Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 67', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 430 Functions of Complex Variable . Cauchy-type integral. Suppose that C is an arbitrary curve without cusps not necessarily closed. Let an arbitrary function f Ç which is assumed to be finite and integrable be given on this curve. The integral F z - f Ç d 2ni Jc Ç - z is called a Cauchy-type integral. The Cauchy-type integral is a function analytic at any point z that does not lie on C . If the curve C divides the plane into several domains then in general the Cauchy-type integral defines different analytic functions in these domains. One says that the function f Ç satisfies the Holder condition with exponent p 1 at a point Ç Ç0 of the contour C if there exists a constant M such that the inequality If Ç - f Ço l M Ç - olM 0 p 1 holds for all points Ç e C sufficiently close to Ç0. The Holder condition means that the increment of the function is an infinitesimal of order at least p with respect to the increment of the argument. The principal value of the integral is defined as the limit lim i - f - r 0JC-c Ç - Ç0 JC Ç - Ç0 where c is the segment of the curve C between the points of intersection of C with the circle z - Ç0I r. The singular integral in the sense of the Cauchy principal value is defined as the integral given by the formula f Ç dÇ f Ç -f 0 . b - Ç0 . 1099 -7 ---7 7-----dÇ f 0 ln---- inf 0 O r Jc-c Ç - Ç0 Je Ç - Ç0 a- Ç0 where a and b are the endpoints of C and O r 0 as r 0. Theorem. If the function f Ç satisfies the Holder condition with exponent p 1 ata point Ç0 which is a regular nonsingular point of the contour C and does not coincide with its endpoints then the Cauchy-type integral exists at this point as a singular integral and its principal value can be expressed in terms of the usual integral by the formula W -1- fFf dç F f ln. 2ni Je Ç - Ç0 2m Je Ç - Ç0 2 2m a - Ç0 If the curve C is closed then a b and formula becomes F P. 1 i f Ç dÇ_ 1 i f Ç -M0 . f Ç0 no99i F Ç0 I .
