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The University of British Columbia Math 301 Final Examination - April 2009
- [10] Find all values of:
(a) (i^2 )i
(b) (ii)^2
(c) arcsinh(i)
- [18] Evaluate the integrals; justify all steps carefully (a)
K = p.v
Z ∞
0
x^1 /^3 x^2 − 1 dx
(b)
J =
Z ∞
0
x^1 /^2 log(x) 1 + x^2 dx
- [18] (a) Carefully construct a branch of the function
g(z) = [z(1 − z)]^1 /^2
with a branch cut on the interval 0 ≤ x ≤ 1 such that g( 12 (1 + i)) = − √^12.
(b) Evaluate; justify all steps carefully
J =
Z 1
0
x
(^12) (1 − x)
(^12) dx.
- [18] (a) Find the image of the upper half z−plane (y ≥ 0) under the conformal mapping s = f(z)
s = ξ + iη = f (z) =
z − i z + i
(b) Find the image of the interior of the unit circle in the s− plane under the conformal mapping w = g(s)
w = u + iv = g(s) = s (1 − s)^2
[Hint: first show that the image of the circle is real.]
(c) Simplify the combined mapping, defining G(z)
w = u + iv = g(f (z)) = G(z). (1)
(d) What is the complex potential F (z) of a uniform flow parallel to the x− axis in the upper half z− plane? Use the above mapping to find the image of this uniform flow in the w− plane defined by (1). Represent the images parametrically as u(t), v(t), then eliminate v to give the streamlines in the form u = H(v). Sketch several curves.
5. [18]
(a) Find the Fourier transform, justify all steps carefully
g(x) = e−|x|, − ∞ < x < ∞.
(b) Use a Fourier transform to solve
utt + 2ut + u = uxx, − ∞ < x < ∞, 0 < t, u(x, 0) = e−|x|, ut(x, 0) = 0, − ∞ < x < ∞.
Write the solution in the form
u(x, t) =
Z ∞
0
G(x, t, ω)dω.
6. [18]
(a) Find the inverse Laplace transform, justify all steps carefully
G(x, t) = L−^1 (
e−x
√s √ s
(c) Use a Laplace transform to find the solution to the problem:
ut = uxx, 0 < x < ∞, 0 < t, u(x, 0) = 0 , u(0, t) =
t
, t > 0.
You may use the results Z (^) ∞
0
e−r 2 dr =
π 2
Z ∞
0
u
e−audu =
r π a
