1000 Solved Problems in Modern Physics

1.2 Problems 31

1.84 Evaluate

∮

c

4 z^2 − 3 z+ 1

(z−1)^3 dzwhenCis any simple closed curve enclosingz=1.

1.85 Locate in the finitez-plane all the singularities of the following function and
name them:
4 z^3 − 2 z+ 1
(z−3)^2 (z−i)(z+ 1 − 2 i)

1.86 Determine the residues of the following function at the polesz = 1 and
z=−2:
1
(z−1)(z+2)^2

1.87 Find the Laurent series about the singularity for the function:

ex

(z−2)^2

1.88 EvaluateI=

∫∞

0

dx

x^4 + 1

1.2.12 CalculusofVariation...............................

1.89 What is the curve which has shortest length between two points?

1.90 A bead slides down a frictionless wire connecting two pointsAandBas in
the Fig. 1.4. Find the curve of quickest descent. This is known as the Brachis-
tochrome, discovered by John Bernoulli (1696).

Fig. 1.4Brachistochrome

1.91 If a soap film is stretched between two circular wires, both having their planes
perpendicular to the line joining their centers, it will form a figure of revolution
about that line. At every point such asP(Fig. 1.5), the horizontal component
of the surface of revolution acting around a vertical section of the film will be
constant. Find the equation to the figure of revolution.

1.92 Prove that the sphere is the solid figure of revolution which for a given surface
area has maximum volume.