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Closed-form
New post in Closed-form
Integrals of Pullbacks
August 1st, 2022
differential-forms
closed-form
differential-topology
integration
compact-manifolds
Closed form of series with factorial-squared denominator?
June 9th, 2021
sequences-and-series
poisson-distribution
closed-form
factorial
summation
Evaluate $\int_{0}^1 x^{p}(\log x)^q dx$
October 12th, 2020
definite-integrals
integration
calculus
closed-form
How to find the closed formula for $\sum _{i=0}^n {i q^i}$?
September 20th, 2020
closed-form
geometric-series
summation
Closed form of: $\displaystyle \int_0^{\pi/2}x^{n}\ln{(\sin x)}dx $
September 29th, 2020
calculus
closed-form
logarithms
definite-integrals
integration
How to derive the closed form of the sum of $kr^k$
August 31st, 2020
summation
closed-form
sequences-and-series
How to find values of $f(m,k)=\prod\limits_{n=1}^{\infty}\left[\frac{(mn-k)^2}{(mn-k+1)(mn-k-1)}\right]$?
January 4th, 2020
closed-form
infinite-product
calculus
convergence-divergence
sequences-and-series
Infinite Product $\prod_{n=1}^\infty\left(1+\frac1{\pi^2n^2}\right)$
September 4th, 2020
limits
closed-form
infinite-product
sequences-and-series
analysis
A long nasty limit problem: $\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln 2\pi\right]$
April 16th, 2020
closed-form
proof-writing
limits
calculus
Closed form integral $\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$
February 18th, 2020
calculus
special-functions
closed-form
elliptic-integrals
definite-integrals
Improper integral : $\int_0^{+\infty}\frac{x\sin x}{x^2+1}$
August 1st, 2022
integration
definite-integrals
improper-integrals
closed-form
calculus
How do I prove this $\int_{0}^{\infty}{e^{-x^n}-e^{-x^m}\over x\ln{x}}dx={\ln{\left(m\over n\right)}}?$
August 1st, 2022
integration
definite-integrals
improper-integrals
closed-form
calculus
Closed form expression for the product $\prod\limits_{k=1}^{n}\left(1 - \frac{1}{ak}\right)$
June 7th, 2020
closed-form
sequences-and-series
products
How is Faulhaber's formula derived?
August 1st, 2022
exponentiation
closed-form
summation
Trigonometry Olympiad problem: Evaluate $1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$
August 1st, 2022
trigonometry
closed-form
sequences-and-series
Closed-forms of real parts of special value dilogarithm identities from inverse tangent integral function
August 1st, 2022
polylogarithm
calculus
special-functions
closed-form
Finding the value of the infinite sum $1 -\frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \frac{1}{16} + \frac{1}{19} + ... $
August 1st, 2022
sequences-and-series
integration
calculus
closed-form
limits
Closed form for the sum of even fibonacci numbers?
October 17th, 2020
fibonacci-numbers
closed-form
project-euler
summation
Infinite Series $\sum\limits_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$
August 1st, 2022
ordinary-differential-equations
sequences-and-series
real-analysis
closed-form
complex-analysis
Evaluate $\int\frac{1}{\sin(x-a)\sin(x-b)}\,dx$
October 28th, 2020
trigonometry
calculus
closed-form
indefinite-integrals
integration
Evaluate $\int_{0}^{\frac{\pi}{2}}\frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^2}$
August 1st, 2022
closed-form
definite-integrals
trigonometry
integration
Partial sums of falling factorials
August 1st, 2022
combinatorics
closed-form
factorial
summation
Closed form for integral of integer powers of Sinc function
December 20th, 2020
definite-integrals
integration
closed-form
improper-integrals
special-functions
A closed form for $\sum\limits_n(e-(1+1/n)^n)$
August 1st, 2022
calculus
sequences-and-series
closed-form
Closed-form of $\int_0^1 \frac{\ln^2(x)}{\sqrt{x(a-bx)}}\,dx$
August 1st, 2022
definite-integrals
closed-form
calculus
integration
Express solutions of equation $ \tan x= x $ in closed form
August 1st, 2022
closed-form
roots
trigonometry
sequences-and-series
How to deduce a closed formula given an equivalent recursive one?
March 15th, 2020
recurrence-relations
closed-form
Prove ${_2F_1}\left({{\tfrac16,\tfrac23}\atop{\tfrac56}}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$
August 1st, 2022
special-functions
closed-form
hypergeometric-function
calculus
Only once differentiable
August 1st, 2022
derivatives
closed-form
Difficult trigonometric integral. [Solved]
August 1st, 2022
definite-integrals
trigonometry
integration
closed-form
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