Reference no: EM131077626

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 = β^n-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∫^∞(x^1/3dx/(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∞e^ptY(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 lim_x→±∞ |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)e^ptdp

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 f_s can be constructed as

f_s(x) = 1/2l_x-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 c_n be the complex Fourier series coefficients of f, so that f(x) = _-∞∑^∞ c_ne^inx. Calculate the complex Fourier series coefficients of f_s in terms of c_n.

(b) Let a_n and b_n be the Fourier series coefficients of f, so that

f(x) = a₀/2 + _n=1∑^∞a_n cos nx + _n=1∑^∞b_n sin nx.

By direct calculation, or by using the results from part (a), determine the Fourier series coefficients of f_s in terms of a_n and b_n.

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, u_x = v_y and u_y = -v_x, 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) = e^{x^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)!/2^2n-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∑^∞a_n = 1/1-a, show that

_n=0∑^∞naⁿ = a/(1 - a)².

Hence determine the complex Fourier series of

f(x) = λe^ix/(1 - λe^ix)²

where |λ| < 1. Determine f_s 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.