Complex Variables and Applications (8E) - Brown/Churchill. Ex 2, 3. 4 Sec35: Hyperbolic Identities скачать в хорошем качестве

Complex Variables and Applications (8E) - Brown/Churchill. Ex 2, 3. 4 Sec35: Hyperbolic Identities

Complex Variables and Applications (8th Ed) - James Ward Brown and Ruel V. Churchill Ch 3: Elementary Functions 35: Hyperbolic Functions 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 2: Prove that sinh(2z) = 2sinh(z)cosh(z) by starting with (a) definitions sinh(z) = (e^z - e^{-1})/2, cosh(z) = (e^z + e{-1})/2 of sinh(z) and cosh(z); (b) the identity sin(2z) = 2sin(z)cos(z) (Sec. 34) and using relations -sinh(iz) = sin(z), cosh(iz) = cos(z) in Sec. 35. Ex 3: Show how identities cosh^2(z) - sinh^2(z) = 1, cosh(z1 + z2) = cosh(z1)cosh(z2) + sinh(z1)sinh(z2) in Sec. 35 follow from identities sin^2(z) + cos^2(z) = 1, cos(z1 + z2) = cos(z1)cos(z2) - sin(z1)sin(z2) respectively, in Sec. 34. Ex 4: Write sinh(z) = sinh(x + iy) and cosh(z) = cosh(x + iy), and then show how expressions sinh(z) = sinh(x)cos(y) + icosh(x)sin(y), cosh(z) = cosh(x)cos(y) + isinh(x)sin(y), in Sec.35 follow from identities sinh(z1 + z2) = sinh(z1)cosh(z2) + cosh(z1)sinh(z2), cos(z1 + z2) = cos(z1)cos(z2) - sin(z1)sin(z2), respectively, in that section.