Inclination function is a kind of special function commonly-used in thesatellite dynamics. With the technical development and intensive study, theorder of inclination function to be calculated becomes higher and higher. For the high-precision computation of inclination function,this bookletintroduced the available methods, put forward new methods,gave theirFORTRAN program, and studied their stability.
This booklet can be used as a reference for scholars of astronomy and earthscience, and can be also used as a textbook of the graduates.

Preface
1 Introduction
1.1 Introduction of inclination function
1.2 Other definitions of IncFun
1.3 Requirements of satellite dynamics for IncFun
1.3.1 Normalized IncFun and its derivatives
1.3.2 Kernel of IncFun
1.3.3 Calculating order and storage of IncFun
1.4 About this book
2 Expressions of IncFun and its derivative
2.1 Frequently-used notations of IncFun
2.1.1 Normalized IncFun
2.1.2 Quasi-normalized IncFun
2.1.3 Kernel of IncFun
2.1.4 Gooding's notation
2.1.5 Emeljanov's notation
2.2 Series expressions of IncFun
2.2.1 Single summation expression
2.2.2 Dual summation expression
2.2.3 Triple summation expression
2.3 Definite integral expression of IncFun
2.4 Jacobi polynomial expression of IncFun
2.5 Hypergeometric series expression of IncFun
2.5.1 Expressions in three areas
2.5.2 Expression suited to areas A and B
2.5.3 Unified hypergeometric series expression
2.6 d-function expression of IncFun
2.6.1 d-function expression
2.6.2 Expression of inclination matrix element
2.6.3 Timoshkova's expression
2.6.4 Kinoshita's expression
2.6.5 Comparison of four expressions
2.7 Tisserand polynomial expression of IncFun
2.8 Calculating method of the derivatives of IncFun
2.8.1 The 1" method
2.8.2 The 24method
2.8.3 The 3"method
2.9 Primary properties of IncFun
3 Recursion of inclination function
3.1 Classification of recursion
3.1.1 Classification with special function used
3.1.2 Classification with recursion index
3.1.3 Classification with recursive function
3.2 Starting values for recursion of IncFun
3.2.1 Starting values for L-plane recursion
3.2.2 Starting values for M-plane recursion
3.3 Recursion using recursion relations of Legendre polynomial:Giacaglia's
formulae
3.3.1 The first set of formulae
3.3.2 The second set of formulae
3.3.3 The third set of formulae
3.4 Recursion using recursion relations of Jacobi polynomial
3.4.1 Primary recursion relations of Jacobi polynomial
3.4.2 Several practical recursions
3.4.3 Formulae of Allan's recursion
3.4.4 Formulae of Gooding's recursion
3.5 Recursion using recursion formula of hypergeometric series
3.5.1 Important property of recursion formula
3.5.2 Several practical recursions
3.6 Recursion using recursion formula of d-function
3.6.1 Blanco recursion
3.6.2 Risbo recursion
3.7 Stability analysis of recursion
3.8 Preliminary analysis for algorithms with high-precision and high-stability
3.8.1 Difficulty of the Mk(I+)recursion
3.8.2 Methods of overcoming this difficulty
4 Computation method of inclination function
4.1 Analytical method
4.1.1 For case m-k<0
4.1.2 For case m-k≥0
4.1.3 Loss of precision of analytical method and its causes
4.2 Definite integral method
4.2.1 Outline of the method
4.2.2 Computation method of normalized Legendre polynomials
4.2.3 Selection of numerical integral formula
4.3 Kostelecky-Wnuk's method
4.4 Giacaglia's method
4.5 Gooding's method
4.6 Modified Gooding's method
4.7 Emeljanov's method
4.8 Modified Emelianov's method
4.9 Jacobi polynomial method
4.10 d-function method
4.11 L-plane recursion method
5 Computation program
5.1 Analytical method
5.2 Definite integral method
5.3 Kostelecky-Wnuk's method
5.4 Giacaglia's method
5.5 Simplified Gooding's method
5.6 Modified Gooding's method
5.7 Emeljanov's method
5.8 Modified Emeljanov's method
5.9 Jacobi polynomial method
5.10 d-function method
5.11 L-plane recursion method
6 Comparison and evaluation
6.1 Computation accuracy
6.1.1 Assessment standard and accuracy index
6.1.2 Computation results of the accuracy index
6.1.3 Comparison inde