test

Test
Starfall
August 14, 2012
1
Questions
1) Consider the quotient rule for differentiation. (Points: 10)
a) Prove the quotient rule, assuming the product rule and the chain rule.
u(x)
u (x)v(x) - u(x)v (x)
b) Hence, deduce that
=
dx.
v(x)
v2(x)
1 - 4x5
c) Hence solve the integral
dx.
(x5 - x + 1)2
2) The Laplace transform of a function is defined by the improper integral

L(f (x)) =
e-stf (t)dt where the Laplace transform is a function of s, often
0
denoted as F (s). (Points: 30)
a) Deduce that the Laplace transform satisfies L(a + b) = L(a) + L(b) and
L(cf ) = cL(f ) where c is a constant.
b) Find the Laplace transform of the first derivative of a function.
c) Find the Laplace transform of the exponential function, eat, where a is a
constant.
dy
d) Hence solve the differential equation
= 3y by transforming the equation
dx
into the s-space, solving it for y, and then transforming y back to the x-space.
1

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3) Consider the Gamma function, (x) =
tx-1e-tdt, and Stirling's asymp-
0

totic formula for it, (x + 1) xx+1/2e-x 2.
a) Prove that (x + 1) = x! when x N.
b) Deduce that the function defined by f (x) = (x)(1 - x) sin(x) is periodic
with period 1.
c) Prove that the function in (2) is analytic.
d) Deduce that f (x) , and moreover use Liouville's theorem for entire
functions to prove that f (x) = .
e) Hence show that (1/2) =
.
2

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