Common HW 9 errors

The following errors worthy of note were made in your Homework 9 solutions. (See the complete solutions posted to the usual UDrive location.)

  1. Forgot to state explicitly what function is holomorphic where; forgot to name the theorem used (“Cauchy’s Integral Formula”).
  2. Forgot to state explicitly what function is holomorphic where; forgot to name the theorems used (“Cauchy’s Integral Formula for Derivatives” or “Cauchy’s Integral Formula”).
  3. Failed to specify a contour over which the integration takes place [need to say that you use any contour from 0 to z in the disk D1(0)]; forgot to note that the singularities of Arctan, at ±i, are not in the open disk D1(0).
  4. In (a), fallaciously claimed that f is differentiable at 0 because it is continuous at 0 (you need to verify directly that f is differentiable at 0); fallaciously claimed that f is holomorphic at 0 because it is differentiable there (you have to note also that f is differentiable at all z ≠ 0).
  5. On some solutions, found a series expansion on a disk |z − i| < √2 instead of on the “annulus” |z − i| > √2.
  6. (no notable errors)
  7. (l) Forgot that sinh(0) = 0, too [which means the pole z = 0 has order only 5 − 1 = 4, and not 5].
  8. (b) not common, but an error worthy of mention: mistakenly wrote formula as
    lim z →z0 (z − z0)3 (d2/dz2)[f(z)] instead of the correct version lim z →z0 (d2/dz2)[(z − z0)3 f(z)]
  9. no common errors other than miscalculation
  10. no common errors other than miscalculation