TAILIEUCHUNG - Handbook of mathematics for engineers and scienteists part 64

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 64', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Basic Notions 409 If f z and g z are analytic functions in a simply connected domain D and z a and z b are arbitrary points of the domain D then the formula of integration by parts holds i f z dg z f b g b - f a g a - i g z df z . a a If an analytic function z g w determines a single-valued mapping of a curve C onto a curve C then y f z dz f g w g w w dw. Cauchy s theorem for a simply connected domain. If a function f z is analytic in a simply connected domain D bounded by a contour C and is continuous in D then fc f z dz 0. Cauchy s theorem for a multiply connected domain. Ifa function f z is analytic in a multiply connected domain D bounded by a contour r consisting of several closed curves and is continuous in D then r f z dz 0 provided that the sense of all curves forming r is chosen in such a way that the domain D lies to one side of the contour. If a function f z is analytic in an n-connected domain D and continuous in D and C is the boundary of D then for any interior point z of this domain the Cauchy integral formula holds f z A i t d . 2ni Jc C - z Here integration is in the positive sense of C . the domain D lies to the left of C. Under the same assumptions as above formula implies expressions for the value of the derivative of arbitrary order of the function f z at any interior point z of the domain fn z i iC f n 1 dC n 1 2 . . 2ni Jc C - z n 1 For an arbitrary smooth curve C not necessarily closed and for a function f C everywhere continuous on C possibly except for finitely many points at which this function has an integrable discontinuity the right-hand side of formula defines a Cauchy-type integral. The function F z determined by a Cauchy-type integral is analytic at any point that does not belong to C. If C divides the plane into several domains then the Cauchy-type integral generally determines different analytic functions in these domains. Formulas and allow .

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