Full Text
UDC 517.946
ON THE EXPANSION IN EIGENFUNCTIONS OF A SINGULAR BOUNDARY-VALUE PROBLEM FOR LEGENDRE’S EQUATION
N. A. BELOVA, Ya. S. UFLYAND
In the present note we consider the expansion in Legendre functions with complex index which arises in the study of the solution of the differential equation
\[ [(x^2-1)y']' + \lambda y = 0,\qquad 1<x_0<x<\infty \tag{1} \]
with boundary conditions
\[ y(x_0)=0,\qquad y(\infty)<\infty . \tag{2} \]
If one puts \(\lambda=-\nu(\nu+1)\), then the general solution of equation (1), bounded as \(x\to\infty\), can be represented in terms of Legendre functions [1] as
\[ y=AP_\nu(x)+BQ_\nu(x),\qquad \operatorname{Re}\nu>-1, \tag{3} \]
where
\[ Q_\nu(x)=\sqrt{\pi}\, \frac{\Gamma(\nu+1)}{\Gamma\left(\nu+\frac{3}{2}\right)} (2x)^{-\nu-1} F\left(\frac{\nu}{2}+1,\frac{\nu}{2}+\frac{1}{2},\nu+\frac{3}{2},\frac{1}{x^2}\right), \tag{4} \]
\[ P_\nu(x)=\frac{1}{\sqrt{\pi}}\left[ \frac{\Gamma\left(\nu+\frac{1}{2}\right)}{\Gamma(\nu+1)} (2x)^\nu F\left(\frac{1}{2}-\frac{\nu}{2},-\frac{\nu}{2},\frac{1}{2}-\nu,\frac{1}{x^2}\right) +\right. \]
\[ \left. +\frac{\Gamma\left(-\nu-\frac{1}{2}\right)}{\Gamma(-\nu)} (2x)^{-\nu-1} F\left(\frac{\nu}{2}+1,\frac{\nu}{2}+\frac{1}{2},\nu+\frac{3}{2},\frac{1}{x^2}\right) \right]. \tag{5} \]
Under the condition that \(\operatorname{Re}\nu>0\), the eigenfunctions of the problem under investigation will be the solutions \(y=Q_\nu(x)\), and the eigenvalues must be found from the equation
\[ Q_\nu(x_0)=0. \tag{6} \]
We shall show that, in the ranges of variation of \(\nu\) and \(x_0\) under consideration, equality (6) is impossible.
Suppose that there exists a complex root \(\nu\) of equation (6), so that the corresponding eigenvalue has the form \(\lambda=\alpha+i\beta,\ \beta\ne0\). Then, substituting in the integral relation
\[ (\lambda_1-\lambda_2)\int_{x_0}^{x} y_1y_2\,dx = (x^2-1)(y_1y_2'-y_2y_1')\bigg|_{x_0}^{x} \tag{7} \]
\(\lambda_{1,2}=\alpha\pm i\beta,\ y_{1,2}=Q_{\nu_{1,2}}(x)\), and passing to the limit as \(x\to\infty\), we obtain that for \(\operatorname{Re}\nu>-\frac{1}{2}\), \(\beta=0\), whence it follows that no complex eigenvalues exist, i.e. equation (6) has no complex roots.
A direct examination of the hypergeometric series (4) shows that for \(\nu>-1,\ Q_\nu(x_0)\ne0\) for any \(x_0>1\), so that equation (6) also has no real roots. Consequently, for \(\operatorname{Re}\nu>0\) the boundary-value problem under study has only the trivial solution.
Let us now turn to the interval
\[ -1 < \operatorname{Re}\nu \leq 0, \tag{8} \]
for which, evidently, there exist eigenfunctions
\[ y=Q_\nu(x_0)P_\nu(x)-P_\nu(x_0)Q_\nu(x). \tag{9} \]
Thus it has been shown that the boundary-value problem under consideration has a continuous spectrum of eigenvalues filling the strip (8).
If, in particular, one chooses \(\nu=-\dfrac{1}{2}+i\tau\), then the eigenfunctions (9)
\[ y=y(x,\tau)=Q_{-\frac12+i\tau}(x_0)P_{-\frac12+i\tau}(x) - P_{-\frac12+i\tau}(x_0)Q_{-\frac12+i\tau}(x) \tag{10} \]
as well as the eigenvalues
\[ \lambda=\tau^2+\frac14,\qquad 0\leq \tau<\infty, \tag{11} \]
become real.
We now investigate the question of expanding a function \(f(x)\), given on the interval \((x_0,\infty)\), in an integral with respect to the eigenfunctions \(y(x,\tau)\):
\[ f(x)=\int_0^\infty F(\tau)y(x,\tau)\,d\tau. \tag{12} \]
In order to carry out a formal derivation*) of an expression for the quantities \(F(\tau)\), consider the mixed problem for the partial differential equation
\[ \frac{\partial}{\partial x} \left[(x^2-1)\frac{\partial u}{\partial x}\right] = \frac{\partial u}{\partial t}, \qquad x_0<x<\infty,\quad t>0 \tag{13} \]
under the following initial and boundary conditions:
\[ u(x,0)=f(x), \tag{14} \]
\[ u(x_0,t)=0,\qquad u(\infty,t)<\infty. \tag{15} \]
After the Laplace transform
\[ \bar u(x)=\int_0^\infty u(x,t)\exp(-pt)\,dt \tag{16} \]
we arrive at the boundary-value problem for the ordinary differential equation
\[ [(x^2-1)\bar u']'-p\bar u=-f(x),\qquad \bar u(x_0)=0,\qquad \bar u(\infty)<\infty, \tag{17} \]
the solution of which can be represented in the form
\[ \bar u=\frac{1}{Q_\nu(x_0)} \int_{x_0}^{\infty} f(\xi)G(x,\xi,p)\,d\xi, \tag{18} \]
where
\[ G(x,\xi,p)= \begin{cases} \bar u_1(x)\bar u_2(\xi), & x\leq \xi,\\ \bar u_1(\xi)\bar u_2(x), & x\geq \xi, \end{cases} \tag{19} \]
\[ \bar u_1=Q_\nu(x_0)P_\nu(x)-P_\nu(x_0)Q_\nu(x),\qquad \bar u_2=Q_\nu(x), \tag{20} \]
\[ \nu=-\frac12+\sqrt{p+\frac14},\qquad \operatorname{Re}\nu>-\frac12. \tag{21} \]
Application of the Riemann—Mellin inversion formula gives
\[ u=\int_{x_0}^{\infty} f(\xi)\,d\xi\, \frac{1}{2\pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} G(x,\xi,p)\exp(pt)\, \frac{dp}{Q_\nu(x_0)}, \qquad \sigma>-\frac14. \tag{22} \]
*) The establishment of the classes of functions \(f(x)\) will be the subject of a separate publication.
On the Expansion in Eigenfunctions of a Boundary-Value Problem
The complex integral entering into (22) can be transformed into real integrals along the banks of the cut drawn along the segment \(-\infty < p < -\frac14\) of the real axis of the plane of the complex variable \(p\). After certain transformations, taking into account the relation [1]
\[ Q_{-\nu-1}(x)=Q_\nu(x)-\pi \operatorname{ctg}\pi\nu\, P_\nu(x), \tag{23} \]
we obtain
\[ u=\int_{x_0}^{\infty} f(\xi)\,d\xi \int_{0}^{\infty} \exp\left[-\left(\tau^2+\frac14\right)t\right] y(x,\tau)y(\xi,\tau) \frac{\tau\,\operatorname{th}\pi\tau\,d\tau} {\left|Q_{-\frac12+i\tau}(x_0)\right|^2}. \tag{24} \]
Finally, interchanging the order of integration and putting \(t=0\), we arrive at an integral expansion of the form
\[ f(x)=\int_{0}^{\infty} y(x,\tau) \frac{\tau\,\operatorname{th}\pi\tau\,d\tau} {\left|Q_{-\frac12+i\tau}(x_0)\right|^2} \int_{x_0}^{\infty} f(\xi)y(\xi,\tau)\,d\xi . \tag{25} \]
From comparison of the equalities (25) and (13) there follows the desired formula
\[ F(\tau)= \frac{\tau\,\operatorname{th}\pi\tau} {\left|Q_{-\frac12+i\tau}(x_0)\right|^2} \int_{x_0}^{\infty} f(\xi)y(\xi,\tau)\,d\xi, \qquad \tau \geqslant 0. \tag{26} \]
The expression
\[ \bar f(\tau)=\int_{x_0}^{\infty} f(x)y(x,\tau)\,dx \tag{27} \]
may be regarded as an integral transform of the function \(f(x)\) with respect to the eigenfunctions of the singular boundary-value problem (1), (2). In this case (13) and (26) give us the corresponding inversion formula in the form
\[ f(x)=\int_{0}^{\infty} \frac{\tau\,\operatorname{th}\pi\tau} {\left|Q_{-\frac12+i\tau}(x_0)\right|^2} y(x,\tau)\bar f(\tau)\,d\tau . \tag{28} \]
The pair of reciprocal formulas (27), (28) is a generalization of the Mehler–Fock integral transform to the interval \(1<x_0<x<\infty\). Passing in (25) to the limit as \(x_0\to 1\), we arrive at the Mehler–Fock expansion [2–5]
\[ f(x)=\int_{0}^{\infty} P_{-\frac12+i\tau}(x)\operatorname{th}\pi\tau\,d\tau \int_{1}^{\infty} f(\xi)P_{-\frac12+i\tau}(\xi)\,d\xi . \tag{29} \]
Let us note, in conclusion, that formulas analogous to (25) can also be obtained for boundary conditions of the second or third kind at the point \(x=x_0\).
References
- Hobson E. W. The Theory of Spherical and Spheroidal Functions. IL, 1952.
- Lebedev N. N. Some integral transforms of mathematical physics. Abstract of doctoral dissertation, L., 1951.
- Lebedev N. N. Special Functions and Their Applications. Fizmatgiz, 1963.
- Mehler F. G. Math. Annalen, 18, 161, 1881.
- Fock V. A. DAN SSSR, 39, 279, 1943.
Received by the editors
May 14, 1966
Leningrad Electrotechnical Institute
named after Ulyanov (Lenin)