COMPLEX VARIABLES
24.22 The equation of an ellipse in plane polar coordinatesr, θ, with one of its foci at
the origin, is
l
r
=1−cosθ,wherelis 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 atz 0 =−^1 −(−^2 −1)^1 /^2.
By settingz=z 0 +ξand expanding the integrand in powers ofξ, find the
residue atz 0 and hence show that the area is equal toπl^2 (1−^2 )−^3 /^2.
[ In terms of the semi-axesaandbof the ellipse,l=b^2 /aand^2 =(a^2 −b^2 )/a^2 .]24.15 Hints and answers
24.1 ∂u/∂y=−(expx)(ycosy+xsiny+siny);zexpz.
24.3 (a) 1; (b) 1; (c) 1; (d)e−p.
24.5 (a) Analytic, analytic; (b) double pole, single pole; (c) essential singularity, ana-
lytic; (d) triple pole, essential singularity; (e) branch point, branch point.
24.7 (i)x^2 −y^2 , 2 xy; (ii)excosy, exsiny; (iii) coshπxcosπy,sinhπxsinπy;
φ(x, y)=xy+(sinhπxsinπy)/sinhπ.
24.9 Assume thatpr(x)(r=n, n− 1 ,... ,1) has no roots and then argue by the method
of contradiction.
24.11 With 0≤θ 1 < 2 πand−π<θ 2 ≤π,f(z)=(r 1 r 2 )^1 /^2 exp[i(θ 1 +θ 2 −π) ]. The four
values are±i(x^2 −1)^1 /^2 , with the plus sign corresponding to points near the cut
that lie in the second and fourth quadrants.I=π/2.
24.13 The only pole inside the unit circle is atz=ia−i(a^2 −1)^1 /^2 ; the residue is given
by−(i/2)(a^2 −1)−^1 /^2 ; the integral has value 2π[a(a^2 −1)−^1 /^2 −1].
24.15 Factorise the denominator, showing that the relevant simple poles are ati/2and
i.
24.17 (a) The only pole is at the origin with residueπ−^1 ;
(b) each is O[ exp(−πR^2 ∓
√
2 πR)];
(c)thesumoftheintegralsis2i∫R
−Rexp(−πr(^2) )dr.
24.19 Use a contour like that shown in figure 24.16.
24.21 Note thatρlnnρ→0asρ→0foralln.Whenzis on the negative real axis,
(lnz)^2 contains three terms; one of the corresponding integrals is a standard
form. The residue atz=iisiπ^2 /8;I=π^3 /8.