1. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
- Author
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Li Hong, Luo Zhao Sheng, and Xu Hong Yan
- Subjects
nevanlinna theory ,entire function ,system of functional equations ,several complex variables ,30d35 ,35m30 ,Mathematics ,QA1-939 - Abstract
With the help of the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we mainly give a description of the existence and the forms of the transcendental entire solutions of several product-type complex partial differential equations systems (a1fz1+a2fz2)(a3gz1+a4gz2)=eh1,(a1gz1+a2gz2)(a3fz1+a4fz2)=eh2,\left\{\begin{array}{l}({a}_{1}{f}_{{z}_{1}}+{a}_{2}{f}_{{z}_{2}})({a}_{3}{g}_{{z}_{1}}+{a}_{4}{g}_{{z}_{2}})={e}^{{h}_{1}},\\ ({a}_{1}{g}_{{z}_{1}}+{a}_{2}{g}_{{z}_{2}})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{f}_{{z}_{2}})={e}^{{h}_{2}},\end{array}\right. and (a1fz1+a2gz1)(a3fz2+a4gz2)=eh1,(a1fz2+a2gz2)(a3fz1+a4gz1)=eh2,\left\{\begin{array}{l}({a}_{1}{f}_{{z}_{1}}+{a}_{2}{g}_{{z}_{1}})({a}_{3}{f}_{{z}_{2}}+{a}_{4}{g}_{{z}_{2}})={e}^{{h}_{1}},\\ ({a}_{1}{f}_{{z}_{2}}+{a}_{2}{g}_{{z}_{2}})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{g}_{{z}_{1}})={e}^{{h}_{2}},\end{array}\right. where fzt=∂ft∂zt{f}_{{z}_{t}}=\frac{\partial {f}_{t}}{\partial {z}_{t}}, gzt=∂gt∂zt{g}_{{z}_{t}}=\frac{\partial {g}_{t}}{\partial {z}_{t}}, t=1,2t=1,2; aj∈C,j=1,2,3,4{a}_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2,3,4, and h1{h}_{1} and h2{h}_{2} are the polynomials in C2{{\mathbb{C}}}^{2}. More importantly, we list some examples to explain that our results about the forms of solutions of such systems are precise to some extent.
- Published
- 2024
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