Elliptic expansions

(A.116) |

It follows that

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

(A.118) |

for (Gradshteyn and Ryzhik 1980b). Moreover,

In particular,

Let us write

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

where is the orbital eccentricity. Hence,

(A.129) |

Comparison with Equation (A.117) reveals that

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

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),

According to Equation (4.69),

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