MATHEMATICAL PHYSICS

M. G. BELKINA

ASYMPTOTIC REPRESENTATIONS OF SPHEROIDAL FUNCTIONS WITH AZIMUTHAL INDEX \(m=1\)

(Presented by Academician V. A. Fock, 20 II 1957)

1. Let an equation of the form be given

\[ Y''+c^2p(\eta)Y=0,\qquad c\gg 1, \tag{1} \]

where the function \(p(\eta)\) has \(n\) poles of first order and zeros at the points \(\eta_k\). Suppose, further, that it is possible to choose a “standard” \({}^{1}\) equation

\[ y''+P(\varphi)y=0, \tag{2} \]

whose independent solutions \(y_1(\varphi)\) and \(y_2(\varphi)\) are known, and such that the poles and zeros \(\varphi_k\) of the coefficient \(P(\varphi)\) can be put into a one-to-one and monotone correspondence with \(\eta_k\) so that poles correspond to poles, and zeros to zeros of the same order. Then the asymptotic representation of the general solution of equation (1) has the form (\(B_1\) and \(B_2\) are arbitrary constants)

\[ Y(\eta)=\sqrt[4]{\frac{P[\varphi(\eta)]}{p(\eta)}}\{B_1y_1[\varphi(\eta)]+B_2y_2[\varphi(\eta)]\}, \tag{3} \]

where the relation between the independent variables \(\varphi=\varphi(\eta)\) is defined by

\[ \int_{\varphi_k}^{\varphi}\sqrt{P(\varphi)}\,d\varphi = c\int_{\eta_k}^{\eta}\sqrt{p(\eta)}\,d\eta \tag{4} \]

and by the additional conditions

\[ \int_{\varphi_k}^{\varphi_i}\sqrt{P(\varphi)}\,d\varphi = c\int_{\eta_k}^{\eta_i}\sqrt{p(\eta)}\,d\eta \qquad (i=1,2,\ldots,k-1,k+1,\ldots n). \tag{5} \]

The function \(\varphi=\varphi(\eta)\) transforms the corresponding \(\eta_k\) and \(\varphi_k\) into one another, with \(\varphi'(\eta)\ne 0\), \(\varphi'(\eta)\ne \infty\). Conditions (5) can be satisfied if the coefficient \(P(\varphi)\) contains \(n-1\) free parameters (cf. \({}^{1,2}\)).

2. The function \(Y(\eta)\), connected with the angular spheroidal function \(S_{1,l}^{(1)}(c,\eta)\) \({}^{3}\) by the relation

\[ Y(\eta)=\sqrt{1-\eta^2}\,S_{1,l}^{(1)}(c,\eta), \tag{6} \]

satisfies equation (1) with the coefficient of \(Y\)

\[ p(\eta)=1+\frac{\beta}{1-\eta^2},\qquad \beta=\chi-1+\frac{2}{c^2}, \tag{7} \]

where \(\chi\) is connected with the separation constant \(A\), introduced in \({}^{3}\), by the formula

\[ A=-c^2\chi-2. \tag{8} \]

The function \(Y(\eta)\) is finite in the interval \((-1,1)\) and vanishes at its endpoints.

Since equation (1) and the boundary conditions are symmetric with respect to \(\eta=0\), the eigenfunctions of the equation are even or odd, and one may restrict consideration to the interval \((0,1)\) with the boundary conditions (cf. (1))

\[ \begin{aligned} Y(1)&=0,\qquad Y'(0)=0 &&(\text{for even }Y),\\ Y(1)&=0,\qquad Y(0)=0 &&(\text{for odd }Y). \end{aligned} \tag{9} \]

The function (7) has a pole of first order at \(\eta=1\) and a zero at the point \(\eta_1=+\sqrt{1+\beta}\), real for \(\beta\geq -1\). For \(\beta<0\) the zero \(\eta_1\) lies inside the interval \((0,1)\), for \(\beta>0\) outside this interval, and for \(\beta=0\) it merges with the pole, and the function \(p(\eta)\) becomes a constant.

1. As the comparison equation we take equation (2) with coefficient, for \(y\),

\[ P(\varphi)=1+\frac{b}{\varphi}, \tag{10} \]

which takes into account all the singularities of the function \(p(\eta)\). The solutions of this equation are degenerate hypergeometric functions with indices \(k=-i\,\frac{b}{2}\), \(m=\frac12\) and argument \(2i\varphi\) (see, for example, (4), Chap. 16). The relation \(\varphi=\varphi(\eta)\) is specified by

\[ \int_{\varphi_1}^{\varphi}\sqrt{P(\varphi)}\,d\varphi = c\int_{\eta}^{\eta_1}\sqrt{P(\eta)}\,d\eta \tag{11} \]

under the additional condition (5), which puts the poles of the functions \(p(\eta)\) and \(P(\varphi)\) into correspondence and makes it possible to relate the parameters \(b\) and \(\beta=-(1-\eta_1^2)\).

The boundary condition at \(\eta=1\) is satisfied by the solution

\[ Y(\eta)=B\sqrt[4]{\frac{P(\varphi)}{p(\eta)}}\, M_{-i\frac{b}{2},\,\frac12}(2i\varphi). \tag{12} \]

Using the asymptotic representation by the method of B. W. K.,

\[ M_{-i\frac{b}{2},\,\frac12}(2i\varphi)= \tag{13} \]

\[ = \frac{2i e^{3\pi b/4}}{\sqrt[4]{P(\varphi)}}\sqrt{\frac{\sh(\pi b/2)}{\pi b/2}}\, \sin\left( \int_{\varphi_1}^{\varphi}\sqrt{P(\varphi)}\,d\varphi -\frac{b}{2} +\frac{b}{2}\ln\left(-\frac{b}{2}\right) -\arc\Gamma\left(1+i\frac{b}{2}\right) \right), \]

valid for large \(\varphi\), but arbitrary \(b\), we obtain from the boundary condition at \(\eta=0\) formulas for determining the eigenvalues \(\beta\):

\[ c\int_{0}^{\eta_1}\sqrt{p(\eta)}\,d\eta = \left(l+\frac12\right)\frac{\pi}{2} -\arc\chi^{-}(b) \qquad (b<0,\ -1<\beta\leq 0), \]

\[ c\int_{0}^{1}\sqrt{p(\eta)}\,d\eta = \left(l+\frac32\right)\frac{\pi}{2} -\arc\chi^{+}(b) \qquad (b>0,\ 0\leq \beta<\infty). \tag{14} \]

Here the functions \(\chi^{-}(b)\) and \(\chi^{+}(b)\) are defined as the ratio of the asymptotic representation of \(\Gamma\left(1+i\frac{b}{2}\right)\) for \(b<0\) and \(b>0\), respectively, to \(\Gamma\left(1+i\frac{b}{2}\right)\) itself (cf. (5), pp. 568–569).

\[ \chi^{-}(b)= \frac{\sqrt{-\pi b}\exp\left\{\frac{b\pi}{4}+i\left[\frac{b}{2}\ln\left(-\frac{b}{2}\right)-\frac{b}{2}-\frac{\pi}{4}\right]\right\}} {\Gamma\left(1+i\frac{b}{2}\right)} \qquad (b<0), \]

\[ \chi^{+}(b)= \frac{\sqrt{\pi b}\exp\left\{-\frac{b\pi}{4}+i\left[\frac{b}{2}\ln\frac{b}{2}-\frac{b}{2}+\frac{\pi}{4}\right]\right\}} {\Gamma\left(1+i\frac{b}{2}\right)} \qquad (b>0). \tag{15} \]

As \(\beta\to0\) \((b\to0)\), both relations (14) pass into the exact formula

\[ c=(l+1)\frac{\pi}{2}\qquad (\beta=0), \tag{16} \]

and the function (12) into the exact solution of equation (1) for \(\beta=0\). As \(b\to-\infty\) \((\beta<0)\) and as \(b\to\infty\) \((\beta>0)\), relations (14) take the form

\[ c\int_{0}^{\eta_1}\sqrt{p(\eta)}\,d\eta = \left(l+\frac12\right)\frac{\pi}{2} \qquad (b\to-\infty,\ \beta<0); \tag{17} \]

\[ c\int_{0}^{1}\sqrt{p(\eta)}\,d\eta = \left(l+\frac32\right)\frac{\pi}{2} \qquad (b\to\infty,\ \beta>0). \tag{18} \]

The first expression can be obtained by taking into account, with the aid of the Airy equation,

\[ y''-\varphi y=0 \tag{19} \]

only the root \(\eta_1\) for \(\eta_1<1\), and the second with the aid of the equation

\[ y''+\frac{1}{\varphi}y=0,\qquad y=\sqrt{\varphi}\,J_1(2\sqrt{\varphi}) \tag{20} \]

Table 1

Values of \(\kappa_l=\beta_l+1-\dfrac{2}{c^2}\)

(\(J_1\) is the Bessel function), taking into account for \(\eta_1 > 1\) only the pole of the function \(p(\eta)\), if the zeros of the functions \(v(\varphi)\), \(v'(\varphi)\), \(J_1(\varphi)\), \(J'_1(\varphi)\) are replaced by their approximate values\(^6\).

As \(\beta \to 0\), formulas (17) and (18) cease to be valid, i.e., representations by means of Airy and Bessel functions are valid only when the root \(\eta_1\) is sufficiently far from the pole \(\eta = 1\).

The eigenvalues \(\varkappa_l(c)\) in Table 1 were computed: 1) by the exact formulas; 2) by approximate relations with the aid of the expressed hypergeometric functions (14); 3) by Airy functions (17); 4) by Bessel functions. In the last case the roots of \(J_1(\varphi)\) and \(J'_1(\varphi)\) were not replaced by their approximate values, which gives better results. For Airy functions, formula (17) is better than the formula with exact roots \(v(\varphi)\) and \(v'(\varphi)\).

I take this opportunity to thank V. A. Fock and L. A. Weinstein for valuable advice.

Received
8 II 1957

REFERENCES

\(^1\) A. A. Dorodnitsyn, Uspekhi Mat. Nauk, 7, no. 6 (52), 3 (1952).
\(^2\) M. I. Petrashen, Dokl. Akad. Nauk, 50, 147 (1945).
\(^3\) J. A. Stratton, P. M. Morse, L. J. Chu, R. A. Hutner, Elliptic Cylinder and Spheroidal Wave Functions, USA, 1941.
\(^4\) E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, Part II, Moscow–Leningrad, 1934.
\(^5\) V. A. Fock, Radiotekhnika i Elektronika, no. 5, 560 (1956).
\(^6\) V. A. Fock, Tables of Airy Functions, 1945.