Foundations of Space Dynamics. Ashish Tewari
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2.7.1 Legendre Polynomials
To carry out the integration in Eq. (2.78), it is assumed that the body is entirely contained within the radius
Equation (2.80) is an infinite series expansion in polynomials of
where
(2.82)
with
(2.83)
The first few Legendre polynomials are the following:
(2.84)
Clearly the Legendre polynomials satisfy the condition
By writing
(2.85)
The generating function can be used to establish some of the basic properties of the Legendre polynomials, such as the following:
(2.86)
where the prime stands for the derivative with respect to the argument,
(2.87)
The reciprocal of the generating function,
(2.88)
where the positive sign is taken to correspond to the smaller of the two roots. The choice
(2.89)
or
Since
The differentiation of Eq. (2.91) with
where
(2.93)
The gravitational potential is expressed as follows in terms of the Legendre polynomials by substituting Eq. (2.81) into Eq. (2.78):