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**Nevanlinna theory and complex differential equations.**
*(English)*
Zbl 0784.30002

de Gruyter Studies in Mathematics. 15. Berlin: W. de Gruyter. viii, 341 p. (1992).

Differential equations in the complex domain is an area of mathematics admitting several ways of approach. The author’s aim is to show how the Nevanlinna theory can be applied to get insight into the global properties of solutions of ordinary differential equations.

The material of the present book has been organized as follows. The first three chapters contain some background material from function theory, especially the two main theorems of Nevanlinna. The next five chapters are devoted to linear differential equations, and the last six chapters consider nonlinear problems.

In the nonlinear part we can find following subjects: The Riccati differential equation, the Painlevé equations, the Schwarzian differential equation, the theorem of Malmquist and generalizations of this classical result, Hölder’s theorem on the gamma function. It is interesting to compare the corresponding parts in L. Bieberbach’s book on this area and to see the recent progress.

The material of the present book has been organized as follows. The first three chapters contain some background material from function theory, especially the two main theorems of Nevanlinna. The next five chapters are devoted to linear differential equations, and the last six chapters consider nonlinear problems.

In the nonlinear part we can find following subjects: The Riccati differential equation, the Painlevé equations, the Schwarzian differential equation, the theorem of Malmquist and generalizations of this classical result, Hölder’s theorem on the gamma function. It is interesting to compare the corresponding parts in L. Bieberbach’s book on this area and to see the recent progress.

Reviewer: F.Gackstatter (Berlin)

### MSC:

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |

34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |