Sample final questions
1. Consider y''(x) + k2y(x) = f(x), where k > 0, on the interval 0 ≤ x ≤ π/2k subject to the boundary conditions y(0) = y(π/2k) = 0. Find a Green function solution of the form
y(x) = -∞∫∞ G(x, x')f(x')dx'.
2. Let z = x + iy, and let a and b be positive real numbers. Calculate an equation in terms of x and y for the locus of points that satisfy
(a) r¯z - rz¯ = 0, where r = a + ib,
(b) (z2 + z¯2)(b2 - a2) + 2zz¯(b2 + a2) = 4a2b2.
If a = 2 and b = 1, sketch the loci of points for the two cases.
3. Using Lagrange multipliers, find the maximum volume of a box described by |x| < d, |y| < e, and |z| < f subject to the constraint
d2/a2 + e2/b2 + f2/c2 ≤ 1
where a, b, and c are arbitrary positive constants.
4. (a) Calculate the eigenvalues and eigenvectors of
(b) Show by explicit calculation that A2 = βA for some constant β, and hence show that An = βn-1A for any positive integer n.
(c) The power series for the exponential is exp(x) = n=0∑∞ xn/n!.
Calculate exp(λ A), expressing your answer in the form f(λ)I + g(λ)A where f and g are real functions and I is the identity matrix.
5. For sufficiently small x, the Taylor series expansion of a function f is given by
f(x) = n=0∑∞ (f(n)(0)xn/n!).
(a) By using the definition above, calculate a complete Taylor series expansion for log(1 - x). Determine the precise interval of convergence of the series.
(b) Calculate a power series expansion of log(1 - x2 - x3), keeping terms up to and including x6.
6. By using residue calculus and a keyhole contour, or otherwise, evaluate the integral
0∫∞(x1/3dx/(x + 2)2).
7. The parabolic coordinate system (a, b) can be defined as
x = ab, y = a2 - b2/2.
For a given function f(a, b), calculate expressions for ∂xf and ∂yf in terms of a, b, ∂af , and ∂bf .
8. (a) Let the Laplace transform of a function y(t) by Y(p). Starting from the definition, calculate the Laplace transforms of y' and y'' in terms of Y(p), y(0), and y'(0).
(b) Consider the differential equation
y'' + 9y' + 8y = 0
subject to the boundary conditions y(0) = y'(0) = 1. Calculate an expression for the Laplace transform of the solution, Y(p).
(c) Use the Bromwich inversion integral
y(t) = 1/2πic-i∞∫c+i∞eptY(p)dp
to calculate y(t), where c is a positive constant.
9. (a) Show that if a function f(x) has Fourier transform f˜(α), then the function f'(x) has Fourier transform iαf˜(α).
(b) Consider the differential equation
d2f/dx2 + df/dx - 2f = δ(x),
subject to the boundary conditions limx→±∞ |f(x)| = 0. Calculate the Fourier transform of the solution, f˜(α).
(c) By applying residue calculus to the inverse Fourier transform, determine the solution f(x).
10. Given a Laplace transform F(p), the original function can be calculated using the Bromwich inversion integral
f(t) = 1/2πic-i∞∫c+i∞F(p)eptdp
where c is a positive constant taken to be large enough that the contour passes to the right of any singularities. Using residue calculus, calculate the functions f(t) for the cases of
F1(p) = (p + b/(p + b)2 + a2), F2(p) = 2ap/(p2 + a2)2
where a and b are real constants. Sketch the functions for the cases of a = π and b = 1/2.
11. Use residue calculus to evaluate the integral
-∞∫∞ (cos x/(x2 + 1)(x + i))dx.
12. For a function f that is periodic on the interval -π ≤ x < π, a smoothed function fs can be constructed as
fs(x) = 1/2lx-l∫x+lf(y)dy
where l > 0. With this definition, the value of fs at x is equal to average value of f over the range from x - l to x + l.
(a) Let cn be the complex Fourier series coefficients of f, so that f(x) = -∞∑∞ cneinx. Calculate the complex Fourier series coefficients of fs in terms of cn.
(b) Let an and bn be the Fourier series coefficients of f, so that
f(x) = a0/2 + n=1∑∞an cos nx + n=1∑∞bn sin nx.
By direct calculation, or by using the results from part (a), determine the Fourier series coefficients of fs in terms of an and bn.
13. (a) Suppose that z = x + iy, and the complex function f is defined in terms of real and imaginary components as u(x, y) + iv(x, y). By using the Cauchy-Riemann equations, ux = vy and uy = -vx, determine whether the following functions are analytic:
- f(x + iy) = x3 + y2x - i(yx2 + y3),
- f(x + iy) = x3 - 3y2x + i(3yx2 - y3).
(b) Use the Cauchy-Riemann equations to show that
f(x, y) = ex^2-y^2(cos 2xy + i sin 2xy)
is analytic. Determine f as a function of z.
14. Consider building a tunnel from (x, y) = (-a, 0) to (x, y) = (a, 0) where 0 < a < 1. Let the tunnel's shape be given by y(x) where y ≤ 0. If the cost per unit length of building the tunnel is proportional to √(1 + y), determine the shape of the tunnel that gives a locally minimal cost. Show that for a fixed value of a, there are two tunnel shapes of locally minimal cost.
15. Consider the function
f(x, y) = x2 - xy + 2y2 - 3x - 2y.
Find the minimum and maximum values of f in
(a) the triangle T1 defined by 0 ≤ y ≤ x ≤ 3,
(b) the triangle T2 defined by 0 ≤ x ≤ y ≤ 3.
16. Use residue calculus to evaluate the integral
and hence show that
-π∫π cos2ntdt = π(2n)!/22n-1(n!)2.
Additional practice questions
17. Find the values of l, w, and h that maximize the area of the irregular pentagon shown below, subject to the constraint that the length of the perimeter is P.
18. Let
Sketch f and g. Calculate (f ∗ g)(x) for x ≥ 0, and by symmetry deduce the value of (f ∗ g)(x) for x < 0. Show that f ∗ g is continuous and sketch it.
19. Repeat the previous exercise with an alternative form of g,
20. Let a be a real number where |a| < 1. By starting from the geometric series formula, n=0∑∞an = 1/1-a, show that
n=0∑∞nan = a/(1 - a)2.
Hence determine the complex Fourier series of
f(x) = λeix/(1 - λeix)2
where |λ| < 1. Determine fs as in defined in question 12.
21. The parabolic coordinate system (a, b) can be defined as
x = ab, y = a2 - b2/2.
(a) Determine the radial distance r = √(x2 + y2) in terms of a and b.
(b) Consider a particle of mass m whose position is described by (a(t), b(t)), moving in a potential V = -(m/r). Calculate a Lagrangian L(a, b, a?, b).
(c) Calculate the Euler-Lagrange equations for the a and b coordinates. Show that if C is a constant, then a(t) = C is consistent with the equations, and solve for the corresponding b(t) using the condition b(0) = 0. Express your answer as t(b) where time is viewed as a function of b.