# 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.