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Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 50', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 7.2. Definite Integral 311 7.2.11-2. Principal value of a singular integral. 1 . Consider the integral a c b. Evaluating this integral as an improper integral we obtain fC i dx fb dx b - c E1 L_ ln hm0ln 2 0 2 0 7.2.11.2 The limit of the last expression obviously depends on the way in which e1 and e2 tend to zero. Hence the improper integral does not exist. This integral is called a singular integral. However this integral can be assigned a meaning if we assume that there is some relationship between e1 and e . For example if the deleted interval is symmetric with respect to the point c i.e. E1 E2 E 7.2.11.3 we arrive at the notion of the Cauchy principal value of a singular integral. The Cauchy principal value of the singular integral is the number lim 0 dx x - c a c b With regard to formula 7.2.11.2 we have i b - c ln-------- c - a 7.2.11.4 2 . Consider the more general integral f x . -----dx x - c 7.2.11.5 where f x e a b is a function satisfying the Holder condition. Let us understand this integral in the sense of the Cauchy principal value which we define as follows b ----- dx lim J a x - c 0 We have the identity b f x c X Jc e x - c b x - c f x - f c M z b dx J a x - c j a x - c X C Ja x - c moreover the first integral on the right-hand side is convergent as an improper integral because it follows from the Holder condition that f x - f c x - c A --------ry 0 A 1 x - c 1-y and the second integral coincides with 7.2.11.4 . 312 Integrals Thus we see that the singular integral 7.2.11.5 where f x satisfies the Holder condition exists in the sense of the Cauchy principal value and is equal to f x J -----dx x - c f x - f c 2 r b - c ------------dx f c ln---. x - c c - a Some authors denote singular integrals by special symbols like v.p. J valeurprincipale . However this is not necessary because on the one hand if an integral of the form 7.2.11.5 exists as a proper or an improper integral then it exists in the sense of the Cauchy principal value and their values .
