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# Elliptic expansions

The well-known Bessel functions of the first kind, , where is an integer, are defined as the Fourier coefficients in the expansion of : (A.116)

It follows that (A.117)

(Gradshteyn and Ryzhik 1980a). The Taylor expansion of about is (A.118)

for (Gradshteyn and Ryzhik 1980b). Moreover,  (A.119) and  (A.120)

In particular,  (A.121)  (A.122)  (A.123) and  (A.124)

Let us write (A.125)

where is the eccentric anomaly, and the mean anomaly, of a Keplerian elliptic orbit. (See Section 4.11.) It follows that (A.126)

Integrating by parts, we obtain (A.127)

However, according to Equation (4.59), the relationship between the eccentric and the mean anomalies is (A.128)

where is the orbital eccentricity. Hence, (A.129)

Comparison with Equation (A.117) reveals that (A.130)

For the special case , L'Hopital's rule, together with Equations (A.119) and (A.122), yields (A.131)

where denotes a derivative.

The real part of Equation (A.125) gives    (A.132)

where use has been made of Equations (A.120), (A.130), and (A.131). Likewise, the imaginary part of Equation (A.125) yields (A.133)

It follows from Equations (A.121)-(A.124) that   (A.134) and   (A.135)

Hence, from Equation (A.128),   (A.136)

According to Equation (4.69), (A.137)

where is is the radial distance from the focus of the orbit and is the orbital major radius. Thus,   (A.138)

Equations (4.39) and (4.67) imply that (A.139)

where is the true anomaly. Hence, it follows from Equations (A.128) and (A.137), and the fact that when , that (A.140)

From Equation (A.136),   (A.141)

Thus,   (A.142)

and   (A.143)   Next: Matrix eigenvalue theory Up: Useful mathematics Previous: Conic sections
Richard Fitzpatrick 2016-03-31