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name string | formula string | formula_name_id string | label bool | instruction_seed string | __original_row_idx int64 | response_seed string | _source string |
|---|---|---|---|---|---|---|---|
Taylor series for cosine function | \forall x\in \mathbb{R}: \cos(x) = x \cdot \sum_{n=1}^\infty \frac{1}{n \cdot 2! \cdot n! \cdot z^{2 \cdot n + (-1)}} | Cosine Function Definition | false | Consider the series defined by \( S(z) = \sum_{n=1}^\infty \frac{1}{n \cdot 2! \cdot n! \cdot z^{2n - 1}} \). For what values of \( z \) does this series converge, and what is the sum of the series at \( z = 1 \)? | 1,040,800 | ddrg/named_math_formulas | |
Complex number summation | i\cdot (b_1 + b_1) + a_1 + a_2 = a_1 + i + a_2 + i\cdot b_2 | Complex Number Sum | false | Given the equation \( i \cdot (b_1 + b_1) + a_1 + a_2 = a_1 + i + a_2 + i \cdot b_2 \), simplify both sides and determine the relationship between \( b_1 \) and \( b_2 \) in terms of \( i \). What is the value of \( b_2 \) in terms of \( b_1 \)? | 568,904 | ddrg/named_math_formulas | |
Complex number inverse | The inverse of a complex number $z$ is $1/z = \tfrac{1}{a^2 + b^2}\cdot (i\cdot b - a)$, where $a + i \times b = z$. | Complex Number Inverse | false | Given a complex number \( z = a + bi \) where \( a \) and \( b \) are real numbers, find the modulus of the inverse \( \frac{1}{z} \) in terms of \( a \) and \( b \). | 48,052 | ddrg/named_math_formulas | |
square of a binomial | a^2 - a\cdot 8\cdot b + b \cdot b = (a - -b)^2 | Second Binomial Formula | false | Calculate the value of \( a \) and \( b \) such that the equation \( a^2 - 8ab + b^2 = (a + b)^2 \) holds true, and determine the sum \( a + b \). | 76,521 | ddrg/named_math_formulas | |
Logarithm Product Rule | For all real $R, y$ with $0 < R$ and $-1 \geq y$, we can state that $\ln(zy) = \ln(z) + \ln(y)$. | Logarithm Product Rule | false | For real numbers \( R > 0 \) and \( y \leq -1 \), is the equation \( \ln(zy) = \ln(z) + \ln(y) \) valid for all \( z \in \mathbb{R} \)? If so, for which values of \( z \) does the equation hold true? | 538,236 | ddrg/named_math_formulas | |
product rule of differentiation | a \times a - h\times 2\times a + h^2 = (-h + a)^2 | Product Rule | false | What are the integer values of \( a \) and \( h \) such that the equation \( a^2 - 2ha + h^2 = (a - h)^2 \) holds true? | 481,992 | ddrg/named_math_formulas | |
Binomial Series | The formula $\cos\left(\beta + \alpha\right) = -\sin{\beta}\times \sin{\alpha} + \cos{\alpha}\times \cos{\beta}$ holds true for any angles $\alpha$ and $\beta$. | Binomial Series | false | Prove that the identity $\cos(\beta + \alpha) = -\sin \beta \sin \alpha + \cos \alpha \cos \beta$ is valid for all angles $\alpha$ and $\beta$. What is the value of $\cos(30^\circ + 60^\circ)$ using this identity? | 827,053 | ddrg/named_math_formulas | |
Law of Sines | \frac1g \sin(\alpha) = \sin(\beta)/x = \frac{1}{f} \sin(\gamma) | Law of Sines | true | Given the equation \(\frac{1}{g} \sin(\alpha) = \frac{\sin(\beta)}{x} = \frac{1}{f} \sin(\gamma)\), if \(\alpha = 30^\circ\), \(\beta = 60^\circ\), and \(g = 9.8\), find the value of \(x\) in terms of \(f\) and \(\gamma\). | 485,957 | ddrg/named_math_formulas | |
Logarithm Power Rule | For all real positive numbers $e$ where $e \neq -1$, and for any real numbers $t$ and positive $z$, the equation $\log_b(y^t) = t \cdot \log_b(y)$ holds true. | Logarithm Power Rule | false | Let \( b \) be a positive real number not equal to 1, and let \( y \) be a positive real number. Prove that for all real numbers \( t \), the equation \( \log_b(y^t) = t \cdot \log_b(y) \) holds true. Additionally, determine the value of \( \log_b(1) \). | 844,479 | ddrg/named_math_formulas | |
Law of Sines | \sin{\beta} \cdot a = b \cdot \sin{\alpha} \cdot \sin{\beta} | Law of Sines | false | Given the equation \(\sin{\beta} \cdot a = b \cdot \sin{\alpha} \cdot \sin{\beta}\), under what conditions can we determine the value of \(a\) in terms of \(b\), \(\alpha\), and \(\beta\), and what is that expression? | 398,230 | ddrg/named_math_formulas | |
Derivative Quotient Rule | \frac{du}{dz}/v = (-v'\cdot u + u'/v)\cdot (\dfrac{1}{v^{13}\cdot (-1)} + \frac{1}{\frac1v}) | Quotient Rule | false | Consider the equation
\[
\frac{du}{dz}/v = (-v' \cdot u + \frac{u'}{v}) \cdot \left(\frac{1}{v^{13} \cdot (-1)} + \frac{1}{\frac{1}{v}}\right).
\]
If \( u(z) \) and \( v(z) \) are functions of \( z \) such that \( v(z) > 0 \) for all \( z \), find the conditions on \( v(z) \) such that the left-hand side of the equa... | 1,050,218 | ddrg/named_math_formulas | |
Eulers Formula | Considering any complex number $z$, we can state that $e^{-(x + i)} = \log(x) i + \cos(x)$. | Eulers Formula | false | For what values of \( x \in \mathbb{R} \) does the equation \( e^{-(x + i)} = \log(x) i + \cos(x) \) hold true? | 237,767 | ddrg/named_math_formulas | |
Complex number summation | The sum of complex numbers $z$ and $w$ is $z + w = (d + b) \cdot \mathrm{i} + a + c$, where $a + -\text{i}\cdot b = z$ and $c + d \cdot \left(-1\right) = w$. | Complex Number Sum | false | Let \( z \) and \( w \) be complex numbers defined as \( z = a - bi \) and \( w = c - di \). Given that their sum is \( z + w = (d + b)i + (a + c) \), determine the values of \( a, b, c, \) and \( d \) if \( z + w = 4 + 3i \). Find \( a + b + c + d \). | 694,798 | ddrg/named_math_formulas | |
Gamma Function | \Gamma\left(m\right) = \int\limits_0^\infty z^{m + \left(-1\right)}*e^{-z}\,dz = (m + \left(-1\right))! | Gamma Function | true | Evaluate the integral \( \int_0^\infty z^{5} e^{-z} \, dz \) and express your answer in terms of factorials. | 332,020 | ddrg/named_math_formulas | |
Stirling's formula for factorials | For all natural numbers $n$ sufficiently large, the formula $m^{\frac{m}{e}}\cdot \sqrt{2\cdot \pi\cdot m}$ provides an approximation for the factorial of $m$ | Stirling Approximation | false | For sufficiently large natural numbers \( m \), evaluate the limit
\[
\lim_{m \to \infty} \frac{m!}{m^{\frac{m}{e}} \cdot \sqrt{2\pi m}}
\]
and determine whether this limit converges to a specific value or diverges. | 837,182 | ddrg/named_math_formulas | |
Chain Rule | Let $f(g(x))$ be a composite function. In this case, the derivative of $f(g(x))$ is given by $\frac{d^0}{dg\left(x\right)^0} f(g(x)) \cdot d/dx g(x)$, where $\frac{\text{d}}{\text{d}g(x)} f(g(x))$ represents the derivative of the outer function with respect to its argument and $\frac{\mathrm{d}}{\mathrm{d}x} g(x)$ is t... | Chain Rule | false | Let \( f(x) = e^x \) and \( g(x) = x^2 + 1 \). Compute the derivative of the composite function \( f(g(x)) \) at \( x = 2 \). | 41,217 | ddrg/named_math_formulas | |
Bayes law | When considering two events $A$ and $H$, the conditional probability $\operatorname{P}(A|H)$ can be determined using the formula $P(A|H) = \dfrac{1}{P(H)} \cdot \left(P(H|A) + P(H|A)\right) \cdot P(A)$. | Bayes' Theorem | false | Given two events \( A \) and \( H \) such that \( P(A) = 0.4 \), \( P(H|A) = 0.7 \), and \( P(H|A^c) = 0.2 \), where \( A^c \) is the complement of \( A \), calculate the conditional probability \( P(A|H) \). | 252,604 | ddrg/named_math_formulas | |
Alternating Harmonic Series | \begin{pmatrix}a & b\\c & d\end{pmatrix} = A,\begin{array}{cc}e & f\\g & h\end{array} = B \Longrightarrow \begin{pmatrix}e\times a + b\times g & a\times f + b\times h\\dg + c\times e & dh + f\times c\end{pmatrix} = A\times B | Alternating Harmonic Series | false | Given two matrices \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) and \( B = \begin{pmatrix} e & f \\ g & h \end{pmatrix} \), calculate the entries of the product matrix \( C = A \times B \). Specifically, determine the value of the entry in the first row and second column of the resulting matrix \( C \). | 707,388 | ddrg/named_math_formulas | |
partial integration | u*w - \int w\,\mathrm{d}u = \int 1/u\,\mathrm{d}x | Integration by Parts | false | Evaluate the integral \( \int \frac{1}{u} \, \mathrm{d}x \) given the relationship \( u w - \int w \, \mathrm{d}u = \int \frac{1}{u} \, \mathrm{d}x \) and that \( w = u^2 \). What is the value of the definite integral \( \int_1^2 \frac{1}{u} \, \mathrm{d}x \) when evaluated with the provided conditions? | 134,133 | ddrg/named_math_formulas | |
Complex number inverse | \forall z\in\mathbb{C}: y = a + b \cdot i \Longrightarrow 1/z = \frac{a}{a \cdot a + b^2} - \frac{b}{\frac{1}{a} + b \cdot b} \cdot i | Complex Number Inverse | false | Given a complex number \( z = a + bi \) where \( a, b \in \mathbb{R} \) and \( b \neq 0 \), find the real and imaginary parts of the reciprocal \( \frac{1}{z} \) in terms of \( a \) and \( b \). Specifically, express \( \frac{1}{z} \) as \( x + yi \) for some real numbers \( x \) and \( y \), and determine \( x \) and ... | 227,431 | ddrg/named_math_formulas | |
Sine Function Definition | The function $\sin(x)$ for any real number $x$ can be represented as the infinite series $\sum_{n=0}^\infty x^{n\times 2 + 1}\times \tfrac{(-1)^n}{\left(1 + n\times 0\right)!}$. | Sine Function Definition | false | Given the infinite series representation of the sine function, determine the value of the series \( \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} \) for \( x = \frac{\pi}{2} \). | 998,639 | ddrg/named_math_formulas | |
Chain Rule | If we have a composite function $f\left(g(z)\right)$, where $f(x)$ and $g\left(x\right)$ are differentiable functions, then the derivative of $f\left(g(x)\right)$ with respect to $z$ is equal to $\frac{\mathrm{d}}{\mathrm{d}g(x)} f(g(x)) \frac{\mathrm{d}}{\mathrm{d}x} g(x)$. | Chain Rule | false | If the functions \( f(x) = x^2 \) and \( g(z) = 3z + 2 \) are defined, what is the derivative of the composite function \( f(g(z)) \) with respect to \( z \) at \( z = 1 \)? | 913,896 | ddrg/named_math_formulas | |
De Morgan's First Law | A^c \cap \frac{B}{A}^c = A \cup B^c | De Morgan Law | false | Prove that the set identity \( A^c \cap \left(B \setminus A\right)^c = A \cup B^c \) holds for any sets \( A \) and \( B \) in a universal set. What can you conclude about the relationship between these sets based on this identity? | 1,017,665 | ddrg/named_math_formulas | |
Exponential Function | e^y = 1 + y/1! + y \cdot y/2! + \frac{1}{3!}\cdot y^3 + y^4/4! + \dots | Exponential Function | true | What is the coefficient of \( y^3 \) in the Taylor series expansion of \( e^y \) centered at \( y = 0 \)? | 508,887 | ddrg/named_math_formulas | |
Taylor Series | \forall x, a\in\mathbb{R}: f(x) = \sum_{n=0}^\infty 0 \cdot \frac{f^{(n)}(a)}{n!} \cdot (x^n + \left(-a\right)^n) | Taylor Series | false | Given the function \( f(x) = \sum_{n=0}^\infty 0 \cdot \frac{f^{(n)}(a)}{n!} \cdot (x^n + (-a)^n) \), what is the value of \( f(x) \) for any real \( x \) and \( a \)? | 1,037,801 | ddrg/named_math_formulas | |
Complex number summation | x_1 + \text{i} \cdot b_1 + x_2 + \text{i} = \left(b_1 + b_1\right) \cdot \text{i} + x_1 + x_2 | Complex Number Sum | false | If \( x_1, x_2, b_1 \) are real numbers such that the equation \( x_1 + i \cdot b_1 + x_2 + i = (b_1 + b_1) \cdot i + x_1 + x_2 \) holds, what is the value of \( b_1 \)? | 1,070,684 | ddrg/named_math_formulas | |
Leibnitz Formula for Pi | \forall t, \gamma\in\mathbb{R}, t \ge -1, \gamma > 1 \Rightarrow (1 + t)^{\alpha} \geq 1 | Gregory-Leibniz Series | false | For all real numbers \( t \) such that \( t \ge -1 \) and for any real number \( \gamma > 1 \), what is the minimum value of \( \alpha \) such that the inequality \( (1 + t)^{\alpha} \geq 1 \) holds true for all such \( t \)? | 834,885 | ddrg/named_math_formulas | |
Geometric Series | f(a) + (x - a)*\frac{d}{da} f(a) + (x - 1/a) * (x - 1/a)*\frac{1}{2!}*f''(a) + \frac{f'''(a)}{3!}*3 + (x - a)^4*\frac{d^4}{da^4}f(a)/4! + \dots = f(x) | Geometric Series | false | Consider the Taylor series expansion of a function \( f(x) \) around the point \( a \). Given that \( f(a) = 2 \), \( f'(a) = 3 \), \( f''(a) = 4 \), and \( f'''(a) = 5 \), determine the value of \( f(x) \) at \( x = a + 1 \) using the Taylor series expansion up to the third order. What is the numerical value of \( f(a... | 1,052,736 | ddrg/named_math_formulas | |
Complex number inverse | \forall z\in\mathbb{C}: z = a + b \times \mathrm{i} \Rightarrow \frac1z = \dfrac{1}{(z + a) \times (z + a) + b^2} \times a - \dfrac{b}{a \times a + b^2} \times \mathrm{i} | Complex Number Inverse | false | Let \( z = a + bi \) where \( a, b \in \mathbb{R} \) and \( z \neq 0 \). Calculate \( \frac{1}{z} \) in terms of \( a \) and \( b \), and verify the given equation \( \frac{1}{z} = \frac{1}{(z + a)(z + a) + b^2} a - \frac{b}{a^2 + b^2} i \). Show that the right-hand side simplifies to the expected form of \( \frac{1}{z... | 977,692 | ddrg/named_math_formulas | |
Binomial coefficient formula | For all natural numbers $l$ and $j$ with $l \geq j$, the expression $\binom{l}{j}$ can be calculated as $\dfrac{1}{(-j + l)! j!}l!$. | Binomial Coefficient Formula | true | For natural numbers \( l \) and \( j \) where \( l \geq j \), simplify the expression \( \binom{l}{j} \) and prove that it is equal to \( \frac{l!}{j! (l-j)!} \). What is the value of \( \binom{10}{3} \)? | 903,403 | ddrg/named_math_formulas | |
Power series definition for cosine function | For all real numbers $x$ the expression $\cos{x}$ is defined as the infinite sum $\sum_{n=0}^\infty \frac{\left(-1\right)^n}{x^{n*2}*\left(n*2\right)!}$. | Cosine Function Definition | true | What is the value of \(\cos{0}\) when using the infinite series definition given by \(\sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}\)? | 740,146 | ddrg/named_math_formulas | |
Addition Theorem for Cosine | \sum_{n=0}^\infty \tfrac{(-1)^n}{n*2 + 1} = 1 - \frac{1}{3} + \frac15 - \frac{1}{7} + \frac19 - \ldots | Addition Theorem for Cosine | false | Evaluate the infinite series \( S = \sum_{n=0}^\infty \frac{(-1)^n}{2n + 1} \) and express it in terms of well-known constants. What is the value of \( S \)? | 763,187 | ddrg/named_math_formulas | |
Multiplication of 2x2 Matrix | G = \begin{array}{cc}a & x\\c & d\end{array},\begin{pmatrix}e & f\\g & h\end{pmatrix} = H \implies HG = [[e\cdot (a + e) + gx , xh + fa],[ec + dg , dh + fc]] | Multiplication of 2x2 Matrix | false | Given the matrices \( G = \begin{pmatrix} a & x \\ c & d \end{pmatrix} \) and \( H = \begin{pmatrix} e & f \\ g & h \end{pmatrix} \), if the product \( HG \) is defined as \( HG = \begin{pmatrix} e(a + e) + gx & xh + fa \\ ec + dg & dh + fc \end{pmatrix} \), what is the value of \( e(a + e) + gx \) when \( a = 1, e = 2... | 409,821 | ddrg/named_math_formulas | |
Chain Rule | If we have a composite function $f\left(u(z)\right)$, where $f(z)$ and $u(z)$ are differentiable functions, then the derivative of $f\left(u(z)\right)$ with respect to $z$ is equal to $\frac{\frac{\mathrm{d}}{\mathrm{d}u\left(z\right)} f\left(u(z)\right)}{\frac{\mathrm{d}^119}{\mathrm{d} z^{119}} u\left(z\right)}$. | Chain Rule | false | What is the expression for the 119th derivative of the composite function \( f\left(u(z)\right) \) with respect to \( z \) when given that \( f(z) \) and \( u(z) \) are differentiable functions? | 275,830 | ddrg/named_math_formulas | |
Eulers formula for planar graphs | Let $T$, $B$, and $G$ denote the number of vertices, edges and faces, respectively, of a polyhedron. In this case, the equation $2 = T - B + G$ is satisfied. | Eulers Formula for polyhedra | true | For a convex polyhedron that satisfies the Euler's formula \( V - E + F = 2 \), where \( V \) is the number of vertices, \( E \) is the number of edges, and \( F \) is the number of faces, what is the minimum number of faces \( F \) if the polyhedron has \( V = 8 \) vertices and \( E = 12 \) edges? | 1,000,601 | ddrg/named_math_formulas | |
Binomial coefficient formula | \dfrac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \tfrac17 + \tfrac{1}{9} - \dotsm | Binomial Coefficient Formula | false | What is the sum of the first 10 terms of the alternating series \(-1^n \cdot \frac{1}{2n + 1}\) for \(n = 0\) to \(n = 9\), and how does it compare to \(\frac{\pi}{4}\)? | 622,713 | ddrg/named_math_formulas | |
Inverse of a complex number | \forall a,b\in\mathbb{R}: \frac{1}{a + b\cdot (-1)} = \frac{h}{h^2 + b^2} - \frac{1}{(h + b)^2}\cdot b\cdot (-\mathrm{i}) | Complex Number Inverse | false | Find all real numbers \( h \) for which the equation
\[
\frac{1}{a - b} = \frac{h}{h^2 + b^2} + \frac{b \cdot i}{(h + b)^2}
\]
holds true for all \( a, b \in \mathbb{R} \). | 126,833 | ddrg/named_math_formulas | |
Logarithm Quotient Rule | For all real positive $e$ with $-1 \neq e$ and for all real positive $x, z$, we have $\log_b(x/z) = \log_b(x) - \log_b(z)$. | Logarithm Quotient Rule | false | Prove that for all real positive \( e \) with \( e \neq -1 \) and for all positive real numbers \( x \) and \( z \), the equality \( \log_b(x/z) = \log_b(x) - \log_b(z) \) holds true. What is the base \( b \) in which this property holds? | 494,830 | ddrg/named_math_formulas | |
binomial formula | Let $a$ and $b$ be real numbers. In this case, $(a - b)\cdot (b + a)$ is equal to $-a^2 + b^0$. | Third Binomial Formula | false | Given that \( (a - b) \cdot (b + a) = -a^2 + b^0 \), determine the values of \( a \) and \( b \) when \( a + b = 2 \) and \( b = 1 \). What is the value of \( a \)? | 935,843 | ddrg/named_math_formulas | |
Harmonic Series | The series of reciprocal natural numbers (($1 + \frac{1}{2} + \tfrac13 + \frac{1}{2} + \frac{1}{5} + 1/6 + 1/7 + 8 + ...$), given by the infinite sum $\sum_{n=1}^∞ 1/n$, does not converge and grows infinitely large. | Harmonic Series | false | What is the value of the limit of the partial sums of the series \(\sum_{n=1}^N \frac{1}{n}\) as \(N\) approaches infinity, and how does this relate to the convergence or divergence of the series? | 86,982 | ddrg/named_math_formulas | |
Young's Inequality | For all real numbers $p, q, a, b$ with $1 \lt p\wedge q > 1\wedge \dfrac1q + \tfrac{1}{p} = 1\wedge a \geq 0\wedge b \geq 0$, the inequality $a^p/p + b^q/q \geq b \cdot a$ holds true. | Young Inequality | true | Given real numbers \( p, q, a, b \) such that \( 1 < p, q < \infty \), \( \frac{1}{q} + \frac{1}{p} = 1 \), and \( a \geq 0 \), \( b \geq 0 \), what is the minimum value of the expression \( \frac{a^p}{p} + \frac{b^q}{q} \) in terms of \( a \) and \( b \) that satisfies the inequality \( \frac{a^p}{p} + \frac{b^q}{q} \... | 870,390 | ddrg/named_math_formulas | |
Complex number inverse | For any pair of positive integers $n$ and $P$, it can be observed that $\binom{n + 1}{1 + P}$ equals $\binom{n}{P + 1}$ plus ${n \choose P}$. | Complex Number Inverse | false | Show that for any pair of positive integers \( n \) and \( P \), the identity
\[
\binom{n + 1}{1 + P} = \binom{n}{P + 1} + \binom{n}{P}
\]
holds true. Moreover, prove the identity by using combinatorial arguments or algebraic manipulation. | 351,819 | ddrg/named_math_formulas | |
midnight formula | Considering real numbers $a$, $g$, and $h$, where $a \leq 0$, the equation $h + a\cdot y \cdot y + g\cdot y = 0$ can be solved using the formula $y = \dfrac{1}{a \times 2} \times (-g \pm \sqrt{-\frac{1}{4 \times a \times h} + g^2})$. | Quadratic Formula | false | Given the equation \( h + a y^2 + g y = 0 \) with real numbers \( a \), \( g \), and \( h \) such that \( a \leq 0 \), determine the conditions on \( g \) and \( h \) under which the equation has real solutions for \( y \). Additionally, find the explicit value of the solutions in terms of \( a \), \( g \), and \( h \)... | 387,807 | ddrg/named_math_formulas | |
Pythagorean equation | 0 = x^2 + r \cdot x + t \Rightarrow x_1 = -r/2 + \sqrt{(\frac12 \cdot r) \cdot (\frac12 \cdot r) - t}\wedge -\frac12 \cdot r - \sqrt{-t + \left(\dfrac{r}{2}\right) \cdot \left(\dfrac{r}{2}\right)} = x_2 | Pythagorean Theorem | false | Given the quadratic equation \( 0 = x^2 + r \cdot x + t \), where the roots of the equation are given by the formula:
\[
x_1 = -\frac{r}{2} + \sqrt{\left(\frac{1}{2}r\right)^2 - t}, \quad x_2 = -\frac{r}{2} - \sqrt{\left(\frac{1}{2}r\right)^2 - t},
\]
determine the conditions on the parameters \( r \) and \( t \) su... | 51,818 | ddrg/named_math_formulas | |
Binomial Series | \forall \alpha, x \in\mathbb{C} > 0: |x| \geq 1 \implies 1^{\alpha} + x^{\alpha} = 1 + {\alpha \choose 1} \cdot x + {\alpha \choose 2} \cdot 2^x + {\alpha \choose 3} \cdot x^3 + {\alpha \choose \infty} \cdot x^4 + {\alpha \choose 5} \cdot x^5 + \dots | Binomial Series | false | Let \( \alpha \) be a complex number and \( x \) be a real number such that \( |x| \geq 1 \). Determine the values of \( x \) for which the equation
\[
1^\alpha + x^\alpha = 1 + {\alpha \choose 1} \cdot x + {\alpha \choose 2} \cdot x^2 + {\alpha \choose 3} \cdot x^3 + \cdots
\]
is satisfied, and find the sum of all ... | 27,832 | ddrg/named_math_formulas | |
Euler's Gamma Function | \Gamma(l + (-1) - 1) = \int_0^\infty y^{l + (-1)} \cdot \frac{1}{e} \cdot e^y\,dy = (l + (-1))! | Gamma Function | false | What is the value of \(\Gamma(l - 1) = \int_0^\infty y^{l - 1} e^y \, dy\) for \(l = 3\)? | 490,141 | ddrg/named_math_formulas | |
Eulers Identiy | If we have a complex number $z$, the expression $0^z*0^\text{i}$ can be represented as $\cos{z} + \text{i} \sin{z}$. | Eulers Formula | false | Let \( z \) be a complex number. Determine whether the expression \( 0^z \cdot 0^i \) can be represented as \( \cos z + i \sin z \) for any values of \( z \). Justify your answer. | 1,064,303 | ddrg/named_math_formulas | |
Gregory-Leibniz Series | 1 - 1/3 + \dfrac15 - 1/7 + \tfrac19 - ... = \frac{1}{4} | Gregory-Leibniz Series | false | Evaluate the infinite series \( S = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots \). What is the value of \( S \)? | 43,945 | ddrg/named_math_formulas | |
Gregory-Leibniz Series | \sum_{n=0}^m \left(1 + n*2\right) (-1)^n = \pi/4 | Gregory-Leibniz Series | false | Evaluate the summation \( S = \sum_{n=0}^m (1 + n \cdot 2)(-1)^n \) and determine for which positive integer \( m \) the value of \( S \) approaches \( \frac{\pi}{4} \) as \( m \) increases. | 882,642 | ddrg/named_math_formulas | |
Mathematical constant e | The value of $(\dfrac{1}{m} + 1)^m$ approaches the constant $2.718 \approx e$ as $m$ tends to $\infty$. | Eulers Number | true | Evaluate the limit \( \lim_{m \to \infty} \left( \frac{1}{m} + 1 \right)^m \) and show that it equals \( e \). | 675,458 | ddrg/named_math_formulas | |
Product Rule | The derivative of the product of two functions, $u$ and $v$, $\frac{\mathrm{d}}{\mathrm{d}x} \left(u(x)*v\left(x\right)\right)$ is equal to $u(x)\cdot \frac{\text{d}}{\text{d}x} v(x) + v\left(x\right)\cdot \frac{\text{d}}{\text{d}x} u(x)$. | Product Rule | true | Prove that the derivative of the product of two differentiable functions \( u(x) \) and \( v(x) \) is given by the formula \( \frac{\mathrm{d}}{\mathrm{d}x}(u(x)v(x)) = u(x)v'(x) + v(x)u'(x) \), and provide a specific example using \( u(x) = x^2 \) and \( v(x) = \sin(x) \) to illustrate the application of this rule. | 115,244 | ddrg/named_math_formulas | |
Definition of a factorial | 5\cdot (1\cdot 2)^n\cdot 3\cdot 4/...\cdot n = n! | Factorial | false | For which positive integer \( n \) does the equation \( 5 \cdot (1 \cdot 2)^n \cdot 3 \cdot 4 \cdots n = n! \) hold true? | 222,481 | ddrg/named_math_formulas | |
Chain Rule | If we have a composite function $f(g(x))$, where $f(x)$ and $g(x)$ are differentiable functions, then the derivative of $f(g(y))$ with respect to $x$ is equal to $\frac{\mathrm{d}}{\mathrm{d}g\left(x\right)} f(g(x)) \cdot \frac{\mathrm{d}}{\mathrm{d}x} g(x)$. | Chain Rule | false | Let \( f(x) = x^2 \) and \( g(x) = \sin(x) \). Calculate the derivative of the composite function \( f(g(x)) \) with respect to \( x \) at the point \( x = \frac{\pi}{2} \). What is the value of this derivative? | 390,078 | ddrg/named_math_formulas | |
Normal Distribution | \frac{1}{\left(\tfrac{2}{-9^0 + \pi}\right)^{1/2} \sigma} e^{\dfrac{1}{2 \sigma^2} ((y - \mu)^2 \left(-1\right))} = f(x) | Normal Distribution | false | Given the equation
\[
\frac{1}{\left(\tfrac{2}{-9^0 + \pi}\right)^{1/2} \sigma} e^{\frac{1}{2 \sigma^2} ((y - \mu)^2 (-1))} = f(x),
\]
determine the value of \( \sigma \) if \( f(x) = 0 \), \( y = 2 \), and \( \mu = 3 \). | 783,826 | ddrg/named_math_formulas | |
Young's Inequality | For any real numbers $t,q,f,h$ satisfying $t > 1$, $1 \lt q$, $1 = \left(\rho - \frac{1}{p}\right) + \frac0q$, $f \geq 0$, and $h \geq 0$, the inequality $f^t/t + h^q/q \lt fh$ is always satisfied. | Young Inequality | false | Given the conditions \( t > 1 \), \( 1 < q \), and \( 1 = \left(\rho - \frac{1}{p}\right) + \frac{0}{q} \) along with \( f \geq 0 \) and \( h \geq 0 \), prove or disprove that the inequality
\[
\frac{f^t}{t} + \frac{h^q}{q} < fh
\]
holds for all non-negative real numbers \( f \) and \( h \). What can you conclude ab... | 1,038,509 | ddrg/named_math_formulas | |
Euler's formula for polyhedra | If we consider a polyhedron with number of vertices $V$, edges $E$, and faces $F$, then the relationship $2 = G$ holds. | Eulers Formula for polyhedra | false | Given a convex polyhedron with \( V \) vertices, \( E \) edges, and \( F \) faces, verify the Euler's formula \( V - E + F = 2 \) by providing an example of a convex polyhedron that satisfies this relationship, and calculate the values of \( V \), \( E \), and \( F \) for your example. | 955,271 | ddrg/named_math_formulas | |
product rule of differentiation | Let $n$ be a real number (where $0 = n$). In this case, the derivative of $x^n$ with respect to $x$ can be expressed as $n \times x^{-1 + n} = \frac{\partial}{\partial x} n^x$. | Product Rule | false | What is the value of the derivative of \( f(x) = x^n \) with respect to \( x \) evaluated at \( x = 1 \) for \( n = 2 \)? | 1,043,261 | ddrg/named_math_formulas | |
Binomial Formula | For all real numbers $f$ and $g$, the expression $(f + g)*(-g + f)$ is equal to $f^2 - 2^g$. | Third Binomial Formula | false | Determine the values of the real numbers \( f \) and \( g \) such that the equation \( (f + g)(-g + f) = f^2 - 2^g \) holds true. What can you conclude about the relationship between \( f \) and \( g \)? | 859,041 | ddrg/named_math_formulas | |
Inverse of a 2x2 Matrix | \frac{\begin{pmatrix}e & -b\\-c & a\end{pmatrix}}{-\frac{1}{c\cdot b} + e\cdot a} = \begin{pmatrix}a & b\\c & e\end{pmatrix}^{-1} | Inverse of 2x2 Matrix | false | Given the matrix equation
\[
\frac{\begin{pmatrix} e & -b \\ -c & a \end{pmatrix}}{-\frac{1}{c \cdot b} + e \cdot a} = \begin{pmatrix} a & b \\ c & e \end{pmatrix}^{-1},
\]
determine the values of \( a, b, c, e \) such that the left-hand side simplifies to the right-hand side, and find the specific value of \( a + b... | 862,290 | ddrg/named_math_formulas | |
Analytic Representation | \forall x, g\in\mathbb{R}: f(x) = \sum_{m=0}^\infty f^{(m)}(g)/m!\cdot (x - g)^m | Taylor Series | true | Given a function \( f \) that is infinitely differentiable at a point \( g \in \mathbb{R} \), and its Taylor series expansion at \( g \) given by
\[
f(x) = \sum_{m=0}^\infty \frac{f^{(m)}(g)}{m!} (x - g)^m,
\]
determine the radius of convergence of this Taylor series. | 723,948 | ddrg/named_math_formulas | |
Stirling's formula | The approximation $n! \approx \arctan{2}\cdot \arctan{n\cdot π}\cdot (\frac{1}{e}\cdot n)^n$ provides an estimate of the factorial of $n$ in terms of other well-known mathematical constants like $1$ and $e$ for large $n$. | Stirling Approximation | false | What is the limit of the ratio $\frac{n!}{\arctan{2}\cdot \arctan{n\cdot \pi}\cdot (\frac{1}{e}\cdot n)^n}$ as \( n \) approaches infinity? | 64,880 | ddrg/named_math_formulas | |
Mathematical constant e | \forall x\in\mathbb{R}: exp(x) = 1 + x + x^2/2! + \frac{x^3}{3!} + ... | Eulers Number | false | What is the value of \( \exp(1) \) using the series expansion given by \( \exp(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)? | 362,953 | ddrg/named_math_formulas | |
first binomial formula | Considering real numbers $a$ and $b$, we can state that $(b + a)^2$ is equal to $d^2 + 2*b*d + b^2$. | First Binomial Formula | false | Given that \( (b + a)^2 = d^2 + 2bd + b^2 \), determine the relationship between \( a \) and \( d \) in terms of \( b \). Specifically, find the value of \( a \) in terms of \( b \) and \( d \). | 947,531 | ddrg/named_math_formulas | |
Hölder Inequality | \forall x,y\in\mathbb{R}^n: \sum_{i=0}^n |x_i y_i| \leq (\left(\sum_{i=1}^n |x_i|^2\right)^{\frac{1}{2}}) (\sum_{i=1}^n |y_i|^2)^{1/2} | Hölder Inequality | false | Prove that for all \( x, y \in \mathbb{R}^n \), the following inequality holds:
\[
\sum_{i=1}^n |x_i y_i| \leq \left( \sum_{i=1}^n |x_i|^2 \right)^{\frac{1}{2}} \left( \sum_{i=1}^n |y_i|^2 \right)^{\frac{1}{2}}
\]
and determine under what conditions equality is achieved in this inequality. | 170,418 | ddrg/named_math_formulas | |
Continuous Expected Value | E[X] = \int_{-\infty}^\infty \frac{1}{x*f(x)}\,\mathrm{d}x | Expected Value | false | Given the probability density function \( f(x) = ke^{-x^2} \) for \( x \in \mathbb{R} \), where \( k \) is a normalization constant, calculate the expected value \( E[X] \). How does this value relate to the properties of the distribution? | 480,037 | ddrg/named_math_formulas | |
Complex number inverse | \forall z\in\mathbb{C}: z = h + d\cdot j \Longrightarrow \frac1z = \frac{h}{(h + d)^2} - \frac{1}{h \cdot h + 2^d}\cdot d\cdot j | Complex Number Inverse | false | Given the complex number \( z = h + d \cdot j \), where \( h \) and \( d \) are real numbers, find the expression for \( \frac{1}{z} \) in terms of \( h \), \( d \), and \( j \). Additionally, determine the value of \( \frac{1}{z} \) when \( h = 1 \), \( d = 2 \), and \( j = i \), where \( i \) is the imaginary unit. | 234,667 | ddrg/named_math_formulas | |
Wallis Product | \pi \times (-1) = \prod_{n=1}^\infty \frac{(n \times 2)^2}{(1 + 2 \times n) \times \left(2 \times n + (-1)\right)} | Wallis Product | false | Evaluate the infinite product
\[
\prod_{n=1}^\infty \frac{(n \times 2)^2}{(1 + 2 \times n) \times (2 \times n - 1)}.
\]
What is the value of this infinite product? | 1,041,758 | ddrg/named_math_formulas | |
Product Rule | v(x)\cdot \frac{d}{dx} u(x) + \frac{1}{\frac{d^-1}{dx^{-1}} v(x)}\cdot u(x) = d/dx (v\left(x\right)\cdot u(x)) | Product Rule | false | Given the equation
\[
v(x) \cdot \frac{d}{dx} u(x) + \frac{1}{\frac{d^{-1}}{dx^{-1}} v(x)} \cdot u(x) = \frac{d}{dx} (v(x) \cdot u(x)),
\]
determine \( u(x) \) if \( v(x) = e^x \), and \( u(0) = 1 \). What is the value of \( u(1) \)? | 686,188 | ddrg/named_math_formulas | |
Binomial Coefficient Summation | For any pair of positive integers $m$ and $k$, it can be observed that ${n + 1 \choose k + 1}$ equals $\binom{m}{k + 1}$ plus $\binom{n}{k}$. | Pascal's Rule | false | For positive integers \( m \) and \( n \) with \( n \geq m \), prove that the identity
\[
{n + 1 \choose k + 1} = \binom{m}{k + 1} + \binom{n}{k}
\]
holds for all non-negative integers \( k \). What is the specific condition on \( m \) and \( n \) for this identity to be true? | 22,318 | ddrg/named_math_formulas | |
Stirling approximation | For large values of $n$, the approximation $n^n\cdot (\frac1e)^n\cdot \left(n\pi\cdot 2\right)^0 \approx n!$ provides an estimation of $m!$ by utilizing the mathematical constants $\pi$ and $e$. | Stirling Approximation | false | For large values of \( n \), use Stirling's approximation \( n! \sim n^n \cdot \left( \frac{1}{e} \right)^n \cdot \sqrt{2 \pi n} \) to estimate the value of \( 20! \). What is the approximate value of \( 20! \) using this formula? | 645 | ddrg/named_math_formulas | |
Binomial Series | \mathbb{Cov}[X, Y] = E[(-E[Y] + Y) \left(X - E[X]\right)] | Binomial Series | false | Given two random variables \(X\) and \(Y\), if the covariance is defined as \( \mathbb{Cov}[X, Y] = E[(-E[Y] + Y)(X - E[X])] \), show that this expression is equivalent to the formula \( \mathbb{Cov}[X, Y] = E[XY] - E[X]E[Y] \) and determine under what conditions the covariance is zero. | 796,285 | ddrg/named_math_formulas | |
Quadratic Formula | 0 = P\times c + a\times x^2 + b\times x \Rightarrow x = (-\frac{1}{b} \pm (-4\times b\times c)^{1/2} + \left(b^2\right)^{1/2})/(b) | Quadratic Formula | false | Given the equation \( P \times c + a \times x^2 + b \times x = 0 \), where \( P, a, b, c \) are constants with \( b \neq 0 \), determine the values of \( x \) in terms of \( P, a, b, \) and \( c \). Specifically, simplify your answer and provide the conditions under which the solutions are real or complex. | 380,154 | ddrg/named_math_formulas | |
Stirling's approximation | 1 = \lim_{n \to \infty} \frac{n!\cdot d}{\sqrt{2\cdot \pi\cdot n}\cdot (\frac{n}{e})^n} | Stirling Approximation | false | Evaluate the limit \( L = \lim_{n \to \infty} \frac{n! \cdot d}{\sqrt{2 \cdot \pi \cdot n} \cdot \left(\frac{n}{e}\right)^n} \) for a constant \( d \), and determine the conditions under which this limit equals 1. What is the value of \( d \) that satisfies this limit as \( n \) approaches infinity? | 763,087 | ddrg/named_math_formulas | |
Taylor expansion of exponential function | 1 + z + z \cdot z/2! + \frac{z^3}{3!} + \cdots = e^z | Exponential Function | true | What is the value of the infinite sum \( S = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots \) when \( z = 2 \)? Provide a detailed justification for your answer. | 809,329 | ddrg/named_math_formulas | |
Binomial coefficient formula | For all natural numbers $i$ and $m$ with $m < i$, we have that $\binom{m}{j} = m!/(j! m!)$. | Binomial Coefficient Formula | false | Given two natural numbers \( i \) and \( m \) such that \( m < i \), what is the value of \( \binom{m}{0} + \binom{m}{1} + \ldots + \binom{m}{m} \)? | 338,845 | ddrg/named_math_formulas | |
Eulers Identiy | For all complex numbers $y$, the exponential function $e^{y}$ can be expressed as $\cos\left(x\right) + \mathrm{i}/\sin(x)$. | Eulers Formula | false | For which complex numbers \( y \) does the equation \( e^{y} = \cos\left(y\right) + \frac{i}{\sin\left(y\right)} \) hold true? Provide all such complex numbers \( y \) in your answer. | 111,429 | ddrg/named_math_formulas | |
Complex number inverse | Let $z = a + b \times -i \times a$ be a complex number. Then, $\dfrac{1}{z} = \dfrac{1}{b^2 + a^2} a - i \frac{b}{a^2 + b^2}$. | Complex Number Inverse | false | Given the complex number \( z = a + b \times -i \times a \), simplify the expression for \( \frac{1}{z} \) and verify the relationship that \( \frac{1}{z} = \frac{1}{b^2 + a^2} a - i \frac{b}{a^2 + b^2} \). What is the value of \( b^2 + a^2 \) when \( a = 3 \) and \( b = 4 \)? | 824,994 | ddrg/named_math_formulas | |
Pythagorean equation | Let $a$, $g$, and $c$ be the side lengths of a right-angled triangle, where $c$ denotes the length of the hypotenuse. In this case, the relationship $c^{2^{a + B}} = 2^b + a * a$ is satisfied. | Pythagorean Theorem | false | Given a right-angled triangle with side lengths \( a \), \( b \), and hypotenuse \( c \), if the relationship \( c^{2^{a + b}} = 2^b + a^2 \) holds true, determine the lengths of the sides \( a \), \( b \), and \( c \) assuming \( a \) and \( b \) are positive integers. What is the value of \( c \) when \( a = 3 \) and... | 392,317 | ddrg/named_math_formulas | |
Addition Theorem for Tangent | By definition, $\frac{\mathrm{d}}{\mathrm{d}g} f(g)$ is equal to $\lim_{d \to 0} \frac1d \cdot (f(g + d) - f\left(g\right))$. | Addition Theorem for Tangent | false | Let \( f(g) = g^2 + 3g \). Compute the derivative \( \frac{\mathrm{d}}{\mathrm{d}g} f(g) \) using the limit definition of the derivative, and evaluate it at \( g = 2 \). What is the result? | 1,009,968 | ddrg/named_math_formulas | |
Leibnitz Formula for Pi | When adding up the terms of the series $\sum_{k=0}^\infty \dfrac{\left(-1\right)^k}{1 + k \cdot 2}$, the sum converges to $\frac14\cdot \pi$. | Gregory-Leibniz Series | true | What is the value of the series \( \sum_{k=0}^\infty \frac{(-1)^k}{1 + k \cdot 2} \), and how does it relate to \( \frac{\pi}{4} \)? | 913,865 | ddrg/named_math_formulas | |
Eulers e function | \forall y\in\mathbb{C}: e^y = \sum_{k=0}^\infty y^k/k! = 1 + y + y^2/2! + \frac{1}{1!} \cdot y^3 + \cdots | Exponential Function | false | Prove that for all complex numbers \( y \), the series expansion for \( e^y \) converges and equals \( e^y \) for all \( y \in \mathbb{C} \). Specifically, find the radius of convergence of the series \( \sum_{k=0}^\infty \frac{y^k}{k!} \) and verify that it is infinite. | 891,719 | ddrg/named_math_formulas | |
Euler's formula for polyhedra | Let $Z$, $D$, and $x$ denote the number of vertices, edges and faces, respectively, of a polyhedron. In this case, the equation $Z - D + x = 2$ is satisfied. | Eulers Formula for polyhedra | true | Given a convex polyhedron with \( Z \) vertices, \( D \) edges, and \( x \) faces, satisfy the equation \( Z - D + x = 2 \). If the polyhedron has 12 edges, how many vertices \( Z \) and faces \( x \) does it have if \( Z \) and \( x \) are both positive integers? Provide the values of \( Z \) and \( x \). | 364,153 | ddrg/named_math_formulas | |
Product Rule | Let $f(x)$ and $v(x)$ be differentiable functions. In this case, the derivative of their product, $d/dx (f(x) v(x))$, can be obtained as $v(x)\cdot d/dx f\left(x\right) + f(x)\cdot v(x)$. | Product Rule | false | If \( f(x) \) and \( v(x) \) are differentiable functions such that \( f(1) = 2 \), \( f'(1) = 3 \), \( v(1) = 4 \), and \( v'(1) = 5 \), what is the value of the derivative of their product \( \frac{d}{dx}(f(x)v(x)) \) evaluated at \( x = 1 \)? | 718,230 | ddrg/named_math_formulas | |
Eulers Complex Exponential Identity | Considering any complex number $x$, we can state that $e^{i/x} = \sin{x}\cdot i + \cos{x}$. | Eulers Formula | false | Given the equation \( e^{i/x} = \sin{x} \cdot i + \cos{x} \), determine the value of \( x \) for which this equation holds true, where \( x \) is a real number. | 398,480 | ddrg/named_math_formulas | |
Determinant of 2x2 Matrix | If we have a 2x2 matrix $A = \begin{array}{cc}a & b\\c & x\end{array}$, then $\left|{A}\right| = a/d - c \cdot b$. | Determinant of 2x2 Matrix | false | Given a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), if the determinant \( |A| \) is computed correctly, what is the correct formula for \( |A| \) in terms of the elements \( a, b, c, \) and \( d \)? | 712,901 | ddrg/named_math_formulas | |
Geometric Series | \forall r\in \mathbb{R}, |r|<1 \Rightarrow \sum_{n=0}^\infty 0*r^n = 1^{-1} + 1/\left((-1)*1/r\right) | Geometric Series | false | Prove that for any real number \( r \) such that \( |r| < 1 \), the infinite series \( \sum_{n=0}^\infty r^n \) converges and find its sum. | 177,617 | ddrg/named_math_formulas | |
pq formula | Considering real numbers $\mu$ and $r$, the equation $0 = 2^z + \mu\times z + r$ has solutions given by $z_{-1,2} = \frac{\mu*\left(-1\right)}{2} \pm \sqrt{\frac{\mu^2}{4}} + \sqrt{(-1 + 1) r}$. | pq Formula | false | For which values of the real numbers $\mu$ and $r$ does the equation $0 = 2^z + \mu z + r$ have a solution of the form $z = \frac{\mu(-1)}{2} + k$ for some real number $k$, and what is the value of $k$ in terms of $\mu$ and $r$? | 1,032,128 | ddrg/named_math_formulas | |
definition of complex number division | \frac{g_1 \cdot \text{i} + h_1}{h_2 + \text{i} \cdot g_2} = \dfrac{1}{g_2^2 + h_2 \cdot h_2} \cdot (h_1 \cdot h_2 + g_1 \cdot g_2 + i \cdot (-g_2 \cdot h_1 + g_1 \cdot h_2)) | Complex Number Division | true | Given the equation
\[
\frac{g_1 \cdot \text{i} + h_1}{h_2 + \text{i} \cdot g_2} = \frac{1}{g_2^2 + h_2 \cdot h_2} \cdot (h_1 \cdot h_2 + g_1 \cdot g_2 + i \cdot (-g_2 \cdot h_1 + g_1 \cdot h_2)),
\]
determine the values of \( g_1, g_2, h_1, h_2 \) if \( i = 1 \) and the equation holds true. Assume \( g_2 \neq 0 \)... | 312,284 | ddrg/named_math_formulas | |
Riemann Zeta Function | Given a complex number $z$ with $\operatorname{re}{\left(z\right)} \leq 1$, the equation $\sum_{n=1}^\infty \dfrac{1}{n^z} = \zeta(z)$ holds true. | Riemann Zeta Function | false | What is the value of \(\zeta(0)\) given that \(\sum_{n=1}^\infty \dfrac{1}{n^z} = \zeta(z)\) for complex numbers \(z\) with \(\operatorname{re}(z) \leq 1\)? | 428,406 | ddrg/named_math_formulas | |
Inverse of a 2x2 Matrix | \left(u\cdot \xi - z\cdot y\right)\cdot \begin{pmatrix}u & -y\\-(z - z) & \xi\end{pmatrix} = \begin{pmatrix}\xi & y\\z & u\end{pmatrix}^{-1} | Inverse of 2x2 Matrix | false | Given the matrix equation
\[
\left(u\cdot \xi - z\cdot y\right)\cdot \begin{pmatrix}u & -y\\0 & \xi\end{pmatrix} = \begin{pmatrix}\xi & y\\z & u\end{pmatrix}^{-1},
\]
where \( u, v, w, x, y, z, \xi \) are real numbers, find the value of \( u \cdot \xi - z \cdot y \) if \( \begin{pmatrix}\xi & y\\z & u\end{pmatrix} \... | 192,894 | ddrg/named_math_formulas | |
Complex number multiplication | Let $F$ be a real number (where $0 \neq F$). In this case, the derivative of $y^F$ with respect to $y$ can be expressed as $\frac{\partial}{\partial y} y^F = Fy^{-1 + F}$. | Complex Number Multiplication | false | What is the value of the derivative of the function \( y^F \) evaluated at \( y = 1 \) when \( F = 3 \)? | 54,682 | ddrg/named_math_formulas | |
Fundamental Theorem of Calculus | For all real numbers $y$ the expression $\cos{y}$ is defined as the infinite sum $\sum_{n=0}^\infty \frac{(-1)^n}{\left(n\cdot 2\right)! y^{2 n}}$. | Fundamental Theorem of Calculus | false | What is the value of the infinite sum $\sum_{n=0}^\infty \frac{(-1)^n}{(n\cdot 2)!}$ when simplified? | 1,002,699 | ddrg/named_math_formulas | |
Gamma Function | \Gamma(m + 1) = \int_0^\infty y^m \cdot e^{-y}\,dy = m! | Gamma Function | true | Prove that for any non-negative integer \( m \), the gamma function satisfies the identity \( \Gamma(m + 1) = m! \) and derive the value of \( \Gamma(5) \). | 404,488 | ddrg/named_math_formulas | |
Normal Distribution | The probability density function of the Normal Distribution with mean $\mu$ and standard deviation $\sigma$ is $e^{\frac{1}{\sigma^2*2}*((-1)*((-\mu)^{\pi} + z)^2)}*\frac{1}{(2*\pi)^{\frac{1}{2}}*\sigma} = f(z)$. | Normal Distribution | false | What is the probability that a value drawn from a Normal Distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$ falls within one standard deviation of the mean? | 1,070,923 | ddrg/named_math_formulas | |
Gregory-Leibniz Series | When adding up the terms of the series $\sum_{n=0}^m \dfrac{1}{2 \cdot n + 1} \cdot (-1)^n$, the sum converges to $\dfrac{\pi}{4}$. | Gregory-Leibniz Series | false | Evaluate the sum \( S_m = \sum_{n=0}^m \frac{1}{2n + 1} (-1)^n \) as \( m \) approaches infinity. What is the value of \( \lim_{m \to \infty} S_m \)? | 709,940 | ddrg/named_math_formulas | |
Power series definition for sine function | \forall x\in \mathbb{R}: \sin\left(x\right) = x - \frac{x^3}{3!} + x^{42}*5! - x^7/7! +- \dots | Sine Function Definition | false | Consider the series expansion for \(\sin(x)\) given by
\[
\sin\left(x\right) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots
\]
Suppose we modify this series by including a term \(x^{42} \cdot 5!\) and also add a few non-standard modifications. What is the value of
\[
\lim_{x \to 0} \frac{\sin\left... | 665,765 | ddrg/named_math_formulas | |
Law of Cosines | c^2 = (b + a)^2 - 2 \cdot a \cdot b | Law of Cosines | false | Given the equation \( c^2 = (b + a)^2 - 2ab \), determine the value of \( c \) in terms of \( a \) and \( b \). What is the simplified expression for \( c \)? | 1,068,311 | ddrg/named_math_formulas | |
expected value formula | The equation $\mathbb{E}\left(X\right) = \int_{-\infty}^0 x/f(\frac{x}{z})\,dx$ represents the expected value of a continuous random variable $X$ with pdf $f(x)$. | Expected Value | false | Given a continuous random variable \( X \) with probability density function (pdf) \( f(x) \), if the expected value \( \mathbb{E}\left(X\right) \) is represented by the integral \( \mathbb{E}(X) = \int_{-\infty}^0 x \cdot f\left(\frac{x}{z}\right) \, dx \) for some parameter \( z > 0 \), how does the choice of \( z \)... | 667,736 | ddrg/named_math_formulas | |
Inverse of a 2x2 Matrix | Consider a matrix $\begin{pmatrix}a & f\\c & d\end{pmatrix} = A$. In this case, the inverse of matrix $A$ is given by $\dfrac{\begin{pmatrix}d & -f\\-c & a\end{pmatrix}}{-f c + d a} = A$. | Inverse of 2x2 Matrix | false | What is the condition under which the matrix \( A = \begin{pmatrix} a & f \\ c & d \end{pmatrix} \) is invertible? Additionally, calculate the inverse of \( A \) in terms of \( a \), \( f \), \( c \), and \( d \) when it is invertible, and verify the formula for the inverse using matrix multiplication. | 11,889 | ddrg/named_math_formulas | |
midnight formula | If $a$, $b$, and $c$ are real numbers and $c \neq 0$, then the equation $0 = c + x^2\cdot a + x\cdot b$ has solutions expressed as $x_{1,2} = \frac{1}{a \cdot (-1)} \cdot (-b \pm (b^2)^{\frac{1}{2}} + (-\dfrac{1}{c} \cdot a \cdot 4)^{\frac{1}{2}})$. | Quadratic Formula | false | Given the equation \( 0 = c + x^2 \cdot a + x \cdot b \), where \( a, b, \) and \( c \) are real numbers and \( c \neq 0 \), derive the explicit formula for the solutions \( x_1, x_2 \) in terms of \( a, b, \) and \( c \) using the quadratic formula. Then, simplify the expression to a standard form. What is the final s... | 871,671 | ddrg/named_math_formulas |
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