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Analytic Structure of the Solution of Equation (2.5)
A. N. Erugin
§ 1. Preliminary considerations, definitions, notation
Let us give the basic definitions and notation. We shall call a \(T\)-function
\[ T(t)=\sum_{s=1}^{\infty}(a_s\sin st+b_s\cos st), \]
where
\[ \sum_{s=1}^{\infty}(|a_s|+|b_s|)<\infty, \]
and a \(K\)-function \(a_0+T(t)\), \(a_0=\operatorname{const}\ne0\).
Furthermore, it can be shown that if a certain \(T\)-function \(T(t)\) and certain constants \(s, g, c\) are given, then
\[ \underbrace{\int\cdots\int}_{l} T(t)E^s\underbrace{dt\cdots dt}_{l} = E^s\sum_{n=1}^{\infty} \frac{a_{n,l}\sin nt+b_{n,l}\cos nt}{(n^2+g^2s^2)^l}+C, \]
\[ \underbrace{\int\cdots\int}_{l} cE^s\underbrace{dt\cdots dt}_{l} = E^s\frac{C}{(gs)^l}+C, \tag{1.1} \]
where \(C\) is an arbitrary constant, \(E=e^{gt}\), \(a_{n,l}=a_{n,l-1}gs+b_{n,l-1}n\), \(b_{n,l}=b_{n,l-1}gs-a_{n,l-1}n\), \(a_{n,0}=a_n\), \(b_{n,0}=b_n\). Therefore it is clear that if a certain \(K\)-function \(G(t)\) is given, then
\[ \underbrace{\int\cdots\int}_{l} E^sG(t)\underbrace{dt\cdots dt}_{l} = E^s(G)_l+C, \tag{1.2} \]
where \((G)_l\) is a certain \(K\)-function which is readily obtained from formulas (1.1). The proof of the formulas is given by the method of integration by parts.
Next, if for \(t\ge t_0\), \(|G(t)|\le C_1=\operatorname{const}\), then one obtains the estimate, for \(t\ge t_0\), \((g<0)\),
\[ \left|\int_{t''}^{t'} E^sG(t)\,dt\right| \le \frac{2C_1}{|g|s}\,(E(t_0))^s, \qquad (t',\,t''\ge t_0). \]
\(C_1\)
We obtain an estimate for the coefficients of the binomial series, which we shall need later in the proof. Let
\[ \alpha_n=\mathrm{const}, \qquad \alpha_n<\alpha_{n+1}, \qquad (n=1,2,\ldots). \]
Denote
\[ C_{n,s}=\frac{1}{S!}\prod_{l=0}^{s-1}(\alpha_n-l). \]
It is then easy to see that \(s_n\leqslant \alpha_n<S_n=s_n+1\), where \(s_n=\operatorname{Ent}(\alpha_n)\). Moreover, there are two cases:
\[ 1)\ s>s_n+1,\qquad 2)\ s\leqslant s_n+1. \]
For \(s>s_n+1\) we obtain
\[ |C_{n,s}|<\frac{1}{s!}\prod_{\substack{l=1\\ l\ne s_n+2}}^{s}|s_n+2-l| <\frac{(s-s_n)!}{s!}(s_n+1)!<1. \]
Next, for \(s\leqslant s_{n+1}\), applying Stirling’s formula,
\[ n!=\sqrt{2\pi n}\cdot\left(\frac{n}{e}\right)^n\cdot \xi(n),\qquad \xi(n)=\exp\left(\frac{\vartheta}{12n}\right),\qquad 0\leqslant \vartheta<1,\quad \vartheta=\mathrm{const}, \]
and also taking into account that for arbitrary \(S_n\) and \(s\), \((s\leqslant S_n)\),
\[ \left| \frac{\sqrt{S_n}}{\sqrt{(S_n-s)s}} -\frac{\xi(S_n)}{\xi(S_n-s)\xi(s)} \right| \leqslant \mathrm{const}, \]
we shall have
\[ C_{n,s}\leqslant \frac{S_n!}{s!(S_n-s)!} \leqslant c\left(1+\frac{s}{S_n-s}\right)^{S_n-s} \left(1+\frac{S_n-s}{s}\right)^s \leqslant \]
\[ \leqslant c\exp(S_n-s)\exp s=c\exp S_n,\qquad c=\mathrm{const}. \]
Therefore, in both cases the estimate obtained is
\[ |C_{n,s}|\leqslant C_2\exp\alpha_n,\qquad C_2=\mathrm{const}. \tag{C_2} \]
§ 2. Equation of the integral curves
Let us be given the system
\[ x'=\sum_{n=0}^{\infty}\varphi_{1,n}(x,y)=X(x,y),\qquad y'=\sum_{n=0}^{\infty}\varphi_{2,n}(x,y)=Y(x,y), \tag{2.1} \]
where
1) the series \(X,Y\) converge absolutely and uniformly for \(x^2+y^2=r^2\leqslant r_0^2=\mathrm{const}\);
2) \(\varphi_{s,n}(r\cos\vartheta,r\sin\vartheta)=r^{\alpha_{s,n}}\varphi_{s,n}(\cos\vartheta,\sin\vartheta)\), \(\alpha_{s,n}=\mathrm{const}\), \((\varphi_{1,0}(\cos\vartheta,\sin\vartheta))^2+(\varphi_{2,0}(\cos\vartheta,\sin\vartheta))^2\ne0\), \(\alpha_{1,0}=\alpha_{2,0}\), \(\alpha_{s,n}<\alpha_{s,n+1}\), \((n=0,1,2,\ldots)\), \((s=1,2)\);
3) \(\varphi_{s,n}(\cos\vartheta,\sin\vartheta)\) are \(K\)-functions \((s=1,2)\);
4) for all \(\vartheta\),
\[
F(\vartheta)=\varphi_{2,0}(\cos\vartheta,\sin\vartheta)\cos\vartheta
-\varphi_{1,0}(\cos\vartheta,\sin\vartheta)\sin\vartheta\ne0,
\]
i.e. the characteristic equation of the system has complex roots.
Passing to polar coordinates, we obtain the equation of the integral curves of system (2.1) in the form
\[ r'=Q_0(\vartheta)r+\sum_{s=1}^{\infty}Q_s(\vartheta)r^{1+\alpha_s}=Q(r,\vartheta), \tag{2.2} \]
where \(Q_0, Q_s\) are \(K\)-functions, \(a_n\) are constants, \(a_n<a_{n+1}\), and the series \(Q(r,\vartheta)\) converges absolutely and uniformly for \(r\le r_0\). We shall have two cases:
\[ \begin{aligned} \text{I.}\quad & g \equiv \int_0^{2\pi} Q_0\,d\vartheta \ne 0,\\ \text{II.}\quad & g \equiv \int_0^{2\pi} Q_0\,d\vartheta = 0. \end{aligned} \tag{2.3} \]
Let us note that this means that in the case \(g=0\), \(Q_0\) is a \(T\)-function, while in the case \(g\ne0\), \(Q_0\) is a \(K\)-function, and the integral \(\int_{\vartheta_0}^{\vartheta} Q_0\,d\vartheta\) in the case \(g=0\) is a \(K\)-function, while in the case \(g\ne0\) it is the nonperiodic function \(g\vartheta+\Theta\), where \(\Theta\) is a \(K\)-function.
Passing to new variables by the formula
\[ r=z\exp\int_{\vartheta_0}^{\vartheta} Q_0\,d\vartheta, \tag{2.4} \]
we obtain, in the case \(g=0\), an equation whose structure coincides with that of the equation of integral curves of system (2.1), which has purely imaginary roots of the characteristic equation [4], while in the case \(g\ne0\) we obtain the equation of the integral curves in the form
\[ z'=\sum_{s=1}^{\infty} Z_s(\vartheta)\,E^{1+\alpha_s} z^{1+\alpha_s}\equiv Z(z,\vartheta), \tag{2.5} \]
where \(E=e^{g\vartheta}\), \(Z_s=Q_s\exp(1+\alpha_s)\Theta\), i.e., \(Z_s\) is a \(K\)-function. Note that for any sign of \(g\) there is one analytic structure.
§ 3. Analytic structure of the solution. On estimating series of the right-hand side of the equation
We proceed to study the analytic structure of the solution of equation (2.5). We shall seek the solution of equation (2.5) in the form
\[ z=C\left(1+\sum_{n=1}^{\infty} u_n(\vartheta) C^{\gamma_n}\right) = C(1+S(C,\vartheta)), \tag{3.1} \]
where \(C\) is an arbitrary constant, \(\gamma_n\) are constants,
\[ \gamma_n<\gamma_{n+1},\qquad (n=1,2,\ldots). \]
Substituting (3.1) into (2.5), we shall have
\[ z'=C\sum_{n=1}^{\infty} u_n' C^{\gamma_n} = \sum_{n=1}^{\infty} Z_n(EC)^{1+\alpha_n}(1+S(C,\vartheta))^{1+\alpha_n} = \]
\[ = \sum_{n=1}^{\infty} Z_n(EC)^{1+\alpha_n} \left(1+\sum_{s=1}^{\infty} C_{n,s}S^s\right) = \sum_{n=1}^{\infty} Z_n(EC)^{1+\alpha_n} \left(1+\sum_{s=1}^{\infty} U_{n,s}C^{\beta_s}\right) = \]
\[ = \sum_{n=1}^{\infty} U_n C^{\delta_n} = U(C,\vartheta), \tag{3.2} \]
where \(U_{n,s}\) are polynomials in \(u_l\), and
\[ U_{n,s} = U_{n,s}(u_l), \quad (l = 1, 2, \ldots, s), \]
\(U_n\) are polynomials in \(u_l\), and
\[ U_n = U_n(u_l) \quad (l = 1, 2, \ldots, n-1), \]
\(\beta_n, \delta_n\) are linear combinations of the numbers \(1, \alpha_1, \alpha_2, \ldots, \alpha_n\), \(U_n\) \((n \geqslant 1)\) are generalized polynomials in \(E\), i.e.
\[ U_n = \sum_{l=1}^{s_n} \widetilde Q_{n,l}(\vartheta) E^{s_{n,l}}, \tag{3.3} \]
where \(\widetilde Q_{n,l}\) are \(K\)-functions, \(s_n\) are integers,
\[ s_{n,1} = \mathrm{const}, \quad 0 < s_{n,1} < s_{n,l+1}, \quad s_{n,l} \leqslant s_{n,s_n}, \quad (n \geqslant 1,\; 1 \leqslant l \leqslant s_n), \]
and, as \(n \to \infty\), \(s_n, s_{n,s_n} \to \infty\).
Equating the exponents and the coefficients of \(C\), we obtain an infinite system of algebraic equations for determining \(\gamma_n\) and an infinite system of differential equations for determining \(u_n\):
\[ 1 + \gamma_n = \delta_n \quad (n \geqslant 1), \tag{3.4} \]
\[ u_n' = U_n(\vartheta) \quad (n \geqslant 1), \tag{3.5} \]
from which \(\gamma_n, u_n\) are determined successively, because
\[ \delta_n = \delta_n(\alpha_s, \gamma_l), \quad U_n = U_n(Z_s, u_l), \quad (s \leqslant n,\; l \leqslant n-1). \]
Moreover, it is most convenient to find \(u_n\) with the initial data
\[ u_n(\vartheta_0) = 0 \quad (n \geqslant 1), \tag{3.6} \]
because then we shall have \(Z(C,\vartheta_0) = C = z_0\). Thus, the implementation of the first method is obtained.
The structure and properties of a function of the form (3.3) are preserved under integration by virtue of formulas (1.1), (1.2); therefore \(u_n\) is obtained in the form
\[ u_n = \sum_{l=1}^{s_n} Q_{n,l}(\vartheta) E^{s_{n,l}}, \tag{3.7} \]
where \(Q_{n,l}\) are \(K\)-functions, \(s_n\) are integers,
\[ s_{n,l}, \quad Q_{n,0} = \mathrm{const}, \quad 0 = s_{n,0} < s_{n,1} < s_{n,l+1}, \quad s_{n,l} \leqslant s_{n,s_n} \quad (n \geqslant 1,\; l \leqslant s_n), \]
and, as \(n \to \infty\), \(s_n, s_{n,s_n} \to \infty\).
Next, we note that it is not difficult to prove that if the series converges absolutely and uniformly for \(z \leqslant z_0'\), \(\vartheta \geqslant \vartheta_0'\), then there exist such \(\overline Z_n\), \(\vartheta_0 = \mathrm{const}\), that the series
\[ \overline Z(z,\vartheta) = \sum_{n=1}^{\infty} \overline Z_n (zE)^{1+\alpha_n} \tag{3.8} \]
converges absolutely and uniformly for \(z \leq z_0 \leq z'_0,\ \vartheta \geq \vartheta_0 \geq \vartheta'_0\).
Further, it is not difficult also to choose such \(z_0,\vartheta_0\) that the series
\[ G(C,\vartheta)=\sum_{n=1}^{\infty} \overline{Z}_n(Ee)^{1+\alpha_n}C^{\alpha_n} \tag{3.9} \]
converges absolutely and uniformly for \(C \leq z_0,\ \vartheta \geq \vartheta_0\).
§ 4. Proof
Suppose that there exists a series convergent for \(C \leq C_0\),
\[ \overline{z}(C)=C\left(1+\sum_{n=1}^{\infty}\overline{u}_n C^{\gamma_n}\right) = C(1+\overline{S}(C)), \tag{4.1} \]
where \(\overline{u}_n=\mathrm{const},\ |u_n(\vartheta)|\leq \overline{u}_n\) \((n=1,2,\ldots;\ \vartheta \geq \vartheta_0)\).
Substituting \(z\) into (3.2), applying the estimate \((C_2)\) and taking into account the remark of the preceding paragraph, for \(C \leq C_0 \leq z_0,\ \vartheta \geq \vartheta_0\) we shall have
\[ U(C_1,\vartheta)=\sum_{n=1}^{\infty}U_n C^{\delta_n} \leq C_2 \sum_{n=1}^{\infty}\overline{Z}_n(E(\vartheta_0)Ce)^{1+\alpha_n}\times \]
\[ \times\left(1+\sum_{n=1}^{\infty}\overline{S}^{\,s}\right) =\sum_{n=1}^{\infty}\overline{U}_n C^{\delta_n} =\overline{U}(C), \tag{4.2} \]
where
\[ \overline{U}_n=\mathrm{const},\quad |U_n(\vartheta)|\leq \overline{U}_n,\quad (n\geq 1,\quad \overline{U}_n=\sum_{l=1}^{s_n}\overline{Q}_{n,l}(E(\vartheta_0))^{s_{n,l}}, \]
\[ \overline{Q}_{n,l}=\mathrm{const},\quad |\widetilde{Q}_{n,l}(\vartheta)|\leq \overline{Q}_{n,l}, \quad (n\geq 1,\ l\leq s_n). \]
Set, by definition, \(\overline{u}_n=2c\overline{U}_n,\ c=(|g|\alpha_1)^{-1}\) \((n\geq 1)\). It is not difficult to see that, since \(u'_n=U_n,\ |U_n|\leq \overline{U}_n\ (n\geq 1)\), we shall have \(|u_n|\leq \overline{u}_n\) \((n\geq 1)\). Indeed, for \(n=1\), applying the estimate \((C_1)\), we obtain
\[ |u_1|=\left|\int_{\vartheta_0}^{\vartheta}(Ee)^{1+\alpha_1} C_2'\overline{Z}_1\,d\vartheta\right| \leq \frac{2(E(\vartheta_0)e)^{1+\alpha_1}C_2\overline{Z}_1} {|g|(1+\alpha_1)} \leq 2c\overline{U}_1 . \]
For \(n>1\) we obtain
\[ |u_n|\leq 2\sum_{l=1}^{s_n}\overline{Q}_{n,l}(s_{n,l}|g|)^{-1}(E(\vartheta_0))^{s_{n,l}} \leq 2c\overline{U}_n . \]
Further, substituting \(\overline{S}(C)=C^{-1}\overline{z}-1\) in (4.2) and taking into account that \(\delta_n=1+\gamma_n\), from (4.2) we shall have
\[ 2c\overline{U}(C)=C\overline{S}(C)=\overline{z}-C, \]
\[ 2c\overline{U}(C)= -\frac{2CC_2c}{2C-z} \sum_{n=1}^{\infty}\overline{Z}_n(Ce\,E(\vartheta_0))^{1+\alpha_n} = -\frac{2C^2C_2c}{2C-z}\,G(C), \tag{4.4} \]
and from this we obtain the equation
\[ \bar z^{2}-3C\bar z+2C^{2}(1+C_{2c}G(C))=0. \tag{4.5} \]
It is not difficult to see that, as \(C\to 0\), \(G(C)\to 0\), and since from (3.1) it is clear that, as \(C\to 0\), \(\bar z(C,\vartheta)\to C\), of the two solutions of equation (4.5) the one suitable for us is
\[ \bar z=\frac{C}{2}\left(3-(L-8cC_{2}G(C))^{1/2}\right) =\frac{C}{2}\left(2+\sum_{s=1}^{\infty}S_{n}G^{s}\right), \tag{4.6} \]
where
\[ S_{n}=(4C_{2}c)^{n}(n!)^{-1}\prod_{s=1}^{n-1}(2s-1). \]
Thus, from equation (4.5) we obtain \(\bar z\) in the form of a series in powers of \(C\), convergent for sufficiently small \(C\), with constant positive coefficients; and from the method of obtaining (4.5) it is clear that this series is precisely the series (4.1). Further, from the method of obtaining equation (4.5) it is clear that the series (4.6) will be a majorant for (3.1) when \(C\ll C'_0\ll C_0\ll z_0\) and when \(\vartheta\gg \vartheta_0\). Hence we obtain that the series (3.1), for sufficiently small \(C\) and sufficiently large \(\vartheta\), converges absolutely and uniformly with respect to \(\vartheta\). Thus, the series (3.1) gives the analytic structure of the solution of equation (2.5).
§ 5. On the given type of focus
We shall give an analysis of the type of focus and compare the given type of focus with the case of a center and a focus.
In order to give a detailed comparison, it is necessary to carry out a detailed study of the analytic representation of the structure of the solution of the equation of the integral curves. For a thorough study of the analytic representation of the structure of the solution much space is needed; therefore it is more convenient to carry out the detailed study separately, and in this section we shall give an analysis of the type of focus as a whole.
Let us note that in the papers [1] and [2] the analytic structure of the solution of the equation of the integral curves of system (2.1) with holomorphic right-hand sides is studied in the case of a focus, under the condition that the roots of the characteristic equation of the system are purely imaginary; and in the paper [3] a property of the qualitative picture of the solution of this system is indicated, called the E-property, consisting in the fact that to a system possessing the E-property there corresponds a certain curve such that, the smaller the domain being studied, the relatively less the integral curves differ from this curve; it is also indicated that such a feature is characteristic of the system both in the case of a focus and in the case of a center, and in the case of a focus the E-property manifests itself even in a certain sense more strongly, because in the case of a center the E-property is only a property of the qualitative picture as a whole, while in the case of a focus the E-property is a property of the qualitative picture as a whole and a property of each integral curve.
System (2.1) in the present case does not belong to the systems possessing the indicated property, i.e., this feature is not necessarily inherent in the system in the case of a focus.
Next we note that the analytic structure of the solution of the equation of the integral curves of system (2.1) in the case of purely imaginary roots of the charac-
of the characteristic equation is more complicated than in the present case, while the type of qualitative picture of the solution of both equations is a focus.
Let us note here that the analytic structure of the solution in the case of purely imaginary roots of the characteristic equation is more complicated than in the present case, because in the case of [4] the integration formulas are more complicated, which is explained by the peculiarities of the analytic structure of the right-hand side of the equation.
Let us note that the present article is a detailed exposition of the article [5]. The additional conditions for proving the convergence of the series, i.e., of the solution of equation (2.5), which are discussed in the article, arise because the method of proof is different.
References
- Erugin A. N. Izvestiya AN BSSR, series of physico-technical sciences, No. 1, 1960, pp. 27—42.
- Erugin A. N. Izvestiya AN BSSR, series of physico-technical sciences, No. 2, 1960, pp. 16—26.
- Erugin A. N. Izvestiya AN BSSR, series of physico-technical sciences, No. 3, 1960, pp. 17—20.
- Erugin A. N. DAN BSSR, No. 5, 1961, pp. 191—193.
- Erugin A. N. DAN BSSR, No. 7, 1961, pp. 284—285.
Received by the editors
May 4, 1965
Leningrad Branch of the V. A. Steklov Institute of Mathematics