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.
- Harmonic functions:
- 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 exp: definition via power series, how it maps, and basic properties,including:
- 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.