Advanced Calculus fi..

PROBLEMSChapter 8 Functions of a Complex Variable 5491. Verify that the following are analytic functions of z:a) 2x3 - 3x2y - 6xy2 + y3 + i(x3 + 6x2y - 3xy2 - 2y3)b) w =eZ =eXcosy +iexsinyC) w = sinz = sinxcoshy + i cosx sinhy2. Test each of the following for analyticity:a) x3 + y3 + i(3x2y + 3xy2)b) sinxcosy +icosxsinyc) 3x + 5y + i(3y - 5x)3. Prove the following properties directly from the definitions of the functions: .. .a) $- ez = ezdb, az sinz =cosz, zcosz = -sinzc) sin (z + n) = -sin z d) sin (-z) = -sin z, cos (-z) = cos z4. Prove the identity eZ1+Z2 = eZl . eZ2 by application of Theorem 11. [Hint: Let z2 = b,a fixed real number, and zl = z, a variable complex number. Then eL+b = eL - eb is anidentity connecting analytic functions which is known to be true for z real. Hence it istrue for all complex 2. Now proceed similarly with the identity eZ1+Z = eZ1 . eZ.]5. Prove the following identities by application of Theorem 11 (see Problem 4):a) cos (zl + z2) = cos ZI cos z2 - sin zl sin z2b) elz = cos z + i sin zC) (eZ)" = enZ (n = 0, 1,2, . . .)6. Determine where the following functions are analytic (see Problem 3 followingSection 8.2):sin zb) CO~Za) tanz = ,= cos z c) tanh z = sinh zd) ~iy e) i-z~ ez8.7 THE FUNCTIONS log z , aZ , za , sin-' z , cos-' zeLf) sinz+coszThe function w = log z is defined as the inverse of the exponential function z = ew.We write z = reie, in terms of polar coordinates r, 0, and w = u + iv, so thatreiQ - e ~+i~ = eueiv,'11 Ni ~7Accordingly,w =,log2 = logr + i(8 + 2kn) = log lzl + i argz, (8.37)where log r is the real logarithm of r. Thus log z is a multiple-valued function ofz, with infinitely many values except for z = 0. We can select one value of 0 foreach z and obtain a single-valued function, log z = log r + i8; however, 8 cannot bechosen to depend continuously on z for all z # 0, since 0 will increase by 217 eachtime one encircles the origin in the positive direction.If we concentrate on an appropriate portion of the z-plane, we can choose 8 tovary continuously within the domain. For example, the inequalities>UK