Hints and answers

COMPLEX VARIABLES
24.22
The equation of an ellipse in plane polar coordinates r, θ, with one of its foci at
the origin, is
l
= 1 − cos θ,
r
where l is a length (that of the latus rectum) and (0 < < 1) is the eccentricity
of the ellipse. Express the area of the ellipse as an integral around the unit circle
in the complex plane, and show that the only singularity of the integrand inside
the circle is a double pole at z0 = −1 − (−2 − 1)1/2 .
By setting z = z0 + ξ and expanding the integrand in powers of ξ, find the
residue at z0 and hence show that the area is equal to πl 2 (1 − 2 )−3/2 .
[ In terms of the semi-axes a and b of the ellipse, l = b2 /a and 2 = (a2 −b2 )/a2 . ]
24.1
24.3
24.5
∂u/∂y = −(exp x)(y cos y + x sin y + sin y); z exp z.
(a) 1; (b) 1; (c) 1; (d) e−p .
(a) Analytic, analytic; (b) double pole, single pole; (c) essential singularity, analytic; (d) triple pole, essential singularity; (e) branch point, branch point.
(i) x2 − y 2 , 2xy; (ii) ex cos y, ex sin y; (iii) cosh πx cos πy, sinh πx sin πy;
φ(x, y) = xy + (sinh πx sin πy)/ sinh π.
Assume that pr (x) (r = n, n − 1, . . . , 1) has no roots and then argue by the method
of contradiction.
With 0 ≤ θ1 < 2π and −π < θ2 ≤ π, f(z) = (r1 r2 )1/2 exp[ i(θ1 + θ2 − π) ]. The four
values are ±i(x2 − 1)1/2 , with the plus sign corresponding to points near the cut
that lie in the second and fourth quadrants. I = π/2.
The only pole inside the unit circle is at z = ia − i(a2 − 1)1/2 ; the residue is given
by −(i/2)(a2 − 1)−1/2 ; the integral has value 2π[a(a2 − 1)−1/2 − 1].
Factorise the denominator, showing that the relevant simple poles are at i/2 and
i.
(a) The only pole is at the √
origin with residue π −1 ;
(b) each is O[ exp(−πR 2 ∓ 2πR)
R];
(c) the sum of the integrals is 2i −R exp(−πr2 ) dr.
Use a contour like that shown in figure 24.16.
Note that ρ lnn ρ → 0 as ρ → 0 for all n. When z is on the negative real axis,
(ln z)2 contains three terms; one of the corresponding integrals is a standard
form. The residue at z = i is iπ 2 /8; I = π 3 /8.
24.15 Hints and answers
24.7
24.9
24.11
24.13
24.15
24.17
24.19
24.21
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