Complex Variables and Applications (8E) - Brown/Churchill. Ex 29.1, 30.1, 2: Exponential/Logarithmic

Complex Variables and Applications (8th Ed) - James Ward Brown and Ruel V. Churchill Ch 3: Elementary Functions Remark 1: In this video, we use the notations exp z = e^z. However, you will find later that exp z and e^z are different. Remark 2: 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. 29: The Exponential Function Ex 1: Show that (a) exp(2 + 3 pi i) = -e^2, exp(2 - 3 pi i) = -e^2; (b) exp((2 + pi i)/4) = sqrt{e/2} (1 + i); (c) exp(z + pi i) = -exp z. 30: The Logarithmic Function 31: Branches and Derivatives of Logarithms Ex 1: Show that (a) Log(-ei) = 1 - pi i/2; (b) Log(1 - i) = (1/2) ln(2) - pi i/4. Ex 2: Show that (a) log(e) = 1 + 2n pi i, n in Z; (b) log(i) = (2n + 1/2)pi i, n in Z; (c) log(-1 + sqrt{3}i) = ln(2) + 2(n + 1/3)pi i, n in Z.