• What: Comprehensive final examination
• When: Tuesday, May 25, 3:30-5:30
• Where: Our classroom, Olin 319
• How: You may use the textbook and your notes.

If I were to give a thorough comprehensive final exam, I would probably devise problems to follow our trail through the course: complex number representation, complex arithmetic, algebra in the complex field, functions of a complex variable and ways of "picturing" them, limits, derivatives, integrals, fundamental theorems, and infinite series. But as a practical matter--the exam period is just two hours--I'll have to be selective.

Probable problems

• An insultingly simple problem on complex arithmetic and complex number representation
• A Cauchy-Riemann-equations problem
• Several contour integration problems
  (We've learned several techniques during the semester. Probably the most useful and comprehensive result is Cauchy's residue theorem.)
• At least one series problem

Possible additional problems will relate to these REVIEW TOPICS:

• Complex numbers
  • Rectangular (a + bi) representation
  • Polar representation
• Complex arithmetic: Addition, subtraction, multiplication, division
• Complex-valued functions of a complex variable
• The definition of the derivative
• Calculation of the derivative, f '(z), ...
  • if f(z) is given as a function of z;
  • if f(z) = u(x,y) + i v(x,y) is given;
  • if f(z) = u + i v where u and v are given as functions of the modulus r and argument theta.
• Necessary conditions for differentiability: the Cauchy-Riemann equations
• Sufficient conditions for differentiability: the Cauchy-Riemann equations plus regularity conditions
• Harmonic functions and harmonic conjugates
• Definitions and derivatives of elementary functions of a complex variable
  • Calculation of values
  • Solving equations
  • Mappings of sets
  • Verification of identities
  • Calculation of derivatives
• Integration of a complex-valued function of a real variable
  Fundamental Theorem of Calculus in this case: p. 114 (4)
• Contours and contour integrals: definitions and calculations
• An upper bound on the modulus of a contour integral
• Theorem: Let f be continuous on domain D; f has an antiderivative in D iff all integrals of f along contours in D are path independent iff all integrals of f around closed contours in D are 0.
  Fundamental Theorem of Calculus in this case: p. 136 (1)
• Cauchy-Goursat Theorem: If f is analytic on and inside a simple closed contour C, then its integral around C is zero.
• Tricks with contours
  • Reversing direction: an integral along C is cancelled by an integral along -C.
  • Principle of deformation of paths, p. 152
• Cauchy integral formula
• Theorem: If f is analytic, then f has derivatives of every order.
  Formula: p. 161 (4).
  Handy integration formula: p. 161 (5)
• Neat and curious consequences:
  • The only bounded entire functions are the constant functions.
  • The Fundamental Theorem of Algebra (proved by complex analysis!)
  • The Maximum Modulus Principle: A nonconstant analytic function f does not have a maximum modulus in a domain D, but its modulus will attain a maximum on the boundary B if f is continuous on R := D union B and R is closed and bounded.
• Series
  • Power series, Taylor series, Maclaurin series
  • Series for e^z, sin z, cos z, sinh z, cosh z, 1/(1-z)
  • Laurent series
  • Basic theorems on convergence of series
• Residues and poles
  • Residues
  • Cauchy's residue theorem
  • Types of singular points
  • Residues at poles
  • Zeros and poles