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*)

- d(exp
- Complex Log and multi-valued log: definition, basic properties, how Log maps.
- Complex power function
*z*.^{c} - 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
- sin
^{2}z + cos^{2}z = 1

- exp (
- 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 C
_{R}(*z*_{0}) of 1/(*z*–z_{0}) and of 1/(*z*–z_{0}).^{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.

- Integral over real interval [