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Prof. D. A. Owen
Physics Dept.
Ben Gurion Univ.
June 6, 2007
-Mathematical Physics (203-2-4221) Problem set #31. (a) Consider the Fourier Series for
+1 0<<π
f() = 
-1 π<<2π
f(  + 2π )=f(). Just to the right of =0, the sum of the first n
looks as shown below.
Find n , the "overshoot" of the first maximum.
(b) Show that
lim n  0.18
n
(Evaluate it precisely). This is know as the Gibbs phenomena.
terms
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2. A function f(x) has the series expansion

f(x)=

cnxn
n!
n=o

Write the function g(y) = cnyn in closed form in terms of f(x).
n=o
3. By using the integral representation of

1
Jo(x)=2π 
 cos(x cos ) d
o
find the Laplace transform of Jo(x).
4. The radioactive nuclei decay successively in series, so that the numbers
Ni(t) of the three types obey the equations
dN1
dt = -1N1
dN2
dt = 1N1 -2N2
dN3
dt = 2N2 -3N3
If initially N1=N, N2=0, N3=n, find N3(t) by using Laplace transforms.
5. A function f(z) has the following properties:
(a) f(z) is analytic except for :(1) a branch line from 0 to + along the
real axis; (2) a simple pole of residue 2 at z=-2.
(b) f(z)  0 as |z| 
(c) f(z) is real on the negative real axis.
x
(d) For x>0, Im f(x+i) =
. Find the function F(x) = f(x+i)
1+x2
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6. Consider the subtracted dispersion relation:
Re F(x) = Re F(xo) +
(x-xo)
π

ImF(x')

P (x'-x )(x'-x)dx'
o

-
(cf. MW 5-20). Suppose that in fact |F(x)|  0 as |x|   .
(a) Derive from the above subtracted dispersion relation the "sum rule"

1
ImF(x') dx'

Re F(0)=π P 
x

-
(b) How can this sum rule be obtained from the unsubtracted dispersion relation

1  ImF(x')
Re F(x) = π P  (x'-x) dx'

-

1  ReF(x')
Im F(x) = -π P  (x'-x) dx' ?

-
(c) Suppose now that F(x) approaches zero at infinity so rapidly that
|xF(x)|  0, |x|  . Derive a further sum rule from the unsubtracted
dispersion relation by analogy with part (a).
7. Find the normal modes and normal frequencies for linear vibrations of the CO 2
molecule (i.e. vibrations in the line of the molecule).
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8. (a) Let a and b be any two vectors in a linear vector space and define
c = a + b where  is as scalar. By requiring that c.c 0 for all , derive
the Cauchy-Schwartz inequality
(a.a)(b.b)  |(a.b)|2
When does the equality hold?
(b) In an infinite-dimensional space, questions of convergence arise
and the expansion x = xiei of an arbitrary vector x in terms of
the
base vectors e1, e2,... may lack meaning. A useful result can,
however, be
derived: Assume ei.ek= ik, define xk=ek.x, and define
n
x(n) =
 xiei .
i=1
the Cauchy-Schwartz inequality (Part (a)) and derive the inequality
n
|xi|2
x.x
i=1
This result which is valid for any n no matter how large, is known as
Bessel's inequality.
Apply