p. 667: 18.1, 18.2, 18.3, 18.4, 18.5, 18.16, 18.17

Solutions to HW #2: *.ps  *.pdf

1. Evaluate the integrals of exp(1/z), sin(1/z) and cos(1/z) around the contour |z|=1.
   
   
2. Evaluate the integral using contour integration on an appropriately chosen contour.
   
   
3. Evaluate the integral and show that it is equal to . Thence evaluate the sums of n^–2 and n^–4 from n=1 to infinity. The P means “take the Cauchy principal value”.

Due: 10/17/2000, 17:00 my office (Rm 312D)

Solutions to HW #3: *.ps  *.pdf

1. What is the differential equation satisfied by the function
   y(x) = x^a J_±m(bx^g ) ?
   Use this result to find the eigenvalues of the swinging chain. (This is a uniform suspended chain whose lower end is free—thus the restoring force is gravitational. The displacement of a point along the chain obeys a linear, second-order pde that we discussed in class:
   where x=0 is the lower—free—end of the chain and x=L is the upper end.)
   
   
2. Energy in a star is produced in nuclear reactions initiated by collisions. If the number of collisions per unit time, of particles with CM kinetic energy between E and E+dE is NEe^–E/Q, where N is a constant and Q=k_BT is the absolute temperature in energy units; and if the probability that a collision will result in a reaction is Me^–a/ÖE (again M and a are constants) use the method of steepest descents to estimate the rate of nuclear reactions assuming (Q/a²)^1/6«1.
   
   
3. Apply the Gram-Schmidt orthogonalization method to the monomials x⁰, x¹, and x² to derive the first 3 Hermite polynomials H₀, H₁, and H₂, where these polynomials are orthogonal on the interval (–¥, +¥) with respect to the weight function e^–x². Do not bother to normalize the polynomials.
   
   
4. The Schrödinger equation of the hydrogenic atom is
   
   Estimate the ground state energy using the trial function e^–lr².
   
   
5. Solve the integral equation
   
   
   
6. Consider the operator defined by the kernel
   
   
   1. Show that it is Hermitian.
   2. Show that its eigenvalue spectrum is countably infinite.
   3. Show that it represents a positive-definite operator.
   4. Show it is bounded.
   5. Show that it is compact.
   
   
   
7. Evaluate the integral
   
   for arbitrary values of the dimensionality d. Hint: evaluate the angular factor, W(d), using the identity
   
   and using the definition of the G function.
   
   What is the nature of the singularity as d goes to 4? (The parameter L is a “running scale” with dimensionality the same as D.)
   You will need to use the properties of the G function for this.
   

Due: December 6, 2000, 16:00 my office (Rm 312D)

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