integration

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21: Peter A. Clarkson

… ►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. …

22: Bernard Deconinck

… ►He has worked on integrable systems, algorithms for computations with Riemann surfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations. …

23: 1.1 Special Notation

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$x,y$	real variables.
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$L^2(X,d\alpha )$	the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $d\alpha$.
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24: 2.6 Distributional Methods

… ►Motivated by Watson’s lemma (§2.3(ii)), we substitute (2.6.2) in (2.6.1), and integrate term by term. …Inserting (2.6.2) into (2.6.1) and integrating formally term-by-term, we obtain … ►The Stieltjes transform of $f(t)$ is defined by … ►On substituting (2.6.15) into (2.6.26) and interchanging the order of integration, the right-hand side of (2.6.26) becomes … ►In terms of the convolution product …

25: 14.32 Methods of Computation

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Numerical integration (§3.7) of the defining differential equations (14.2.2), (14.20.1), and (14.21.1).

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26: Mark J. Ablowitz

… ►Widespread interest in Painlevé equations re-emerged in the 1970s and thereafter partially due to the connection with IST and integrable systems. …

27: Mourad E. H. Ismail

… ►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …

28: 2.5 Mellin Transform Methods

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§2.5(i) Introduction

… ►We now apply (2.5.5) with , and then translate the integration contour to the right. … ►Let $f(t)$ and $h(t)$ be locally integrable on $(0,\mathrm{\infty })$ and …Also, let … ►Put $x=1/\zeta$ and break the integration range at $t=1$, as in (2.5.23) and (2.5.24). …

29: 1.4 Calculus of One Variable

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§1.4(iv) Indefinite Integrals

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Integration by Parts

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§1.4(v) Definite Integrals

… ►If the limit exists then $f$ is called Riemann integrable. … ►

Square-Integrable Functions

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30: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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§1.18(ii) $L^2$ spaces on intervals in $ℝ$

… ►For a Lebesgue–Stieltjes measure $d\alpha$ on $X$ let $L^2(X,d\alpha )$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $d\alpha$, … ►We integrate by parts twice giving: … ►Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. … ►The well must be deep and broad enough to allow existence of such $L^2$ discrete states. …