Complex Variables and Applications (8E) - Brown/Churchill. Ex 31.3, 4, 5: Branches of Logarithms скачать в хорошем качестве

Complex Variables and Applications (8E) - Brown/Churchill. Ex 31.3, 4, 5: Branches of Logarithms

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 3: Show that (a) Log(1 + i)^2 = 2Log(1 + i); (b) Log(-1 + i)^2 is not equal to 2Log(-1 + i). Ex 4: Show that (a) log(i^2) = 2log(i) when log(z) = ln r + i theta (r greater than 0, theta in (pi/4, 9pi/4); (b) log(i^2) is not equal to 2log(i) when log(z) = ln r + i theta (r greater than 0, theta in (3pi/4, 11pi/4). Ex 5: Show that (a) the set of values of log(i^{1/2}) is (n + 1/4)pi i, n in Z and that the same true of (1/2)log(i); (b) the set of values of log(i^2) is not the same as the set of values of 2log(i).