## Math 642 Midterm Review – Spring 2015

The midterm for Math 642 will be held on Friday, March 13. The test
covers these sections from the text: 5.1, 5.2, 5.4; 6.1, 6.2.3, 6.2.4, 6.4, and
6.5.
**Calculus of Variations**

- Section 5.1
- Know the difference between Frechet and Gateaux derivatives.
- Be able to derive the Euler-Lagrange equations, using variational
calculations, for consrtained and unconstrained problems that are
subject to various boundary conditions.
- 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
- Hamilton's principle; Lagrangians
- Legendre transformations and Hamiltonians
- 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
- Be able to use variation methods for finding eigenvalues and
eigenfunctions in a Sturm-Liouville problem, subject to various
boundary conditions.
- Rayleigh-Ritz principle
- Courant-Fischer minimax theorem

** Complex variables and special functions**

- Section 6.2
- Cauchy's theorem, Cauchy's integral formula
- Taylor and Laurent series
- Residues and the residue theorem
- Isolation of zeroes and applications to deriving identities and
extensions of anaytic functions
- Maximum principle -- be able to prove it.
- Be able to stae the Phragmen-Lindelof theorem and to apply it to
problems similar to ones in the homework.

- Section 6.4
- Jordan's lemma
- Contour integration, with and without cuts

- Section 6.5
- 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.)

- Bessel functions
- Solve with method of Frobenius
- Generating function
- Integral formulas for Bessel functions with integer order

- 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).