term-by-term integration
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21—30 of 150 matching pages
21: Peter A. Clarkson
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►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations.
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22: Bernard Deconinck
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►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.
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23: 1.1 Special Notation
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real variables. | |
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the space of all Lebesgue–Stieltjes measurable functions on which are square integrable with respect to . | |
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24: 2.6 Distributional Methods
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►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
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►The Stieltjes
transform of is defined by
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►On substituting (2.6.15) into (2.6.26) and interchanging the order of integration, the right-hand side of (2.6.26) becomes
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►In terms of the convolution product
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25: 14.32 Methods of Computation
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26: Mark J. Ablowitz
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►Widespread interest in Painlevé equations re-emerged in the 1970s and thereafter partially due to the connection with IST and integrable systems.
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27: Mourad E. H. Ismail
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►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.
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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 and be locally integrable on and …Also, let … ►Put and break the integration range at , 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
… ►Integration by Parts
… ►§1.4(v) Definite Integrals
… ►If the limit exists then is called Riemann integrable. … ►Square-Integrable Functions
…30: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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