UDC 517.946.9

EXPANSION FORMULAS FOR A BOUNDARY VALUE PROBLEM WITH A COMPLEX PARAMETER

E. A. OSTROVSKII

The application of M. L. Rasulov’s residue method [1] to the solution of mixed problems is connected with the question of expanding arbitrary functions of a certain class in a series in the residues of the solutions of the corresponding spectral problem.

In [1, 2] such formulas were obtained for the case of distinct and nonzero roots of the characteristic equation in the sense of Birkhoff.

In the present paper we obtain expansion formulas for arbitrary sufficiently smooth functions in a series in the residues of the solution of the spectral problem for linear differential equations of second order with discontinuous coefficients under sufficiently general boundary conditions, for the case when on one of the intervals the roots of the characteristic equation in the sense of Birkhoff are equal to zero and, consequently, identically coincide.

A simpler problem for somewhat simpler differential equations and for a particular case of boundary conditions was considered in [6], where stronger restrictions were imposed on the functions being expanded.

Consider the problem of finding a solution of the equations

\[ \sum_{l=0}^{2} C_{0l}^{(1)}(x)\,\frac{d^{l}y^{(1)}}{dx^{l}}-\lambda y^{(1)}=F^{(1)}(x), \tag{1} \]

\[ \sum_{\substack{k+l\le 2\\ k\le 1}} \lambda^{k} C_{kl}^{(2)}(x)\,\frac{d^{l}y^{(2)}}{dx^{l}}-\lambda^{2}y^{(2)}=F^{(2)}(x) \tag{2} \]

under the boundary conditions

\[ \sum_{l=0}^{3}\left\{ \alpha_{sl}^{(0,1)}\left.\frac{d^{l}y^{(1)}}{dx^{l}}\right|_{x=a_1} + \beta_{sl}^{(0,1)}\left.\frac{d^{l}y^{(1)}}{dx^{l}}\right|_{x=b_1} \right\} + \]

\[ + \sum_{\substack{k+l\le 3\\ k\le 1}} \lambda^{k}\left\{ \alpha_{sl}^{(k,2)}\left.\frac{d^{l}y^{(2)}}{dx^{l}}\right|_{x=a_2} + \right. \]

\[ \left. + \beta_{sl}^{(k,2)}\left.\frac{d^{l}y^{(2)}}{dx^{l}}\right|_{x=b_2} \right\} =\gamma_s \tag{3} \]

\[ (s=1,\,2,\,3,\,4), \]

where \((a_i,b_i)\) are mutually disjoint intervals having common endpoints, \(\alpha_{sl}^{(k,i)}\), \(\beta_{sl}^{(k,i)}\), \(\gamma_s\) are constant numbers, and

\[ \alpha_{s3}^{(0,i)}=\alpha_{sli}^{(i)}C_{02}^{(i)}(a_i),\qquad \beta_{s3}^{(0,i)}=\beta_{sli}^{(i)}C_{02}^{(i)}(b_i), \]

\[ \alpha_{s2}^{(1,2)}=\alpha_{s12}^{(2)}C_{11}^{(2)}(a_2),\qquad \beta_{s2}^{(1,2)}=\beta_{s12}^{(2)}C_{11}^{(2)}(b_2), \tag{4} \]

\(F^{(i)}(x)\) are continuous functions.

Assume that the following conditions are satisfied:

\(1^\circ.\) The functions \(C_{kl}^{(i)}(x)\) are continuously differentiable \(4+k+l\) \((k+l\le 2)\) times on \([a_i,b_i]\), and \(C_{02}^{(i)}(x)>0\) for \(x\in[a_i,b_i]\).

\(2^\circ.\)

\[ A= \begin{vmatrix} \alpha_{111}^{(1)}&\beta_{111}^{(1)}&\alpha_{112}^{(2)}&\beta_{112}^{(2)}\\ \cdot&\cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot&\cdot\\ \alpha_{411}^{(1)}&\beta_{411}^{(1)}&\alpha_{412}^{(2)}&\beta_{412}^{(2)} \end{vmatrix} \ne 0. \]

It is known [3] that under condition \(1^\circ\) the homogeneous equations corresponding to equations (1), (2) have a fundamental system of particular solutions \(y_k^{(i)}(x,\lambda)\) \((i,k=1,2)\), which may be chosen as entire functions of \(\lambda\).

Solving problem (1)—(3) by the method of variation of arbitrary constants, we obtain [1]

\[ y^{(i)}(x,\lambda)=\frac{\Delta^{(i)}(x,\lambda)}{\Delta(\lambda)}+ \]

\[ +\sum_{j=1}^{2}\int_{a_j}^{b_j}G^{(i,j)}(x,\xi,\lambda)F^{(j)}(\xi)\bigl(C_{02}^{(j)}(\xi)\bigr)^{-1}\,d\xi \tag{5} \]

\[ (i=1,2), \]

where

\[ \Delta^{(1)}(x,\lambda)= \begin{vmatrix} 0&y_1^{(1)}(x,\lambda)&y_2^{(1)}(x,\lambda)&0&0\\ -\gamma_1&\multicolumn{4}{c}{\dashline\quad}\\ \vdots&&\multicolumn{3}{c}{\Delta(\lambda)}\\ -\gamma_4&&&& \end{vmatrix}, \tag{6} \]

\[ \Delta^{(2)}(x,\lambda)= \begin{vmatrix} 0&0&0&y_1^{(2)}(x,\lambda)&y_2^{(2)}(x,\lambda)\\ -\gamma_1&\multicolumn{4}{c}{\dashline\quad}\\ \vdots&&\multicolumn{3}{c}{\Delta(\lambda)}\\ -\gamma_4&&&& \end{vmatrix}, \tag{7} \]

\[ G^{(1,j)}(x,\xi,\lambda)=\frac{1}{\Delta(\lambda)} \begin{vmatrix} g^{(1,j)}(x,\xi,\lambda)&y_1^{(1)}(x,\lambda)&y_2^{(1)}(x,\lambda)&0&0\\ L_1^{(j)}(g^{(j)})_x&\multicolumn{4}{c}{\dashline\quad}\\ \vdots&&\multicolumn{3}{c}{\Delta(\lambda)}\\ L_4^{(j)}(g^{(j)})_x&&&& \end{vmatrix}, \tag{8} \]

\[ G^{(2,j)}(x,\xi,\lambda)=\frac{1}{\Delta(\lambda)} \begin{vmatrix} g^{(2,j)}(x,\xi,\lambda)&0&0&y_1^{(2)}(x,\lambda)&y_2^{(2)}(x,\lambda)\\ L_1^{(j)}(g^{(j)})_x&\multicolumn{4}{c}{\dashline\quad}\\ \vdots&&\multicolumn{3}{c}{\Delta(\lambda)}\\ L_4^{(j)}(g^{(j)})_x&&&& \end{vmatrix}, \tag{9} \]

\[ g^{(i,j)}(x,\xi,\lambda)= \begin{cases} g^{(i)}(x,\xi,\lambda), & \text{for } j=i,\\ 0, & \text{for } j\ne i, \end{cases} \tag{10} \]

\[ g^{(i)}(x,\xi,\lambda)=\pm\frac{1}{2W^{(i)}(\xi,\lambda)} \begin{vmatrix} y_1^{(i)}(\xi,\lambda) & y_2^{(i)}(\xi,\lambda)\\ y_1^{(i)}(x,\lambda) & y_2^{(i)}(x,\lambda) \end{vmatrix} \tag{11} \]

\[ +\ \text{for } a_i\le \xi\le x\le b_i,\qquad -\ \text{for } a_i\le x\le \xi\le b_i, \]

\[ W^{(i)}(x,\lambda)= \begin{vmatrix} y_1^{(i)}(x,\lambda) & y_2^{(i)}(x,\lambda)\\ \dfrac{dy_1^{(i)}(x,\lambda)}{dx} & \dfrac{dy_2^{(i)}(x,\lambda)}{dx} \end{vmatrix}, \tag{12} \]

\[ \Delta(\lambda)= \begin{vmatrix} U_{11}^{(1)}(\lambda) & U_{12}^{(1)}(\lambda) & U_{11}^{(2)}(\lambda) & U_{12}^{(2)}(\lambda)\\ \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot\\ U_{41}^{(1)}(\lambda) & U_{42}^{(1)}(\lambda) & U_{41}^{(2)}(\lambda) & U_{42}^{(2)}(\lambda) \end{vmatrix}, \tag{13} \]

\[ U_{sk}^{(1)}(\lambda)=L_s^{(1)}\bigl(y_k^{(1)}(x,\lambda)\bigr)= \]

\[ =\sum_{l=0}^{3} \left\{ \alpha_{sl}^{(0,1)} \left.\frac{d^l y_k^{(1)}}{dx^l}\right|_{x=a_1} + \beta_{sl}^{(0,1)} \left.\frac{d^l y_k^{(1)}}{dx^l}\right|_{x=b_1} \right\}, \tag{14} \]

\[ U_{sk}^{(2)}(\lambda)=L_s^{(2)}\bigl(y_k^{(2)}(x,\lambda)\bigr)= \]

\[ = \sum_{\substack{k+l\le 3\\ k\le 1}} \lambda^k \left\{ \alpha_{sl}^{(k,2)} \left.\frac{d^l y_k^{(2)}}{dx^l}\right|_{x=a_2} + \beta_{sl}^{(k,2)} \left.\frac{d^l y_k^{(2)}}{dx^l}\right|_{x=b_2} \right\}, \tag{15} \]

\(L_s^{(i)}(g^{(i)})_x\) is the result of applying the operator \(L_s^{(i)}\) to \(g^{(i)}\) as a function of \(x\).

It is easy to see that, under condition \(1^\circ\), we are in the conditions of applicability of Tamarkin’s theorem on the asymptotic representation of a fundamental system of solutions of homogeneous equations [4]. According to this theorem, the corresponding homogeneous differential equations (1), (2) have a fundamental system of particular solutions \(y_k^{(i)}(x,\lambda)\) \((i,k=1,2)\), admitting, together with derivatives up to the third order, the asymptotic representations

\[ \frac{d^l y_k^{(1)}(x,\lambda)}{dx^l} = (\sqrt{\lambda})^l \left\{ \sum_{s=0}^{4}\eta_{ksl}^{(1)}(x)(\sqrt{\lambda})^{-s} + E_{kl}^{(1)}(x,\lambda)\lambda^{-\frac52} \right\} \exp\left(\sqrt{\lambda}\int_{a_1}^{x}\varphi_k^{(1)}(\alpha)\,d\alpha\right), \tag{16} \]

\[ \frac{d^l y_k^{(2)}(x,\lambda)}{dx^l} = \lambda^l \left\{ \sum_{s=0}^{4}\eta_{ksl}^{(2)}(x)\lambda^{-s} + E_{kl}^{(2)}(x,\lambda)\lambda^{-5} \right\} \exp\left(\lambda\int_{a_2}^{x}\varphi_k^{(2)}(\alpha)\,d\alpha\right). \tag{17} \]

in the entire \(\lambda\)-plane, where \(\varphi_k^{(1)}(x)\) are the roots of the characteristic equation

\[ C_{02}^{(1)}(x)\Theta^2-1=0, \tag{18} \]

\[ \varphi_k^{(1)}(x)=\mu_k S(x),\qquad S(x)=\bigl(C_{02}^{(1)}(x)\bigr)^{-\frac12}>0 \quad (k=1,2), \tag{19} \]

where \(\mu_1=-1,\ \mu_2=1\) in the upper \(\lambda\)-half-plane, and \(\mu_1=1,\ \mu_2=-1\) in the lower \(\lambda\)-half-plane, while for \(\sqrt{\lambda}\) that branch is chosen for which \(\operatorname{Im}\sqrt{\lambda}>0\); \(\varphi_k^{(2)}(x)\) are the roots of the characteristic equation

\[ C_{02}^{(2)}(x)\Theta^2+C_{11}^{(2)}(x)\Theta-1=0, \tag{20} \]

and for the left \(\lambda\)-half-plane

\[ \varphi_1^{(2)}(x)=\Theta_1,\qquad \varphi_2^{(2)}(x)=\Theta_2, \tag{21} \]

whereas for the right one—

\[ \varphi_1^{(2)}(x)=\Theta_2,\qquad \varphi_2^{(2)}(x)=\Theta_1, \tag{22} \]

\[ \Theta_1=\left(-C_{11}^{(2)}(x)+ \sqrt{\bigl(C_{11}^{(2)}(x)\bigr)^2+4C_{02}^{(2)}(x)}\right) \bigl(2C_{02}^{(2)}(x)\bigr)^{-1}>0, \]

\[ \tag{23} \]

\[ \Theta_2=\left(-C_{11}^{(2)}(x)- \sqrt{\bigl(C_{11}^{(2)}(x)\bigr)^2+4C_{02}^{(2)}(x)}\right) \bigl(2C_{02}^{(2)}(x)\bigr)^{-1}<0, \]

\[ \eta_{k00}^{(1)}(x)= \frac{ \exp\left(-\displaystyle\int_{a_1}^{x} C_{01}^{(1)}(\alpha)\bigl(2C_{02}^{(1)}(\alpha)\bigr)^{-1}\,d\alpha\right) }{ \sqrt{2\varphi_k^{(1)}(x)} }, \tag{24} \]

\[ \eta_{k00}^{(2)}(x)= \frac{ \exp\left( -\displaystyle\int_{a_2}^{x} \frac{C_{01}^{(2)}(\alpha)\varphi_k^{(2)}(\alpha)+C_{10}^{(2)}(\alpha)} {2C_{02}^{(2)}(\alpha)\varphi_k^{(2)}(\alpha)+C_{11}^{(2)}(\alpha)} \,d\alpha \right) }{ \sqrt{2\varphi_k^{(2)}(x)+C_{11}^{(2)}(x)\bigl(C_{02}^{(2)}(x)\bigr)^{-1}} }, \tag{25} \]

\[ \eta_{k0l}^{(i)}(x)=\bigl(\varphi_k^{(i)}(x)\bigr)^l\eta_{k00}^{(i)}(x) \quad (i,\ k=1,2;\ l=0,1,2,3), \tag{26} \]

and the functions \(\eta_{knl}^{(i)}(x)\) are \(5-n\) \((n=0,1,2,3,4)\) times continuously differentiable.

For brevity we shall use the following notation of Birkhoff [5]:

\[ [f(x)]=f(x)+\lambda^{-1}E(x,\lambda), \tag{27} \]

where \(E(x,\lambda)\) is bounded for large \(|\lambda|\). Taking (27) and (19) into account, we rewrite formulas (16), (17) in the form

\[ \frac{d^l y_k^{(1)}(x,\lambda)}{dx^l} =(\sqrt{\lambda})^l[\eta_{k0l}^{(1)}(x)] \exp\left(\mu_k\sqrt{\lambda}\int_{a_1}^{x} S(\alpha)\,d\alpha\right), \tag{28} \]

\[ \frac{d^l y_k^{(2)}(x,\lambda)}{dx^l} =\lambda^l[\eta_{k0l}^{(2)}(x)] \exp\left(\lambda\int_{a_2}^{x}\varphi_k^{(2)}(\alpha)\,d\alpha\right). \tag{29} \]

Substituting (28), (29) into (14), (15) and taking into account (4), (18), (20), (26), we obtain

\[ U_{sk}^{(1)}(\lambda)=\lambda^{\frac{3}{2}}\left\{ \alpha_{s11}^{(1)}\left[\eta_{k01}^{(1)}(a_1)\right]+ \right. \]
\[ \left. +\beta_{s11}^{(1)}\left[\eta_{k01}^{(1)}(b_1)\right]\exp\left(\mu_k\sqrt{\lambda}\,\omega^{(1)}\right)\right\}, \tag{30} \]

\[ U_{sk}^{(2)}(\lambda)=\lambda^3\left\{ \alpha_{s12}^{(2)}\left[\eta_{k01}^{(2)}(a_2)\right]+ \right. \]
\[ \left. +\beta_{s12}^{(2)}\left[\eta_{k01}^{(2)}(b_2)\right]\exp\left(\nu_k\lambda\omega_k^{(2)}\right)\right\}, \tag{31} \]

where

\[ \omega^{(1)}=\int_{a_1}^{b_1} S(\alpha)\,d\alpha,\qquad \omega_k^{(2)}=\left|\int_{a_2}^{b_2}\varphi_k^{(2)}(\alpha)\,d\alpha\right|, \tag{32} \]

and \(\nu_1=1,\ \nu_2=-1\) in the left \(\lambda\)-half-plane, \(\nu_1=-1,\ \nu_2=1\) in the right one. Substituting (30), (31) into (13) and expanding the determinant, we obtain

\[ \Delta(\lambda)=\lambda^9\exp\left(\mu_2\sqrt{\lambda}\,\omega^{(1)} +\nu_2\lambda\omega_2^{(2)}\right)H(\lambda), \tag{33} \]

where

\[ H(\lambda)=A\left\{ \left[\eta_{101}^{(1)}(a_1)\right]\left[\eta_{201}^{(1)}(b_1)\right]- \right. \]
\[ \left. -\left[\eta_{201}^{(1)}(a_1)\right]\left[\eta_{101}^{(1)}(b_1)\right] \exp\left(2\mu_1\sqrt{\lambda}\,\omega^{(1)}\right) \right\}\times \]
\[ \times\left\{ \left[\eta_{101}^{(2)}(a_2)\right]\left[\eta_{201}^{(2)}(b_2)\right]- \right. \]
\[ \left. -\left[\eta_{201}^{(2)}(a_2)\right]\left[\eta_{101}^{(2)}(b_2)\right] \exp\left(\lambda\nu_1\left(\omega_1^{(2)}+\omega_2^{(2)}\right)\right) \right\}. \tag{34} \]

Under condition \(2^\circ\), the equation \(\Delta(\lambda)=0\) is equivalent to the equation \(H(\lambda)=0\), which splits into two equations

\[ H_1(z)= \left[\eta_{101}^{(1)}(a_1)\eta_{201}^{(1)}(b_1)\right]- \]
\[ -\left[\eta_{201}^{(1)}(a_1)\eta_{101}^{(1)}(b_1)\right] \exp\left(2\mu_1 z\,\omega^{(1)}\right)=0, \tag{35} \]

\[ H_2(\lambda)= \left[\eta_{101}^{(2)}(a_2)\eta_{201}^{(2)}(b_2)\right]- \]
\[ -\left[\eta_{201}^{(2)}(a_2)\eta_{101}^{(2)}(b_2)\right] \exp\left\{\nu_1\lambda\left(\omega_1^{(2)}+\omega_2^{(2)}\right)\right\}=0, \tag{36} \]

where

\[ z=\sqrt{\lambda}. \tag{37} \]

For the exponential polynomials \(H_1(z)\) and \(H_2(\lambda)\) with asymptotically constant coefficients, Lemma 1 of Ch. III [1] is applicable, according to which they have a countable set of zeros located in strips \(D_h,\ D_h'\) of bounded width \(h\), centered at the origins of the corresponding planes, whose boundaries are parallel to the imaginary axes. These zeros admit the asymptotic representations:

\[ |z_\nu|=\frac{\pi}{\omega^{(1)}}\,\nu\left(1+O\left(\frac{1}{\nu}\right)\right), \tag{38} \]

\[ |\lambda_k|=\frac{2\pi k}{\omega_1^{(2)}+\omega_2^{(2)}}\left(1+O\left(\frac{1}{k}\right)\right). \tag{39} \]

If from the strips \(D_h, D'_h\) one removes the interiors of small circles

\[ |z-z_\nu|\leq \delta, \tag{40} \]

\[ |\lambda-\lambda_k|\leq \delta, \tag{41} \]

then in the remaining parts of the \(z\)- and \(\lambda\)-planes the inequalities

\[ |H_1(z)|\geq C_{1\delta}, \tag{42} \]

\[ |H_2(\lambda)|\geq C_{2\delta}, \tag{43} \]

hold, where \(C_{i\delta}\) are constants depending on the choice of \(\delta\).

Obviously, under the mapping (37), the neighborhoods (40) of the \(z\)-plane pass into certain neighborhoods \(O(\lambda_\nu)\) of the values \(\lambda_\nu\), located inside the parabola

\[ \lambda_1=-\frac{h^2}{4}-\frac{\lambda_2^2}{h^2}, \tag{44} \]

where \(\lambda=\lambda_1+\sqrt{-1}\lambda_2\). Consequently, outside the neighborhoods \(O(\lambda_\nu)\) of the \(\lambda\)-plane, the function \(H_1(\sqrt{\lambda})\) remains bounded below by a positive number

\[ \left|H_1(\sqrt{\lambda})\right|\geq C_{1\delta}. \tag{45} \]

By virtue of the inequalities (43), (45), we conclude that outside the neighborhoods (41) and \(O(\lambda_\nu)\) the function \(H(\lambda)\) remains bounded below:

\[ |H(\lambda)|\geq C_\delta. \tag{46} \]

Here it is obvious that for the zeros \(H(\lambda)\) corresponding to the values (38), the asymptotic representation

\[ |\lambda_\nu|=\left(\frac{\pi}{\omega^{(1)}}\right)^2\nu^2\left(1+O\left(\frac{1}{\nu}\right)\right) \tag{47} \]

holds.

Thus we have proved

Theorem 1. Under conditions \(1^\circ, 2^\circ\), the solution of problem (1)—(3) has two infinite sequences of poles \(\lambda_k\) and \(\lambda_\nu\), located respectively inside a strip of bounded width \(h\), parallel to the imaginary axis of the \(\lambda\)-plane with center at the origin, and inside the parabola (44). If these poles are numbered in the order of increasing moduli, then for them the asymptotic representations (39), (47) hold.

If from the \(\lambda\)-plane one removes the interiors of the neighborhoods (41) and \(O(\lambda_\nu)\), then in the remaining part the inequality (46) holds.

Let us now deal with asymptotic estimates of the solution of problem (1)—(3) outside the neighborhoods (41) and \(O(\lambda_\nu)\). Expanding the determinants (6), (7) with respect to the elements of the first row and the first column, we obtain

\[ \Delta^{(i)}(x,\lambda) = y_1^{(i)}(x,\lambda)\sum_{k=1}^{4}\gamma_k\Delta_{k,\,2i-1}(\lambda) + \]

\[ + y_2^{(i)}(x,\lambda)\sum_{k=1}^{4}\gamma_k\Delta_{k,\,2i}(\lambda), \tag{48} \]

where \(\Delta_{ks}(\lambda)\) are the cofactors of the corresponding elements of the determinant \(\Delta(\lambda)\).

Substituting into \(\Delta_{ks}(\lambda)\), in place of \(U_{sk}^{(i)}(\lambda)\), their asymptotic representations (30), (31) and expanding these determinants, we obtain the following estimates:

\[ \begin{aligned} \Delta_{k1}(\lambda) &= \lambda^{7}\sqrt{\lambda}\, \exp\!\left(\mu_{2}\omega^{(1)}\sqrt{\lambda}+\nu_{2}\omega_{2}^{(2)}\lambda\right) D_{1}(\lambda),\\ \Delta_{k2}(\lambda) &= \lambda^{7}\sqrt{\lambda}\, \exp\!\left(\nu_{2}\omega_{2}^{(2)}\lambda\right) D_{2}(\lambda),\\ \Delta_{k3}(\lambda) &= \lambda^{6} \exp\!\left(\mu_{2}\omega^{(1)}\sqrt{\lambda}+\nu_{2}\omega_{2}^{(2)}\lambda\right) D_{3}(\lambda),\\ \Delta_{k4}(\lambda) &= \lambda^{6} \exp\!\left(\mu_{2}\omega^{(1)}\sqrt{\lambda}\right) D_{4}(\lambda), \end{aligned} \tag{49} \]

where \(D_k(\lambda)\) \((k=1,2,3,4)\) are functions bounded in modulus in the entire \(\lambda\)-plane outside the neighborhoods (41) and \(O(\lambda_\nu)\).

From (48), with the aid of (28), (29), (33), (46), and (49), we obtain the following estimates, valid in the entire \(\lambda\)-plane outside the neighborhoods (41) and \(O(\lambda_\nu)\):

\[ \left| \frac{1}{\Delta(\lambda)} \frac{d^m \Delta^{(1)}(x,\lambda)}{dx^m} \right| \le |\sqrt{\lambda}|^{\,m-3}\times \]

\[ {}\times \max_{\gamma_s} \left\{ L_{1\delta} \left| [\eta_{10m}^{(1)}(x)] \exp\!\left( \mu_1\sqrt{\lambda}\int_{a_1}^{x} S(\alpha)\,d\alpha \right) \right| +\right. \]

\[ \left. +\,L_{2\delta} \left| [\eta_{20m}^{(1)}(x)] \exp\!\left( \mu_1\sqrt{\lambda}\int_{x}^{b_1} S(\alpha)\,d\alpha \right) \right| \right\}; \tag{50} \]

\[ \left| \frac{1}{\Delta(\lambda)} \frac{d^m \Delta^{(2)}(x,\lambda)}{dx^m} \right| \le |\lambda|^{\,m-3}\times \]

\[ {}\times \max_{\gamma_s} \left\{ L_{3\delta} \left| [\eta_{10m}^{(2)}(x)] \exp\!\left( \lambda\int_{a_2}^{x}\varphi_1^{(2)}(\alpha)\,d\alpha \right) \right| +\right. \]

\[ \left. +\,L_{4\delta} \left| [\eta_{20m}^{(2)}(x)] \exp\!\left( -\lambda\int_{x}^{b_2}\varphi_2^{(2)}(\alpha)\,d\alpha \right) \right| \right\}, \tag{51} \]

where \(L_{k\delta}\) are numbers depending only on \(\delta\).

Thus, the following has been proved.

Theorem 2. Under conditions \(1^\circ\), \(2^\circ\), if the interiors of the neighborhoods (41) and \(O(\lambda_\nu)\) are removed from the \(\lambda\)-plane, then in the remaining part the estimates (50), (51) hold.

Let us find a suitable asymptotic representation of the second term of (5). From (11) we have

\[ G^{(i)}(x,\xi,\lambda) = \pm \frac{1}{2} \left\{ y_1^{(i)}(x,\lambda)z_1^{(i)}(\xi,\lambda) + \right. \]

\[ \left. + y_2^{(i)}(x,\lambda)z_2^{(i)}(\xi,\lambda) \right\} \quad (i=1,2) \tag{52} \]

\[ +\ \text{ for } a_i\le \xi \le x \le b_i,\qquad -\ \text{ for } a_i\le x \le \xi \le b_i, \]

where

\[ z_{1}^{(i)}(\xi,\lambda)=-y_{2}^{(i)}(\xi,\lambda)\bigl(W^{(i)}(\xi,\lambda)\bigr)^{-1}, \]
\[ z_{2}^{(i)}(\xi,\lambda)=y_{1}^{(i)}(\xi,\lambda)\bigl(W^{(i)}(\xi,\lambda)\bigr)^{-1}. \tag{53} \]

Taking into account (12), (28), (29), from (53) we obtain

\[ z_{s}^{(1)}(\xi,\lambda)=(-1)^{s}\lambda^{-1/2} \left\{ \sum_{k=0}^{4}\zeta_{sk}^{(1)}(\xi)(\sqrt{\lambda})^{-k} + (\sqrt{\lambda})^{-5}E_{s}^{(1)}(\xi,\lambda) \right\} \exp\left(-\mu_{s}\sqrt{\lambda}\int_{a_{1}}^{\xi}S(\alpha)\,d\alpha\right), \tag{54} \]

\[ z_{s}^{(2)}(\xi,\lambda)=(-1)^{s}\lambda^{-1} \left\{ \sum_{k=0}^{4}\zeta_{sk}^{(2)}(\xi)\lambda^{-k} + \lambda^{-5}E_{s}^{(2)}(\xi,\lambda) \right\} \exp\left(-\lambda\int_{a_{2}}^{\xi}\varphi_{s}^{(2)}(\alpha)\,d\alpha\right), \tag{55} \]

where

\[ \zeta_{s0}^{(1)}(\xi)=\{2\mu_{2}S(\xi)\eta_{s00}^{(1)}(\xi)\}^{-1}, \]

\[ \zeta_{s0}^{(2)}(\xi)=\left\{\eta_{s00}^{(2)}(\xi)\left(\varphi_{2}^{(2)}(\xi)-\varphi_{1}^{(2)}(\xi)\right)\right\}^{-1}, \tag{56} \]

and the functions \(\zeta_{sk}^{(i)}(\xi)\) are \(5-k\) times continuously differentiable on \([a_i,b_i]\) \((k=0,1,2,3,4)\).

Following Birkhoff [5], we transform the determinants appearing in the numerators of formulas (8), (9). To do this, the columns

\[ \begin{pmatrix} U_{11}^{(j)}(\lambda)\\ \vdots\\ U_{41}^{(j)}(\lambda) \end{pmatrix}, \qquad \begin{pmatrix} U_{12}^{(j)}(\lambda)\\ \vdots\\ U_{42}^{(j)}(\lambda) \end{pmatrix} \]

are multiplied respectively by \(-\dfrac12 z_{1}^{(j)}(\xi,\lambda)\), \(-\dfrac12 z_{2}^{(j)}(\xi,\lambda)\), and added to the first column; then, taking into account (10), (14), (15), and (52), we obtain

\[ G^{(1,j)}(x,\xi,\lambda)=\frac{1}{\Delta(\lambda)} \left| \begin{array}{c|cccc} g_{0}^{(1,j)}(x,\xi,\lambda) & y_{1}^{(1)}(x,\lambda) & y_{2}^{(1)}(x,\lambda) & 0 & 0\\ \hline g_{1}^{(j)}(\xi,\lambda) & \multicolumn{4}{c}{\multirow{3}{*}{\Delta(\lambda)}}\\ \vdots & & & &\\ g_{4}^{(j)}(\xi,\lambda) & & & & \end{array} \right|, \tag{57} \]

\[ G^{(2,j)}(x,\xi,\lambda)=\frac{1}{\Delta(\lambda)} \left| \begin{array}{c|cccc} g_{0}^{(2,j)}(x,\xi,\lambda) & 0 & 0 & y_{1}^{(2)}(x,\lambda) & y_{2}^{(2)}(x,\lambda)\\ \hline g_{1}^{(j)}(\xi,\lambda) & \multicolumn{4}{c}{\multirow{3}{*}{\Delta(\lambda)}}\\ \vdots & & & &\\ g_{4}^{(j)}(\xi,\lambda) & & & & \end{array} \right|, \tag{58} \]

where

\[ g_{0}^{(i,j)}(x,\xi,\lambda)= \begin{cases} y_{1}^{(i)}(x,\lambda)z_{1}^{(i)}(\xi,\lambda), & \text{if } a_i\le \xi\le x\le b_i,\quad j=i,\\[4pt] -y_{2}^{(i)}(x,\lambda)z_{2}^{(i)}(\xi,\lambda), & \text{if } a_i\le x\le \xi\le b_i,\quad j=i,\\[4pt] 0, & \text{if } j\ne i. \end{cases} \tag{59} \]

\[ g_s^{(1)}(\xi,\lambda)=L_s^{(1)}(g^{(1)})_x+\frac{1}{2}U_{s1}^{(1)}(\lambda)z_1^{(1)}(\xi,\lambda)- \]

\[ -\frac{1}{2}U_{s2}^{(1)}(\lambda)z_2^{(1)}(\xi,\lambda)= \]

\[ =z_1^{(1)}(\xi,\lambda)\sum_{l=0}^{3}\beta_{sl}^{(0,1)} \left.\frac{d^l y_1^{(1)}}{dx^l}\right|_{x=b_1} - \]

\[ -z_2^{(1)}(\xi,\lambda)\sum_{l=0}^{3}\alpha_{sl}^{(0,1)} \left.\frac{d^l y_2^{(1)}}{dx^l}\right|_{x=a_1}, \tag{60} \]

\[ g_s^{(2)}(\xi,\lambda)=L_s^{(2)}(g^{(2)})_x+ \]

\[ +\frac{1}{2}U_{s1}^{(2)}(\lambda)z_1^{(2)}(\xi,\lambda) -\frac{1}{2}U_{s2}^{(2)}(\lambda)z_2^{(2)}(\xi,\lambda)= \tag{61} \]

\[ =z_1^{(2)}(\xi,\lambda)\sum_{\substack{k+l\le 3\\ k\le 1}} \lambda^k\beta_{sl}^{(k,2)} \left.\frac{d^l y_1^{(2)}}{dx^l}\right|_{x=b_2} - \]

\[ -z_2^{(2)}(\xi,\lambda)\sum_{\substack{k+l\le 3\\ k\le 1}} \lambda^k\alpha_{sl}^{(k,2)} \left.\frac{d^l y_2^{(2)}}{dx^l}\right|_{x=a_2}. \]

Taking into account (4), (18), (20), (26), (28), (29), (54), (55), from (59)—(61) we obtain

\[ \frac{\partial^l g_0^{(1,1)}(x,\xi,\lambda)}{\partial x^l}= \]

\[ =-(\sqrt{\lambda})^{\,l-1} \begin{cases} [\eta_{10l}^{(1)}(x)]\,[\xi_{10}^{(1)}(\xi)] \exp\left(\mu_2\sqrt{\lambda}\displaystyle\int_x^\xi S(\alpha)\,d\alpha\right), & a_1\le \xi\le x\le b_1,\\[1.2em] [\eta_{20l}^{(1)}(x)]\,[\xi_{20}^{(1)}(\xi)] \exp\left(\mu_1\sqrt{\lambda}\displaystyle\int_x^\xi S(\alpha)\,d\alpha\right), & a_1\le x\le \xi\le b_1, \end{cases} \tag{62} \]

\[ \frac{\partial^l g_0^{(2,2)}(x,\xi,\lambda)}{\partial x^l}= \]

\[ =-\lambda^{l-1} \begin{cases} [\eta_{10l}^{(2)}(x)]\,[\xi_{10}^{(2)}(\xi)] \exp\left(-\lambda\displaystyle\int_x^\xi \varphi_1^{(2)}(\alpha)\,d\alpha\right), & a_2\le \xi\le x\le b_2,\\[1.2em] [\eta_{20l}^{(2)}(x)]\,[\xi_{20}^{(2)}(\xi)] \exp\left(-\lambda\displaystyle\int_x^\xi \varphi_2^{(2)}(\alpha)\,d\alpha\right), & a_2\le x\le \xi\le b_2. \end{cases} \tag{63} \]

\[ g_s^{(1)}(\xi,\lambda)=-\lambda\left\{[\xi_{10}^{(1)}(\xi)]\,[\beta_{s11}^{(1)}\eta_{101}^{(1)}(b_1)]\times\right. \]

\[ \left. \times \exp\left(\mu_2\sqrt{\lambda}\int_{b_1}^{\xi} S(\alpha)\,d\alpha\right)+ [\xi_{20}^{(1)}(\xi)]\,[\alpha_{s11}^{(1)}\eta_{201}^{(1)}(a_1)] \exp\left(\mu_1\sqrt{\lambda}\int_{a_1}^{\xi} S(\alpha)\,d\alpha\right) \right\}, \tag{64} \]

\[ g_s^{(2)}(\xi,\lambda)=-\lambda^2\left\{[\xi_{10}^{(2)}(\xi)]\,[\beta_{s12}^{(2)}\eta_{101}^{(2)}(b_2)]\times\right. \]

\[ \left. \times \exp\left(-\lambda\int_{b_2}^{\xi}\varphi_1^{(2)}(\alpha)\,d\alpha\right)+ [\xi_{20}^{(2)}(\xi)]\,[\alpha_{s12}^{(2)}\eta_{201}^{(2)}(a_2)] \exp\left(-\lambda\int_{a_2}^{\xi}\varphi_2^{(2)}(\alpha)\,d\alpha\right) \right\}. \tag{65} \]

Differentiating the equalities (57), (58) \(l\) times \((l=0,1,2,3)\) and expanding the determinants with respect to the elements of the first row and first column by means of (28), (29), (33), (46), (49), (62)—(65), we conclude that throughout the \(\lambda\)-plane outside neighborhoods of the spectrum the following asymptotic representations hold:

\[ \frac{\partial^l G^{(1,1)}(x,\xi,\lambda)}{\partial x^l}= \]

\[ =-(\sqrt{\lambda})^{\,l-1} \left\{ \begin{array}{c} [\eta_{101}^{(1)}(x)]\,[\xi_{10}^{(1)}(\xi)] \exp\left(\mu_2\sqrt{\lambda}\int_x^{\xi} S(\alpha)\,d\alpha\right) \\[0.5em] \text{for } a_1\le \xi\le x\le b_1, \\[0.8em] [\eta_{201}^{(1)}(x)]\,[\xi_{20}^{(1)}(\xi)] \exp\left(\mu_1\sqrt{\lambda}\int_x^{\xi} S(\alpha)\,d\alpha\right) \\[0.5em] \text{for } a_1\le x\le \xi\le b_1 \end{array} \right\} + \]

\[ +(\sqrt{\lambda})^{\,l-1} \left\{ [\xi_{10}^{(1)}(\xi)] \exp\left(\mu_2\sqrt{\lambda}\int_{b_1}^{\xi} S(\alpha)\,d\alpha\right) E_{l1}^{(1,1)}(x,\lambda)+ \right. \]

\[ \left. +[\xi_{20}^{(1)}(\xi)] \exp\left(\mu_1\sqrt{\lambda}\int_{a_1}^{\xi} S(\alpha)\,d\alpha\right) E_{l2}^{(1,1)}(x,\lambda) \right\}, \tag{66} \]

\[ \frac{\partial^l G^{(2,1)}(x,\xi,\lambda)}{\partial x^l}= \]

\[ =\lambda^{l-2} \left\{ [\xi_{10}^{(1)}(\xi)] \exp\left(\mu_2\sqrt{\lambda}\int_{b_1}^{\xi} S(\alpha)\,d\alpha\right) E_{l1}^{(2,1)}(x,\lambda)+ \right. \]

\[ \left. +[\xi_{20}^{(1)}(\xi)] \exp\left(\mu_1\sqrt{\lambda}\int_{a_1}^{\xi} S(\alpha)\,d\alpha\right) E_{l2}^{(2,1)}(x,\lambda) \right\}, \tag{67} \]

\[ \frac{\partial^{l}G^{(2,2)}(x,\xi,\lambda)}{\partial x^{l}} = \]

\[ = -\lambda^{l-1} \left\{ \begin{array}{l} [\eta_{10l}^{(2)}(x)]\,[\zeta_{10}^{(2)}(\xi)]\, \exp\left(-\lambda\displaystyle\int_{x}^{\xi}\varphi_{1}^{(2)}(\alpha)\,d\alpha\right) \\[0.8em] \hfill \text{for } a_{2}\leq \xi \leq x \leq b_{2} \\[0.8em] [\eta_{20l}^{(2)}(x)]\,[\zeta_{20}^{(2)}(\xi)]\, \exp\left(-\lambda\displaystyle\int_{x}^{\xi}\varphi_{2}^{(2)}(\alpha)\,d\alpha\right) \\[0.8em] \hfill \text{for } a_{2}\leq x \leq \xi \leq b_{2} \end{array} \right\} + \]

\[ +\lambda^{l-1} \left\{ [\zeta_{10}^{(2)}(\xi)]\, \exp\left(-\lambda\displaystyle\int_{b_{2}}^{\xi}\varphi_{1}^{(2)}(\alpha)\,d\alpha\right) E_{l1}^{(2,2)}(x,\lambda) + \right. \]

\[ \left. + [\zeta_{20}^{(2)}(\xi)]\, \exp\left(-\lambda\displaystyle\int_{a_{2}}^{\xi}\varphi_{2}^{(2)}(\alpha)\,d\alpha\right) E_{l2}^{(2,2)}(x,\lambda) \right\}, \tag{68} \]

\[ \frac{\partial^{l}G^{(1,2)}(x,\xi,\lambda)}{\partial x^{l}} = (\sqrt{\lambda})^{\,l+1} \left\{ [\zeta_{10}^{(2)}(\xi)]\times \right. \]

\[ \times \exp\left(-\lambda\displaystyle\int_{b_{2}}^{\xi}\varphi_{1}^{(2)}(\alpha)\,d\alpha\right) E_{l1}^{(1,2)}(x,\lambda) + \]

\[ \left. + [\zeta_{20}^{(2)}(\xi)]\, \exp\left(-\lambda\displaystyle\int_{a_{2}}^{\xi}\varphi_{2}^{(2)}(\alpha)\,d\alpha\right) E_{l2}^{(1,2)}(x,\lambda) \right\}, \tag{69} \]

where

\[ E_{lk}^{(1,j)}(x,\lambda) = [\eta_{10l}^{(1)}(x)]\times \]

\[ \times \exp\left(\mu_{1}\sqrt{\lambda}\displaystyle\int_{a_{1}}^{x} S(\alpha)\,d\alpha\right) D_{k1}^{(1,j)}(\lambda) + \]

\[ + [\eta_{20l}^{(1)}(x)] \exp\left(\mu_{1}\sqrt{\lambda}\displaystyle\int_{x}^{b_{1}} S(\alpha)\,d\alpha\right) D_{k2}^{(1,j)}(\lambda), \]

\[ E_{lk}^{(2,j)}(x,\lambda) = [\eta_{10l}^{(2)}(x)]\times \]

\[ \times \exp\left(\lambda\displaystyle\int_{a_{2}}^{x}\varphi_{1}^{(2)}(\alpha)\,d\alpha\right) D_{k1}^{(2,j)}(\lambda) + \]

\[ + [\eta_{20l}^{(2)}(x)] \exp\left(-\lambda\displaystyle\int_{x}^{b_{2}}\varphi_{2}^{(2)}(\alpha)\,d\alpha\right) D_{k2}^{(2,j)}(\lambda) \quad (k,j=1,2), \tag{70} \]

\(E_{lk}^{(i,j)}(x,\lambda)\), \(D_{ks}^{(i,j)}(\lambda)\) \((i,j,k,s=1,2;\ l=0,1,2,3)\) in the part of the \(\lambda\)-plane remaining after deleting the inner neighborhoods \(O(\lambda_{\nu})\) and (41), are bounded by some number \(L_{\delta}\), depending only on \(\delta\).

Thus the following has been proved.

Theorem 3. Under conditions \(1^\circ, 2^\circ\), if from the \(\lambda\)-plane one removes the interiors of the neighborhoods (41) and \(O(\lambda_\nu)\), then in the remaining part the estimates (66)—(69) hold.

With the help of the estimates obtained, we shall prove the following theorem.

Theorem 4. Under conditions \(1^\circ, 2^\circ\), if \(\Phi^{(i)}(x)\) for \(x\in [a_i,b_i]\) have continuous derivatives up to the second order and \(\Phi^{(2)}(a_2)=\Phi^{(2)}(b_2)=0\), then the following expansion formulas hold:

\[ \frac{-1}{2\pi\sqrt{-1}} \sum_{n=1}^{\infty}\int_{C_n} d\lambda \int_{a_1}^{b_1} G^{(1,1)}(x,\xi,\lambda)\times \]

\[ \times \Phi^{(1)}(\xi)\bigl(C_{02}^{(1)}(\xi)\bigr)^{-1}\,d\xi = \Phi^{(1)}(x), \]

\[ \frac{-1}{2\pi\sqrt{-1}} \sum_{n=1}^{\infty}\int_{C_n} d\lambda \int_{a_2}^{b_2} G^{(1,2)}(x,\xi,\lambda)\times \]

\[ \times \Phi^{(2)}(\xi)\bigl(C_{02}^{(2)}(\xi)\bigr)^{-1}\,d\xi =0, \]

\[ \frac{-1}{2\pi\sqrt{-1}} \sum_{n=1}^{\infty}\int_{C_n}\lambda^s\,d\lambda \int_{a_2}^{b_2} G^{(2,2)}(x,\xi,\lambda)\times \]

\[ \times \Phi^{(2)}(\xi)\bigl(C_{02}^{(2)}(\xi)\bigr)^{-1}\,d\xi = \begin{cases} 0, & \text{for } s=0,\\ \Phi^{(2)}(x), & \text{for } s=1, \end{cases} \]

\[ \frac{-1}{2\pi\sqrt{-1}} \sum_{n=1}^{\infty}\int_{C_n}\lambda^s\,d\lambda \int_{a_1}^{b_1} G^{(2,1)}(x,\xi,\lambda)\times \]

\[ \times \Phi^{(1)}(\xi)\bigl(C_{02}^{(1)}(\xi)\bigr)^{-1}\,d\xi =0 \quad \text{for } s=0,1, \]

where \(C_n\) is a simple closed contour enclosing one pole \(\lambda_\nu\) of the solution of problem (1)—(3).

Proof. Draw a sequence of circles \(O_\nu\), with center at the origin of coordinates of the \(\lambda\)-plane, of radii \(r_\nu\), not intersecting the neighborhoods \(O(\lambda_\nu)\) and (41). The possibility of choosing such circles follows from what was stated above. Obviously, on \(O_\nu\) the estimates of Theorems 2 and 3 are valid. Let us compute the limits of the integrals

\[ J_\nu^{(1,1)}(\Phi)= \frac{-1}{2\pi\sqrt{-1}} \int_{O_\nu} d\lambda \int_{a_1}^{b_1}G^{(1,1)}(x,\xi,\lambda)\times \]

\[ \times \Phi^{(1)}(\xi)\bigl(C_{02}^{(1)}(\xi)\bigr)^{-1}\,d\xi, \tag{71} \]

\[ J_\nu^{(1,2)}(\Phi)= \frac{-1}{2\pi\sqrt{-1}} \int_{O_\nu} d\lambda \int_{a_2}^{b_2}G^{(1,2)}(x,\xi,\lambda)\Phi^{(2)}(\xi) \bigl(C_{02}^{(2)}(\xi)\bigr)^{-1}\,d\xi, \tag{72} \]

\[ J_{\nu s}^{(2,2)}(\Phi)= \frac{-1}{2\pi\sqrt{-1}} \int_{O_\nu}\lambda^s\,d\lambda \int_{a_2}^{b_2}G^{(2,2)}(x,\xi,\lambda)\times \]

\[ \times \Phi^{(2)}(\xi)\left(C_{02}^{(2)}(\xi)\right)^{-1}\,d\xi, \tag{73} \]

\[ J_{\nu s}^{(2,1)}(\Phi)= \frac{-1}{2\pi\sqrt{-1}} \int_{O_\nu}\lambda^s\,d\lambda \int_{a_1}^{b_1}G^{(2,1)}(x,\xi,\lambda)\times \]

\[ \times \Phi^{(1)}(\xi)\left(C_{02}^{(1)}(\xi)\right)^{-1}\,d\xi \tag{74} \]

as \(\nu\to\infty\).

Substituting, in place of \(G^{(i,j)}(x,\xi,\lambda)\), their asymptotic representations (66)—(69), integrating by parts the terms containing \(\tau_{sk}^{(i)}(\xi)\), \(2-k\) \((k=0,1)\) times and taking into account (56), (70) and the conditions of the theorem, we obtain

\[ \int_{a_1}^{b_1}G^{(1,1)}(x,\xi,\lambda)\Phi^{(1)}(\xi) \left(C_{02}^{(1)}(\xi)\right)^{-1}\,d\xi = \]

\[ = -\lambda^{-1}\Phi^{(1)}(x)+\lambda^{-\frac32}W_0^{(1,1)}(x,\lambda) +\lambda^{-1}W_1^{(1,1)}(x,\lambda)\times \]

\[ \times \exp\left(\mu_1\sqrt{\lambda}\int_{a_1}^{x}S(\alpha)\,d\alpha\right) +\lambda^{-1}W_2^{(1,1)}(x,\lambda)\times \]

\[ \times \exp\left(\mu_1\sqrt{\lambda}\int_{x}^{b_1}S(\alpha)\,d\alpha\right), \tag{75} \]

\[ \int_{a_2}^{b_2}G^{(1,2)}(x,\xi,\lambda)\Phi^{(2)}(\xi) \left(C_{02}^{(2)}(\xi)\right)^{-1}\,d\xi =\lambda^{-\frac32}W_0^{(1,2)}(x,\lambda), \tag{76} \]

\[ \int_{a_2}^{b_2}G^{(2,2)}(x,\xi,\lambda)\Phi^{(2)}(\xi) \left(C_{02}^{(2)}(\xi)\right)^{-1}\,d\xi = -\lambda^{-2}\Phi^{(2)}(x)+ \]

\[ +\lambda^{-3}W_0^{(2,2)}(x,\lambda) +\lambda^{-2}W_1^{(2,2)}(x,\lambda) \exp\left(\lambda\int_{a_2}^{x}\varphi_1^{(2)}(\alpha)\,d\alpha\right)+ \]

\[ +\lambda^{-2}W_2^{(2,2)}(x,\lambda) \exp\left(-\lambda\int_{x}^{b_2}\varphi_2^{(2)}(\alpha)\,d\alpha\right), \tag{77} \]

\[ \int_{a_2}^{b_1}G^{(2,1)}(x,\xi,\lambda)\Phi^{(1)}(\xi) \left(C_{02}^{(1)}(\xi)\right)^{-1}\,d\xi =\lambda^{-3}W_0^{(2,1)}(x,\lambda), \tag{78} \]

where \(W_s^{(i,j)}(x,\lambda)\) \((i,j=1,2;\ s=0,1,2)\) are functions bounded in modulus in \(O_\nu\) for all \(x\in[a_i,b_i]\).

Substituting (75) into (71) and passing to the limit as \(\nu\to\infty\), we shall have

\[ \lim_{\nu\to\infty}J_\nu^{(1,1)}(\Phi)= \frac{-1}{2\pi\sqrt{-1}} \lim_{\nu\to\infty}\int_{O_\nu}d\lambda \int_{a_1}^{b_1}G^{(1,1)}(x,\xi,\lambda)\times \]

\[ \times \Phi^{(1)}(\xi)\left(C_{02}^{(1)}(\xi)\right)^{-1}\,d\xi = \]

\[ = \Phi^{(1)}(x)-\frac{1}{2\pi\sqrt{-1}}\lim_{\nu\to\infty}\int_{O_\nu}\lambda^{-1}W^{(1,1)}_1(x,\lambda)\times \]

\[ \times \exp\left(\mu_1\sqrt{\lambda}\int_{a_1}^{x} S(\alpha)\,d\alpha\right)d\lambda- \]

\[ -\frac{1}{2\pi\sqrt{-1}}\lim_{\nu\to\infty}\int_{O_\nu}\lambda^{-1}W^{(1,1)}_2(x,\lambda) \exp\left(\mu_1\sqrt{\lambda}\int_{x}^{b_1} S(\alpha)\,d\alpha\right)d\lambda. \tag{79} \]

The limits of the integrals over \(O_\nu\) on the right-hand side of (79) are equal to zero uniformly with respect to \(x\in(a_1,b_1)\), by virtue of a known lemma (see [1], Ch. III). Then, on the basis of the residue theorem, from (79) we obtain the first formula of the theorem.

Substituting (76) and (72) and passing to the limit as \(\nu\to\infty\), we obtain

\[ \lim_{\nu\to\infty} J_\nu^{(1,2)}(\Phi) = -\frac{1}{2\pi\sqrt{-1}}\lim_{\nu\to\infty}\int_{O_\nu} d\lambda \int_{a_2}^{b_2} G^{(1,2)}(x,\xi,\lambda)\times \]

\[ \times \Phi^{(2)}(\xi)\bigl(C_{02}^{(2)}(\xi)\bigr)^{-1}\,d\xi=0 \]

uniformly with respect to \(x\in(a_2,b_2)\). Hence, on the basis of the residue theorem, we obtain the second formula of the theorem.

The third and fourth formulas are proved analogously.

References

1. Rasoulov M. L. The Method of Contour Integration. Moscow, Nauka, 1964.
2. Rasoulov M. L. Mathematics Collection, 48 (72), 3, 1959.
3. Goursat É. A Course of Mathematical Analysis, vol. III, part I. Gostekhizdat, 1939.
4. Tamarkin Ya. D. On certain general problems in the theory of ordinary differential equations and the expansion of arbitrary functions in series. Petrograd, 1917.
5. Birkhoff G. D. On the asymptotic character of the solution of certain linear differential equations containing a parameter, Trans. Am. Math. Soc., 9, 1908.
6. Gaiduk S. K. Dissertation. Minsk, 1965.

Received by the editors
May 25, 1966

Belorussian Polytechnic Institute