Complex Variables and Applications (8E) - Brown/Churchill Ex 7, Sec74, Ex 9, Sec76: Residues & Poles скачать в хорошем качестве

Complex Variables and Applications (8E) - Brown/Churchill Ex 7, Sec74, Ex 9, Sec76: Residues & Poles

Complex Variables and Applications (8th Ed) - James Ward Brown and Ruel V. Churchill Ch 6: Residues and Poles Remark: Note that different textbooks adopt different notations. In this textbook, arg z is the argument of the complex number z and Arg z is the principal value of the argument of z defined in (-pi, pi]. And log z is the multiple-valued logarithmic function and Log z is the principal branch of log z defined in (-pi, pi]. The relations arg z = Arg z + 2n pi, log z = Log z + i(2n pi), where n ranges over all the integers. 73: Residues and Poles 74: Examples Ex 7: Let z0 be an isolated singular point of a function f and suppose that f(z) = phi(z)/(z - z0)^m, where m is a positive integer and phi(z) is analytic and nonzero at z0. By applying the extended form (6), Sec. 51, of Cauchy integral formula to the function phi(z), show that Res_{z = z0} f(z) = phi^{(m-1)}(z0)/(m-1)!, as stated in the theorem of Sec. 73. 75: Zeros of Analytic Functions 76: Zeros and Poles Ex 9: Let p and q denite functions that are analytic at a point z0, where p(z0) is not equal to 0 and q(z0) = 0. Show that if the quotient p(z)/q(z) has a pole of order m at z0, then z0 is a zero of order m of q.