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Topics listed below are from Chapters 3–5 (except sec. 4.2) and from Sections 6.1–6.4; they are covered in Homework Sets 4–8.
- Differentiability:
- Definition of differentiable and holomorphic (analytic, in text’s parlance).
- Differentiable implies continuous.
- Cauchy-Riemann equations (you do not need to remember polar form!).
- Testing for differentiability: using Cauchy-Riemann equations or definition of derivative as limit of difference quotient.
- Holomorphic function with zero derivative on a domain is constant.
- 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:
- 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:
- Complex exp: definition via power series and basic properties,including:
- d(exp z)/dz = exp z
- exp(z + w) = (exp z)(exp w)
- Complex Log and multi-valued log: definition and basic properties.
- Complex power function zc.
- Complex sin and cos: definition via power series 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 link to TrigFormulas.pdf on the Notes page about what trig formulas to remember and how, and how to use Mathematica to get others.
- Complex integrals:
- 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, proof, 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.
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