Complex Variables and Applications (8E) - Brown/Churchill. Ex 31.6, 7, 8, 9: Branches/Derivatives скачать в хорошем качестве

Complex Variables and Applications (8E) - Brown/Churchill. Ex 31.6, 7, 8, 9: Branches/Derivatives

Complex Variables and Applications (8th Ed) - James Ward Brown and Ruel V. Churchill Ch 3: Elementary Functions 30: The Logarithmic Function 31: Branches and Derivatives of Logarithms 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. Ex 6: Given that the branch log z = ln r + i theta (r greater than 0, theta in (alpha, alpha + 2pi) of the logarithmic function is analytic at each point z in the stated domain, obtain its derivative by differentiating each side of the identity e^{log z} = z, (z not 0) and using the chain rule. Ex 7: Find all roots of the equation log z = i pi/2. Ex 8: Suppose that the point z = x + iy lies in the horizontal stipe y in (alpha, alpha + 2pi). Show that when the branch log z = ln r + i theta (r greater than 0, theta in (alpha, alpha + 2pi) of the logarithmic function is used, log(e^z) = z. Ex 9: Show that (a) the function f(z) = Log(z - i) is analytic everywhere except on the portion x not greater than 0 of the line y = 1; (b) the function f(z) = Log(z + 4) /(z^2 + i) is analytic everywhere except at the points (1 - i)/sqrt{2}, -(1 - i)/sqrt{2} and on the portion x not greater than - 4 of the real axis.