Series Reversion and Applications (UMAP)

This module applies advanced calculus to astronomy and engineering. Five conceptually different techniques for finding the coefficients of a reverted series are given. In the case of certain nonlinear ordinary differential equations, the coefficients of the series to be reverted are derivative operators. Applications, equations of composition, exercises and their solutions show how solutions to complicated functional equations can be simplified by reversion techniques. A FORTRAN program and bibliogrpahy are included. The primary objective of this module is to provide the applications oriented student with a mathematical took which may be useful in solving problems in applied science, engineering, and mathematics.

Table of Contents:

INTRODUCTORY REMARKS

1. PRELIMINARIES AND THE EQUATION OF COMPOSITION

2. FINDING COEFFICIENTS OF REVERTED SERIES BY DIFFERENTIATION

3. THE METHOD OF RESIDUES

4. THE METHOD OF SUCCESSIVE APPROXIMATIONS

5. HISTORICAL NOTE

6. THE USE OF MATRIX ALGEBRA

7. THE SOLUTION OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS

8. REVERSION OF SERIES WITH LEAD COEFFICIENT ZERO

9. SUMMARY AND SUGGESTIONS FOR FURTHER READING

APPENDIX A: MULTINOMINAL EXPANSION THEOREM

APPENDIX B: RECURSION FORMULA, FORTRAN PROGRAM

10. REFERENCES

11. BIBLIOGRAPHY

12. ANSWERS TO EXERCISES