Advanced Calculus fi..

552 Advanced Calculus, Fifth Editioncontaining the points dZ 1. For each z other than dZ 1, one has two chbices of 4-and then an infinite sequence of choices of the logarithm, differing by multiplesof 2ni.PROBLEMS1. Obtain all values of each of the following:a) log2 b) log i c) log(1 - i ) d) ife) (1 + i)'" f) iJZ g) sin-' 1 h) cos-' 22. Prove the formula (8.43). [Hint: If w = sin-' z, then 2iz = elw - e-lW; multiply by efand solve the resulting equation as a quadratic for elW.]3. a) Evaluate sin-' 0, cos-' 0.b) Find all roots of sin z and cos z [compare part (a)].4. Show that each branch of log z is analytic in each domain in which 0 varies continuouslyand that(dldz) log z = 112.[Hint: Show from the equations x = r cos8, y = r sin0 that a0lax = -y/r2, a8/ay =x/r2. Show that the Cauchy-Riemann equations hold for u = log r, v = 8.15. Prove the following identities in the sense that for proper selection of values of the multiplevaluedfunctions concerned the equation is correct for each allowed choice of the variables:a) log(zl . z2) = log zl + log z2 (zi # 0, z2 # 0)b) elogz = z (Z # 0)C) log eZ = zd) log z1z2 = z2 log zi (zi # 0)6. For each of the following, determine all analytic branches of the multiple-valued functionin the domain given:a) logz, x c 0 b) fir x>O7. Prove that for the analytic function za (principal value),8. Plot the functions u = Re ( A)and v = Im (A) as functions of x and y and show thetwo branches described in the text.9. Let w = u + i v = f (z), z = x + iy, be analytic in domain D.a) Show that a(u, v)/a(x, y) =' 1 f'(z)12.b) Apply the inverse mapping theorem of Section 2.12 to conclude that if fl(zo) # 0,where zo is in D, and f (zo) = wo, then there is an inverse z = g(w) of f defined in aneighborhood Do of wo, mapping Do onto a neighborhood of zo.C) Show that g(w) is analytic in Do. [Hint: Find the corresponding Jacobian matrixand show that the Cauchy-Riemann equations in u, v are satisfied by x(u, v) =Re dw), y(u, v) = Im g(w).ld) Show that gl(w) = llf '(z) at w = f (z) and that gl(w) # 0 in Do.