Evaluation of Integrals and Sums Depending on the Functions ѱ, E, F

J. M. Burgers
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J. M. Burgers: University of Maryland, Institute for Fluid Dynamics and Applied Mathematics

Chapter Chapter VI in The Nonlinear Diffusion Equation, 1974, pp 84-123 from Springer

Abstract: Abstract In this chapter we consider integrals of the type 34.1 ∫ − ∞ + ∞ d x 1 ∫ x 1 + ∞ d x 2 Q ( x 2 , x 1 ) E ( x 2 ) Ψ ( x 2 ; x 1 ) F ( x 1 ) , $$\int\limits_{ - \infty }^{ + \infty } {d{x_1}\int\limits_{{x_1}}^{ + \infty } {d{x_2}Q\left( {{x_2},{x_1}} \right)E\left( {{x_2}} \right)\Psi \left( {{x_2};{x_1}} \right)F\left( {{x_1}} \right)} } ,$$ in which Q is a function symmetric in x 2 and — x 1 . Since Ѱ(x 2 ; x 1 )becomes infinite of order (x 2 – x 1 ) -3/2 when x 2 and x 1 approach one another, it is necessary that Q shall vanish at least as (x 2 – x 1 ) [this condition was satisfied in (26.3), (27.1), (27.5)].

Date: 1974
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DOI: 10.1007/978-94-010-1745-9_7

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