Singularities/Residues/Applications 757
where
0 ::=; Argwl/m < 27r/m
How~ver, to w = 0 'there corresponds only one inverse image, namely,
z = 0.
Exercise 9.10
Find the largest neighborhood of the origin where w = f(z) = z^2 + z is
one-to-one. Express z = f-^1 (w) in terms of a series in powers of w.
Bibliography
- E. Badell and M. 0. Gonzalez, Calculo de una integral de fase por el metodo
de variable compleja, Rev. Soc. Cub. Ci. Fis. Mat., 1 (1942), 37-41.
- J. Bass, Exercises in Mathematics, Academic Press, New York, 1966.
- W. R. Derrick, Introductory Complex Analysis and Applications, Academic
Press, New York, 1972.
- J. Farrell and B. Ross, Note on evaluating certain real integrals by Cauchy's
residue theorem, Amer. Math. Monthly, 68 (1961), 151-152.
- B. A Fuchs and B. V. Shabat, Functions of a Complex Variable and Some of
Their Applications, Vol. 1, Pergamon Press, Oxford, 1964.
- I. Glicksberg, A remark on RoucM's theorem, Amer. Math. Monthly, 83
(1976), 186-187.
- F. Gomes Teixeira, Sur les series ordennees suivant les puissances d'une
fonction donnee, J. Reine Angew. Math., 122 (1900), 97-123.
- M. 0. Gonzalez, Una generalizaci6n de la formula de Cauchy y su relaci6n con
el desarrollo en serie de Biirmann, Rev. Cienc. Lima (Peru), 41, 549-553.
- F. P. Greenleaf, Introduction to Complex Variables, W. B. Saunders,
Philadelphia, 1972.
- A. S. B. Holland, Complex Function Theory, Elsevier North-Holland, New
York, 1980.
- A. Hurwitz, Uber die Nullstellen der Besse'shen Funktionen, Math. Ann., 33
(1889), 246-266.
- J. L. Lagrange, Mem. Acad. Berlin, 24 (1770), Oeuvres, 2, 25.
- E. Lindelof, Le Calcul des Residues, Gauthier-Villars, Paris, 1905.