Complex Variables and Applications (8E) - Brown/Churchill. Ex 31.10, 11: Logarithmic Function скачать в хорошем качестве

Complex Variables and Applications (8E) - Brown/Churchill. Ex 31.10, 11: Logarithmic Function

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. Prob 10: Show in two ways that the function ln(x^2 + y^2) is harmonic in every domain that does not contain the origin. Prob 11: Show that Re(log(z - 1)) = (1/2) ln((x - 1)^2 + y^2), for z not 1. Why must this function satisfy Laplace's equation when z is not equal to 1?