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Über verallgemeinerte Differentialgleichungen von Aczél-Jabotinsky im Komplexen. (Parameterabhängigkeit und holomorphe Fortsetzung von Integralen). (On generalized differential equations of Aczél-Jabotinsky in the complex domain. (Parameter-dependence and holomorphic continuation of integrals.)). (German) Zbl 0731.34004

Author’s summary: “We consider differential equations of the form \[ (1)\quad \frac{dw}{dz}=\frac{\delta w^ m(1+\delta_ 1w+\delta_ 2w^ 2+...)}{\gamma z^ n(1+\gamma_ 1z+\gamma_ 2z^ 2+...)} \] where \(\delta\neq 0\), \(\gamma\neq 0\), m,n\(\geq 1\), and where the power series in the numerator and denominator on the right hand side converge. We are looking for integrals \[ (2)\quad w(z)=\rho z+c_ 2z^ 2+... \] which “enter the singularity (0,0) holomorphically”. Firstly, it is proved, that there exist formal integrals of (1) if and only if certain algebraic relations for the coefficients \(\gamma_ i,\delta_ j\) are satisfied, and if \(m=n\). But then each formal integral (2) is also convergent, hence a locally analytic solution. Moreover, there exists a continuum of solutions of (1) if this equation is solvable in the above sense, since the coefficient \(c_ m(c_ 1=\rho)\) (if \(m=1)\) may be chosen arbitrarily. It is shown that the functions \((z,c_ m)\mapsto w(z,c_ m)\) are holomorphic in regions \(| c_ m| <\eta\), \(| z| <\delta (\eta)\). Here \(w(z,c_ m)\) denotes the unique integral (2) with the given coefficient \(c_ m\), \(\eta >0\) is arbitrary, \(\delta (\eta)>0\) depending on \(\eta\). Applications are given to the third Aczél- Jabotinsky equation in iteration theory and to holomorphic continuation of integrals of (1) into the singularity (0,0).”
Reviewer: G.Frank (Berlin)

MSC:

34M99 Ordinary differential equations in the complex domain
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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