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Exponentials in integrals

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#calculus #integration #exponential-function #gamma-function
209
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$\frac{\Gamma\left(x+\frac{1}{x}+1\right)}{2}\geq \left(\Gamma(x+1)\Gamma\left(\frac{1}{x}+1\right)\right)x^{-\frac{1}{2\pi}\ln\left(x\right)},x>0$

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#inequality #logarithms #gamma-function
70
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Calculation of a derivative of a function related to the Euler Gamma function

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#special-functions #gamma-function #polygamma
31
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Legendre Duplication by Logarithmic Derivatives

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#complex-analysis #gamma-function
115
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Integral of Choose functions

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#integration #combinatorics #gamma-function
36
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Result involving monic polynomial, its roots and the gamma function

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#complex-analysis #polynomials #roots #gamma-function
85
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Behavior of Incomplete Gamma function with negative s, as x goes to zero

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#integration #definite-integrals #power-series #special-functions #gamma-function
72
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How to prove that $\big(\operatorname{sinc}(x) + \operatorname{sinc}(x-1) \big)^{*r} = \frac{\Gamma{(r+1)}}{\Gamma{(x+1)}\Gamma{(r-x+1)}}$

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#binomial-coefficients #gamma-function #convolution #interpolation #hypergeometric-function
31
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Stirlings approximation for $\Gamma(x+\alpha)$, when $\alpha$ is very small but $(x+\alpha)$ is not necessarily small.

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#gamma-function
107
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Looking for a function to solve for $k$ in a binomial coefficient. $n \choose k$

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#binomial-coefficients #gamma-function
80
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Evaluate $\zeta(0)$

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#number-theory #analytic-number-theory #gamma-function #riemann-zeta
60
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Differentiating a Bessel function power series

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#power-series #gamma-function #bessel-functions
117
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Factorial equation and no solution existing :A conjecture

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#inequality #examples-counterexamples #factorial #gamma-function
169
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Conjecture on the representation of $\log$-convex functions

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#convex-analysis #lebesgue-measure #gamma-function #holder-inequality
159
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Does a closed-form expression exist for $ \int_0^\infty \ln(x) \operatorname{sech}(x)^n dx $?

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#integration #closed-form #gamma-function #pi #polygamma

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