{z^{7}}{7}}+\cdots } This series, sometimes called the Gregory–Leibniz series, equals π 4 {\textstyle {\frac {\pi }{4}}} when evaluated with z = 1 {\displaystyle z=1}...

Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno (1855...

Gottfried Wilhelm Leibniz or Leibnitz (1 July 1646 [O.S. 21 June] – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist...

}^{\infty }e^{-2\pi iy\xi }f'(y)\,dy\\&=\left[e^{-2\pi iy\xi }f(y)\right]_{-\infty }^{\infty }-\int _{-\infty }^{\infty }(-2\pi i\xi e^{-2\pi iy\xi })f(y)\...

y&=\int _{0}^{\pi -\tan ^{-1}\!\left({\frac {3}{4}}\right)}((5\sin t)(-5\sin t)+(5\cos t)(5\cos t))\,\mathrm {d} t\\&=\int _{0}^{\pi -\tan ^{-1}\!\left({\frac...

(1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili...

d^{3}\mathbf {r} '\;{\frac {\operatorname {div} \mathbf {v} (\mathbf {r} ')}{4\pi \left|\mathbf {r} -\mathbf {r} '\right|}}.} The source-free part, B, can be...

{\begin{aligned}y&=\arctan(u)\\\implies u&=\tan(y)&&\quad {\text{ in the range }}-\pi /2<y<\pi /2{\text{ and }}\\\implies u'&=\sec ^{2}y\cdot y'&&\quad {\text{ by applying...

reflection formula Γ ( z ) Γ ( 1 − z ) = π sin ⁡ π z . {\displaystyle \Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin \pi z}}.} However, this formula cannot be...

and their sums (with topical reference to the Newton and Leibniz series for π {\displaystyle \pi } )". Proceedings of the Royal Society. 182 (989): 113–129...

{\begin{aligned}(-2)^{3+4i}&=2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2+3(\pi +2k\pi ))+i\sin(4\ln 2+3(\pi +2k\pi )))\\&=-2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2)+i\sin(4\ln...

derived from their respective arctangent formulas above by setting x = y = 1 and multiplying by 4. The Leibniz formula for π: π = 4 1 + 1 2 2 + 3 2 2 + 5 2...

notation comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities...

} in joules per kilogram, by ∂ 2 φ ∂ x j ∂ x j = 4 π G ρ . {\displaystyle {\partial ^{2}\varphi \over \partial x^{j}\,\partial x^{j}}=4\pi G\rho \,.} Using...

e + π , e π , π e , π π , e π 2 , e e {\displaystyle e+\pi ,e\pi ,\pi ^{e},\pi ^{\pi },e^{\pi ^{2}},e^{e}} ) are themselves transcendental? The four exponentials...

(12 September 2002). Philosophers at War: The Quarrel Between Newton and Leibniz. Cambridge University Press. p. 24. ISBN 978-0-521-52489-6. Retrieved 1...