Table Of Integrals Series And Products

By K. Vogler. 2016

13,452 citations recorded by the source corpus.

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In Applied Mathematics

Citations in the corpus

Consistent Kernel Mean Estimation for Functions of Random Variables (2016, neutral): “Thus, using (Gradshteyn and Ryzhik, 2007, 2.321.2)∫ +∞ r=R−‖f‖K rs2−2e−2r dr = e−2(R−‖f‖K) ( s2−2∑ k=0 k!”
Minimax Estimation of Kernel Mean Embeddings (2016, neutral): “…= c−4γ ∫ Rd ( 1 + ∥ ∥ ∥ x c ∥ ∥ ∥ 2 2 )−2γ dx = cd−4γ ∫ Rd ( 1 + ‖x‖22 )−2γ dx = cd−4γ 2πd/2 Γ(d/2) ∫ ∞ 0 ( 1 + r2 )−2γ rd−1dr = cd−4γ πd/2 Γ(d/2) ∫ ∞ 0 (1 + x)−2γ xd/2−1dx = cd−4γ πd/2 Γ(d/2) Γ(d/2)Γ(2γ − d/2) Γ(2γ) , where last identity can be found in (Gradshteyn and Ryzhik, 2000, 3.194.3).”
Posterior propriety in Bayesian extreme value analyses using reference priors (2015, neutral): “…σ ( 1 + ξv σ )−(1+1/ξ) dv, = (−ξ)a/ξ−1σ−a/ξ ∫ 1 0 x−a/ξ(1− x)−(1+1/ξ) dx, = (−ξ)a/ξ−1σ−a/ξΓ(1− a/ξ)Γ(−1/ξ) Γ(1− (a+ 1)/ξ) , (6.2) where we have used integral number 1 in section 3.251 on page 324 of Gradshteyn and Ryzhik (2007), namely ∫ 1 0 xµ−1(1− xλ)ν−1 dx = 1 λ Beta (µ λ , ν ) = Γ(µ/λ)Γ(ν)…”
A new family of distributions with properties, regression models and applications (2018, background): “which is given by = , = 1, , , i i i y z i n μ σ +  (13) where the random error zi has density function (12), = v T i i μ τ is the location parameter of yi, where 1 = ( , , ) , T p τ τ τ  s > 0 and q > 0 are unknown parameters.”
On the Fundamental Tradeoff of Integrated Sensing and Communications Under Gaussian Channels (2022, neutral): “is the digamma function [45], with γ = 0.”
Bayesian Nonparametric Weight Factorization for Continual Learning (2020, methodology): “…the second term, we write the expectation as: In the above equation, the expectation of the log terms can be computed using Gradshteyn and Ryzhik [12] (4.253): The third term involves taking the expectation of log (1 − v α ) which can approximated with a Taylor series: Note that the inﬁnite sum…”
A Tutorial on Near-Field XL-MIMO Communications Toward 6G (2023, neutral): “To resolve the undesired grating lobe issue suffered by modular XL-array, an efﬁcient user grouping method was developed for multi-user transmission scheduling in [129], which avoids allocating the same time-frequency resource block to users located within the grating lobes, so as to maximize the…”
The Burr XII power series distributions: A new compounding family (2015, positive): “…defined by Gm,np,q ( x ∣∣∣∣ a1, . . . , apb1, . . . , bq ) = 1 2πi ∫ L ∏m j=1 (bj + t) ∏n j=1 (1 − aj − t)∏p j=n+1 (aj + t) ∏p j=m+1 (1 − bj − t) x−t dt, where i is the imaginary unit and L denotes an integration path; see Section 9.3 in Gradshteyn and Ryzhik (2000) for a description of this path.”
Performance Analysis of Free-Space Optical Links Over Málaga ($\mathcal{M} $) Turbulence Channels With Pointing Errors (2018, methodology): “The MGF deﬁned as M γ ( s ) E [ e − γs ] , can be expressed, using integration by parts, in terms of CDF as By placing (11) into (15) and utilizing [22, Eq.”
The Dynamics of Learning: A Random Matrix Approach (2018, neutral): “…− 2√c cos θ and with the fact that ∫ π 0 sin2 θ p+q cos θdθ = pπ q2 ( 1− √ 1− q2/p2 ) (see The Dynamics of Learning: A Random Matrix Approach for example 3.644-5 in (Gradshteyn & Ryzhik, 2014)), one gets E = Ẽ +O(α2t2) and V = Ṽ +O(α2t2), where Ẽ ≡ αt 2 g(µ, c) + (‖µ‖4 − c)+ ‖µ‖2 αt = ‖µ‖…”
Classical and Bayesian estimation of $$P(X<Y)$$P(X<Y) using upper record values from Kumaraswamy’s distribution (2014, background): “; .) is Gauss hypergeometric function, see Gradshteyn and Ryzhik (1994)) (formula 3.196(1)).”
Continual Learning using a Bayesian Nonparametric Dictionary of Weight Factors (2021, methodology): “For the second term, we write the expectation as: In the above equation, the expectation of the log term can be computed using Gradshteyn & Ryzhik (2007) (4.253): The third term involves taking the expectation of log (1 − v α ) which can approximated with a Taylor series: Note that the inﬁnite sum…”
Near-Field Communications: A Tutorial Review (2023, methodology): “(10.25 .2)], we have The Laplace transform of f |CN ( h i ,λ i ) | 2 ( x ) is derived as By substituting (259) into (260) and calculating the resulting integral with [100, Eq.”
Exploiting Full/Half-Duplex User Relaying in NOMA Systems (2018, methodology): “where [38, Eq.”
The Horseshoe Estimator: Posterior Concentration around Nearly Black Vectors (2014, background): “The posterior mean can be expressed as: where Φ 1 ( α, β, γ ; x, y ) denotes the degenerate hypergeometric function of two variables (Gradshteyn and Ryzhik, 1965).”