Abstract Generated abstract
This paper examines the connection between inversion formulas for the Sommerfeld integral, used to represent solutions of the two-dimensional Helmholtz equation, and the Kontorovich-Lebedev transformation. Starting from the odd form of the Sommerfeld density and assuming suitable analyticity and decay conditions, it rewrites the density by a Fourier integral and formally interchanges integrations after a contour deformation. The derivation shows that, when the value at the origin vanishes, the Sommerfeld representation can be transformed into an integral over Bessel functions of complex order, with the inverse formula expressed through Hankel functions. Introducing an even spectral function then yields the standard Kontorovich-Lebedev transform pair.
Full Text
MATHEMATICAL PHYSICS
G. D. MALYUZHINETS
RELATION BETWEEN THE INVERSION FORMULAS FOR THE SOMMERFELD INTEGRAL AND THE KONTOROVICH–LEBEDEV FORMULAS
(Presented by Academician M. A. Leontovich on 20 IX 1957)
The Sommerfeld integral
\[ S(r,\varphi)=\frac{1}{2\pi i}\int_{\gamma} e^{-ikr\cos\alpha}s(\varphi+\alpha)\,d\alpha, \]
which gives, in the region \(r>0\), the general solution of the two-dimensional equation of harmonic oscillations \(\Delta S+k^{2}S=0\) \((S\sim e^{-i\omega t})\), may, using the symmetry of the contour \(\gamma=\gamma_1+\gamma_2\) (see Fig. 1), be represented in the form
\[ S(r,\varphi)=\frac{1}{2\pi i}\int_{\gamma} e^{-ikr\cos\alpha} \frac{s(\varphi+\alpha)-s(\varphi-\alpha)}{2}\,d\alpha . \]
Considering this integral for a fixed value of \(\varphi\), we rewrite it briefly as
\[ F(r)=\frac{1}{2\pi i}\int_{\gamma} e^{-ikr\cos\alpha}f(\alpha)\,d\alpha . \tag{1} \]
If \(F(r)=O\{r^{-1+a}\exp(br)\}\) \((a>0)\), then in the class of functions
\[ f(\alpha)=O\{\exp[(1-a)|\operatorname{Im}\alpha|]\}\quad (|\operatorname{Im}\alpha|\to\infty) \]
the integral equation (1), where the loops of the contour \(\gamma\) are wholly situated in the regions of regularity of the function \(f(\alpha)/\sin\alpha\), has a unique odd solution
\[ f(\alpha)=-\frac{ik\sin\alpha}{2}\int_{0}^{\infty} F(r)e^{ikr\cos\alpha}\,dr . \tag{2} \]
Fig. 1
Expressions (1), (2) are the inversion formulas for the Sommerfeld integral \((^3)\), used for solving boundary-value problems in a wedge-shaped domain.
Since the function \(f(\alpha)\) is odd, the integral (1) can be written in the form
\[ F(r)=\frac{1}{\pi i}\int_{\gamma_1} e^{-ikr\cos\alpha}f(\alpha)\,d\alpha . \tag{3} \]
In the special case when the function \(f(\alpha)\) is regular for \(|\operatorname{Im}\alpha|>\mathrm{const}\), it follows from (3) that \((^4)\) \(F(0)=2if(i\infty)\).
We shall show that if \(F(0)\), and consequently also \(f(i\infty)\), is equal to zero, the transformation of the Sommerfeld integral can be reduced to the Kontorovich–Lebedev transformation \((^5)\):
\[ F(r)=\frac{1}{2}\int_{-i\infty}^{i\infty} e^{-i\nu\pi/2} J_\nu(kr)\,\omega(\nu)\,\nu\,d\nu, \tag{4} \]
\[ \omega(\nu)=\int_{0}^{\infty} e^{i\nu\pi/2} H_\nu^{(1)}(kr)\,F(r)\,\frac{dr}{r}\,* . \tag{5} \]
Since the conditions of applicability of formulas (4), (5) have been investigated in \((^{5})\), we shall restrict ourselves to a formal derivation.
To this end, suppose that the analytic function \(f(\alpha)\) is regular in the strip \(|\operatorname{Re}\alpha|<\pi/2+\varepsilon\) and decreases there as \(|\operatorname{Im}\alpha|\to\infty\) like \(f(\alpha)=O(e^{-c|\operatorname{Im}\alpha|})\). Then we represent the function \(f(\alpha)\) by a Fourier integral of the form \((^{4})\)
\[ f(\alpha)=\frac{i}{2}\int_{-i\infty}^{i\infty} g(\nu)e^{-i\nu\alpha}\,d\nu, \tag{6} \]
absolutely convergent in this strip, where the function
\[ g(\nu)=\frac{i}{\pi}\int_{-i\infty}^{i\infty} f(\alpha)e^{i\nu\alpha}\,d\alpha \tag{7} \]
is odd, just as \(f(\alpha)\) is, and moreover the last integral is absolutely convergent in the strip \(|\operatorname{Re}\nu|<c\).
Substituting (6) into (3), we have
\[ F(r)=-\frac{1}{2\pi}\int_{\gamma_1} d\alpha \int_{-i\infty}^{i\infty} g(\nu)e^{-i(kr\cos\alpha-\nu\alpha)}\,d\nu, \tag{8} \]
where we have changed the sign of the integration variable \(\nu\) to the opposite one and used the oddness of \(g(\nu)\).
Regarding temporarily the quantity \(k\) as positive imaginary, we can deform the contour \(\gamma_1\) so that it lies entirely in the strip \(|\operatorname{Re}\alpha|<\pi/2+\varepsilon\). In this case integral (8) becomes absolutely convergent, and the interchange of the order of integration is legitimate.
In the integral
\[ F(r)=-\frac{1}{2\pi}\int_{-i\infty}^{i\infty} g(\nu)\,d\nu \int_{\gamma_1} e^{-i(kr\cos\alpha-\nu\alpha)}\,d\alpha \]
one may return to positive values of the parameter \(k\) and to the original form of the contour.
Hence, since
\[ J_\nu(kr)=-\frac{\exp(i\nu\pi/2)}{2\pi}\int_{\gamma_1} e^{-i(kr\cos\alpha-\nu\alpha)}\,d\alpha, \]
we obtain
\[ F(r)=\int_{-i\infty}^{i\infty} e^{-i\nu\pi/2}J_\nu(kr)g(\nu)\,d\nu. \tag{9} \]
To obtain the inverse formula, splitting integral (2) into parts:
\[
\int_{0}^{\infty}=\int_{0}^{a}+\int_{a}^{\infty},
\]
and integral (7) into parts
\[
\int_{-i\infty}^{i\infty}
=
\int_{-ib}^{ib}
+
\left(
\int_{-i\infty}^{-ib}
+
\int_{ib}^{i\infty}
\right)
\]
and substituting (2) into (7), we have
\[ \underline{\hspace{3cm}} \]
* The authors use \(H_\nu^{(2)}(kr)\).
\[ g(\nu)=-\frac{k}{2\pi}\left(\int_{-i\infty}^{-ib}+\int_{ib}^{i\infty}\right)d\alpha\int_a^\infty F(r)e^{i(kr\cos\alpha+\nu\alpha)}\sin\alpha\,dr+A, \]
where by \(A\) are denoted integrals tending to zero as \(a,b\to 0\). The first term on the right-hand side for \(\operatorname{Im} k>0\) is an absolutely convergent integral, in which the order of integration may be interchanged.
Therefore, in the limit we obtain
\[ g(\nu)=-\frac{k}{2\pi}\int_0^\infty F(r)\,dr\int_{-i\infty}^{i\infty} e^{i(kr\cos\alpha+\nu\alpha)}\sin\alpha\,d\alpha, \]
or, since
\[ \int_{-i\infty}^{i\infty} e^{i(kr\cos\alpha+\nu\alpha)}\sin\alpha\,d\alpha = -\frac{\pi\nu}{kr}e^{i\nu\pi/2}H_\nu^{(1)}(kr), \]
\[ g(\nu)=\frac{\nu}{2}e^{i\nu\pi/2}\int_0^\infty F(r)H_\nu^{(1)}kr\,\frac{dr}{r}. \tag{10} \]
If in (9) and (10), instead of the odd function \(g(\nu)\), we introduce the even function \(\omega(\nu)=\dfrac{2}{\nu}g(\nu)\), we obtain the Kontorovich–Lebedev transformation formulas (4), (5).
Acoustics Institute
Academy of Sciences of the USSR
Received
5 IX 1957
CITED LITERATURE
- Ph. Frank, R. Mises, Differential and Integral Equations of Mathematical Physics, Ch. XX, L.—M., 1937.
- G. D. Malyuzhinets, “Some generalizations of the method of reflections in the theory of diffraction of sinusoidal waves,” Abstract of doctoral dissertation, Ch. IV, FIAN, Publishing House of the Academy of Sciences of the USSR, 1950.
- G. D. Malyuzhinets, DAN, 118, No. 6 (1958).
- G. D. Malyuzhinets, Akusticheskii zhurnal, 1, Nos. 2 and 3 (1955) (errata 2, No. 2 (1956)).
- M. I. Kontorovich, N. N. Lebedev, ZhETF, 8, Nos. 10 and 11 (1938).