The multivariate Faà di Bruno formula and multivariate Taylor expansions with explicit integral remainder term

ANZIAM Journal

Volumn 48, Issue 3, 2007, Pages 327-341

  Leipnik, Roy B ^a   Pearce, Charles E M ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF ADELAIDE   (Australia)

Author keywords

Differentiation theory; Fa di Bruno formula; Integral remainder term; Multivariate composite functions; Multivariate Taylor series

Indexed keywords

EID: 36348969994     PISSN: 14461811     EISSN: None     Source Type: Journal    
DOI: 10.1017/s1446181100003527     Document Type: Article

References (24)

1. R. Abraham and J. W. Robbin, Transverse Mappings and Flows (W. A. Benjamin, New York, 1967).
2. L. F. A. Arbogast, Du Calcul des Dérivations (Levrault, Strasbourg, 1800).
3. L. Bieberbach, Differential Geometry (Johnson Reprint Co., New York, 1968).
4. G. M. Constantine and T. H. Savits, "A multivariate Faà di Bruno formula with applications", Trans. Amer. Math. Soc. 348 (1996) 503-520.
5. A. D. D. Craik, "Prehistory of FaA di Bruno's formula", Amer Math. Month. 112 (2005) 119-130.
6. J. A. Dieudonne, Infinitesimal Calculus (Houghton Miffin, New York, 1971).
7. F. Faà Bruno, "Sullo sviluppo delle funzione", Annali di Scienze Matematiche e Fisiche 6 (1855) 479-480.
8. F. Faà di Bruno, "Note sur une nouvelle formule de calcul différentiel", Quart. J. Math. 1 (1857) 359-360.
9. H. Flanders, "From Ford to Faà", Amer Math. Month. 108 (2001) 559-561.
10. R. H. Fowler, Statistical Mechanics (Cambridge University Press, Cambridge, 1936).
11. H. W. Gould, "The generalized chain rule of differentiation with historical notes", Utilitas Mathematica 61 (2002) 97-106.
12. W. P. Johnson, "The curious history of Faà di Bruno's formula", Amer Math. Month. 109 (2002) 217-234.
13. R. Leipnik and T. Reid, "Multivariable Faà di Bruno formulas", Elec. Proc. 9th Ann. Intern. Conf. Tech. Colleg. Math. (1996), available at http://archives.math.utk.edu/ICTCM/ EP-9.html#C23.
14. E. Lukács, "Application of Faà di Bruno's formula in mathematical statistics", Amer Math. Month. 62 (1955) 340-348.
15. R. Mishkov, "Generalization of the formula of Faà di Bruno for a composite function with a vector argument", Intern. J. Math. Sci. 24 (2000) 481-491.
16. S. Noschese and P. E. Ricci, "Differentiation of multivariable composite functions and Bell polynomials", J. Comput. Anal. Appl. 5 (2003) 333-340.
17. Ch. J. de la Vallée Poussin, Cours d'Analyse Infinitésimal (Dover, New York, 1946).
18. G.-C. Rota, "The number of partitions of a set", Amer Math. Month. 71 (1964) 498-504.
19. G.-C. Rota, "Geometric probability", Math. Intell. 20 (1998) 11-16.
20. L J. Schwatt, Introduction to Operations with Series (Chelsea, New York, 1962).
21. L. Takács, Introduction to the Theory of Queues (Oxford University Press, Oxford, 1962).
22. H. Whitney, Geometric Integration Theory (Princeton University Press, Princeton, 1957).
23. C. S. Withers, "A chain rule for differentiation with application to multivariate hermite polynomials", Bull. Austral. Math. Soc. 30 (1984) 247-250.
24. W. C. Yang, "Derivatives are essentially integer partitions", Discr Math. 222 (2000) 235-245.