Symbolic asymptotics: Functions of two variables, implicit functions
This paper presents an algorithmic approach to finding the limiting asymptotic behaviour of exp-log functions of two variables and hence of implicit functions of one variable. We work in \({\mathcal H}(x,y)\), the field obtained by closing \(\mathbb{K}(x,y)\) under application of exp and log, where \(\mathbb{K}\) is the field of real elementary constants. Proof that the algorithm works and terminates relies on the theory of Hardy fields. A Hardy field is a subring of the ring of germs at \(\infty\) of \(C^\infty\) functions which is a field closed under differentiation. It can be ordered by setting \(f>g\) when \(f(x)-g(x)>0\) for sufficiently large \(x\). Elements of Hardy fields tend to (possibly infinite) limits. Two elements \(f\) and \(g\) of a Hardy field tending to infinity are said to be comparable if there exists a positive integer \(n\) such that \(f<g^n\) and \(g<f^n\); thus the non-zero elements can be decomposed into comparability classes. A partial nested form of a positive function tending to zero or infinity consists essentially of a composition of exponentials and logarithms, whose length is essentially its nesting depth. A nested form is a partial nested form satisfying two extra conditions, and a nested expansion is obtained by repeatedly giving a nested form for the residual ``\(o(1)"\) part. Let \(h(x,y)\in {\mathcal H}(x,y)\) and suppose the continuous real function \(y(x)\) satisfies \(h(x,y (x))=0\). Then it is proved that \(y\) has a nested form whose length is bounded in terms of the structure of \(h\). For handling functions with finite limits, the \(z\)-function notation of [\textit{J. Shackell}, J. Symb. Comput. 10, No. 6, No. 6, 611-632 (1990; Zbl 0727.26002)], is used, where \(z\exp_n(t)\), \(z\log_n(t)\) and \(z\text{pow}_n(r,t)\) give essentially Taylor remainder terms of order \(t^{n+1}\) divided by \(t^n\), which thus have linear zeros at \(t=0\). Asymptotic estimates as \(x\to \infty\) of bivariate exp-log functions \(h(x,y(x))\) depend on the relative growth of \(x\) and \(y\). For example, \(\exp(xy)-x \sim\exp(xy)\) if \(y\nrightarrow 0\) or \(\lim\log y/\log x>-1\), otherwise \(\exp(xy)-x\sim-x\) unless \(\lim\log y/\log x=-1\), in which case the estimate is denoted ``?''. The latter is important because it is the only case in which \(h(x,y)=0\) can define \(y(x)\) implicitly. Formally a bivariate asymptotic estimate is a set of pairs (condition, expression) such that the conditions are mutually exclusive and together cover all the possibilities. The conditions involve comparability classes together with conditions on limits of logarithms, and expressions are either question marks or monomials in elements of \({\mathcal H}(x,y)\), which can be obtained using \(z\)-functions. A constructive proof that a bivariate asymptotic estimate can be computed is given by building a tower of differential fields \(\mathbb{K}(x,y)= {\mathcal F}_0\subset {\mathcal F}_1\subset \cdots\subset {\mathcal F}_k\). By working up the tower, a set \({\mathcal D}_j= \{({\mathcal T}^j_i, {\mathcal C}^j_i)\), \(i=1,\dots, \beta_j\}\) is obtained for each \(j=0,\dots,k\). Each \({\mathcal T}_i^j\) is a set of elements of \({\mathcal F}_j\) such that \({\mathcal F}_j\) is contained in the set of \(z\)-functions on \({\mathcal T}^j_i\), and each \({\mathcal C}_i^j\) is a conjunction of conditions as above where \(i\) labels all the mutually exclusive cases. Details and examples of the construction algorithm are given. The bivariate algorithm leads to the following algorithm to find the asymptotic behaviour of implicit functions. Input \(h(x,y)=0\), where \(h\) is a bivariate asymptotic estimate of \(h\). 1. Compute a bivariate asymptotic estimate of \(h\). 2. Select the conditions that lead to a question-mark estimate. 3. Use these conditions to either compute a partial nested form of the solution or reduce the problem to a simpler one and iterate. 4. In order to calculate the remaining terms in the sequence which is the nested form, make the first of these a new dependent variable. Then substitute for \(y\) and iterate. It is proved that the algorithm finds nested expansions of all the continuous real solutions \(y(x)\) of \(h(x,y(x))=0\). (It may also produce non-solutions, which can usually be rejected by increasing the order of the computation.) Examples are given.
scientific article; zbMATH DE number 1204279