b\\c & e\end{pmatrix} \Rightarrow \left|{X}\right| = e \cdot a - c \cdot b[SEP]
[CLS]Determinant of 2x2 Matrix: X = \begin{pmatrix}a & b\\c & e\end{pmatrix} \Rightarrow \left|{X}\right| = e \cdot a - c \cdot b[SEP]
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[CLS]Taylor approximation: The expansion $v(x) = \sum_{k=0}^\infty \frac{1}{k!} \cdot v^{(k)}\left(a\right) \cdot \left(x - a\right)^k$ expresses the function $v\left(x\right)$ as an infinite sum of its derivatives evaluated at $a$ multiplied by powers of $x - a$ divided by $k!$.[SEP]
[CLS]Taylor approximation: The expansion $v(x) = \sum_{k=0}^\infty \frac{1}{k!} \cdot v^{(k)}\left(a\right) \cdot \left(x - a\right)^k$ expresses the function $v\left(x\right)$ as an infinite sum of its derivatives evaluated at $a$ multiplied by powers of $x - a$ divided by $k!$.[SEP]
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[CLS]Binomial Series: (1 + z)^{\alpha} = 1 + z \cdot \binom{\alpha}{1} + z \cdot z \cdot \binom{\alpha}{2} + \binom{\alpha}{3} \cdot z^3 + \binom{\alpha}{4} \cdot z^4 + \binom{\alpha}{5} \cdot z^5 + \cdots[SEP]
[CLS]Binomial Series: -1 ..., z)^{\alpha})} = 1 + z $\cdot \binom{\alpha}{1} +cz \|_{ taking z \| addition \binom}^{alpha}{862} $\ \876}_\alpha}{3}}$. \cdot z^3 + \287_{\Ui}{)}} $|\cdot z^{-4 + *776)^{-alpha{(5})^ \cdot z^435 + \cdots[SEP]
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