Riley hobson bence · Problem 24.22

Applications of complex variables

Mathematics — Methods · complex-integration-ellipse-area

Problem

The equation of an ellipse in plane polar coordinates

r, \theta

, with one of its foci at the origin, is

\frac{l}{r} = 1 - \epsilon \cos \theta,

where

l

is a length (that of the latus rectum) and

\epsilon

(

0 < \epsilon < 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

z_0 = \epsilon^{-1} - (\epsilon^{-2} - 1)^{1/2}

. By setting

z = z_0 + \xi

and expanding the integrand in powers of

\xi

, find the residue at

z_0

and hence show that the area is equal to

\pi l^2 (1 - \epsilon^2)^{-3/2}

. [ In terms of the semi-axes

a

and

b

of the ellipse,

l = b^2/a

and

\epsilon^2 = (a^2 - b^2)/a^2

. ]

Related theory

Complex Integration And Cauchy S Theorem

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