Math 642 Midterm Review – Spring 2015

Calculus of Variations

• Section 5.1
  1. Know the difference between Frechet and Gateaux derivatives.
  2. Be able to derive the Euler-Lagrange equations, using variational calculations, for consrtained and unconstrained problems that are subject to various boundary conditions.
  3. Know and be able to use the coordinate invariance for the Euler-Lagrange equations. (See class notes for 1/30/15. You will not be asked to prove this.)

• Section 5.2
  1. Hamilton's principle; Lagrangians
  2. Legendre transformations and Hamiltonians
  3. Be able to derive equations of motion for simple mechanical systems, such as a mass subject to a central force (radial potential) or a pendulum.

• Section 5.4
  1. Be able to use variation methods for finding eigenvalues and eigenfunctions in a Sturm-Liouville problem, subject to various boundary conditions.
  2. Rayleigh-Ritz principle
  3. Courant-Fischer minimax theorem

Complex variables and special functions

• Section 6.2
  1. Cauchy's theorem, Cauchy's integral formula
  2. Taylor and Laurent series
  3. Residues and the residue theorem
  4. Isolation of zeroes and applications to deriving identities and extensions of anaytic functions
  5. Maximum principle -- be able to prove it.
  6. Be able to stae the Phragmen-Lindelof theorem and to apply it to problems similar to ones in the homework.
• Section 6.4
  1. Jordan's lemma
  2. Contour integration, with and without cuts
• Section 6.5
  1. Gamma function
     • Definition and extension to ${\mathbb C}\! \setminus \!{\mathbb N}_{\le 0}$
     • Fundamental indentity (Keener, Eqn. 6.25) -- Be able to sketch a proof.)
  2. Bessel functions
     • Solve with method of Frobenius
     • Generating function
     • Integral formulas for Bessel functions with integer order
  3. Legendre polynomials
     • Orthogonality and completeness -- just know these facts
     • Rodrigue's formula and Schläfli's integral formula
     • Generating function

Structure of the exam

The test will have two parts. The in-class part will have 4 or 5 questions. You will be asked to state definitions and to do problems similar to assigned homework problems and examples done in class. In addition, you will be asked prove or sketch a proof for a major theorem or lemma from the material covered by this test. Take-home part will consist of computations and proofs of theorems.

Updated 3/11/15 (fjn).