The University of Massachusetts Amherst

Exam 2 coverage

Exam 2 (in our LGRT 219 classroom) covers the same content as Homework sets 6–8. Sections from the text, and the topics included, are as follows:

  • Differentiability (Section 3.3):
    • Harmonic functions:
      • Definition in terms of 2nd partials and Laplace’s equation.
      • For holomorphic f on domain D, Re f and Im f are harmonic.
      • Constructing a harmonic conjugate of a harmonic function.
  • Sequences and series (Sections 4.1 and 4.3–4.4):
    • Idea of what convergence of sequences and of series means.
    • Geometric series: where converges and what sum is.
    • Ratio Test.
    • Finding radius of convergence of power series.
  • Elementary functions (Chapter 5):
    • Complex exp: definition via power series, how it maps, and basic properties,including:
      • d(exp z)/dz = exp z
      • exp(z + w) = (exp z)(exp w)
    • Complex Log and multi-valued log: definition, basic properties, how Log maps.
    • Complex power function zc.
    • Complex sin and cos: definition via power series, how they map, and basic properties, including:
      • exp (i z) = cos z + i sin z
      • derivatives of sin and cos
      • sin2 z + cos2 z = 1
    • The other trig functions and the inverse trig functions.
      See the document TrigFormulas.pdf about what trig formulas to remember and how.
  • Complex integrals (Sections 6.1–6.4):
    • Integral over real interval [a, b] of complex-valued function f(t): calculating such integrals and basic properties.
    • Smooth curves, piecewise smooth curves, contours, closed curves, simple closed curves: definitions; parameterizing.
    • Definition of contour integral as a limit of sums.
    • Calculating a contour integral by parameterizing the contour.
    • Statement and use of Cauchy’s Theorem (i.e., Cauchy-Goursat Theorem).
    • Special contour integrals: formulas for integral around CR(z0) of 1/(z–z0) and of 1/(z–z0)n.
    • Deformation of Contour Theorem: statement and use.
    • Extended Cauchy Theorem: statement and use.
    • Fundamental theorems: existence of antiderivatives on simply connected domain; Fundamental Theorem of Calculus to evaluate contour integrals.