May 11, 1972 - A modified algorithm for computing Hankel (Bessel) functions of complex index Y and complex argument z, Re z > 0, to any required accuracy, ...
COMPUTATION OF HANKEL (BESSEL) FUNCTIONS OF COMPLEX INDEX AND ARGUMENT BY NUMERICAL INTEGRATION OF A SCHLiiFLI CONTOUR INTEGRAL* G. F. REMENETS Leningrad (Received 11 May 1972)
A modified algorithm for computing Hankel (Bessel) functions of complex index Y and complex argument z, Re z > 0, to any required accuracy, is described. Check computations on the basis of the algorithm are quoted for the cases Izl, lvl=O(lO), O(10’). The algorithm described below is a modification of the algorithm for computing Hankel functions by means of the Schlafli integral form [I] and the Sommerfeld integral form [24] . In [l] , Hankel functions of large index and argument were computed by utilizing the asymptotic form of the function in terms of the Hankel function with index l/3. The latter functions were in turn computed by means of the Schlafli integral form [5]. However, the asymptotic forms can be avoided and the Hankel functions of the 1st and 2nd kind FIG. 1 computed directly in terms of the Schlifli integral form or in terms of the equivalent Sommerfeld form [ 51, by bringing the contour of integration into coincidence with the contour of descent of the saddle-point method (see [6], Part II, Section 78; and [7]). This is the idea behind the algorithm employed in [2] . We have modified the algorithm of [2] in such a way that instead of performing the numerical integration along a step-line approximating the contour of steepest descent, it is performed along the contour itself [4] . This results in a widening of the range of the program for computing the Hankel (Bessel) functions of complex index and argument. It was shown in [7] that the question as to which of the particular solutions of Bessel’s equation corresponds to the result of the numerical integration described above, is answered by the position of the saddle-point on the complex plane and demands special *Zh. vjkhisl. Mat. mat. Fiz., 13,6,
1417-1424,
1973.
58
Computation of Hankel (Bessel) functions of complex index and argument
59
investigation. The topology of the constant-phase contours in the cases where the cylindrical functions have pure imaginary or pure real arguments was also examined in [7]. In the present paper we discuss the topology of the constant phase contours for the case Re z > 0. Special attention is paid to cases where the two saddle-points are connected by a contour of constant phase. This case represents the “divide” between the two versions, in one of which numerical integration along the contour of constant phase gives the Hankel function, and in the other, the Bessel function.
General Working Expressions The Hankel functions of the 1st and 2nd kinds are given in the Schlafli integral form by [5] Hi” (z)= - f
fp(z)=
fS
s e'/rZ(h-l/A)h-(Y+l)dh, I?1
Re z>O,
e’l,z(A-llA)h-(v+l)dh,
Re z>O. rr The contours IYiand I’2 leave from zero into the right-hand half-plane then go to infinity in the left-hand half-plane (Fig. 1). A cut is made along the negative semi-axis. As the phase function we take the expression - (v+l)
In h.
(3)
From the condition df/dA = 0 we find the saddle-points
(4)
where s = 1,2, and A1 = l/AZ. It can be seen from Eq. (4) that two saddle-points merge into one only when z=~+l, and they can only merge at the point X = 1. Noting that the parameters z and v and the variable X, are complex, we put Z=Zi+iZ*,
where i is the square root of -1.
v=ji+ijz,
h=teiq,
60
G.
F. Remenets
The real and imaginary parts of the phase function f(X) can be written in the new variables as
sincp- VI+
I) ln (t) S&P,
sin cp-jz In(t) - (j,+1)
cp.
The contours of steepest descent and steepest rise are given by the equations Im f(h) =Im f(k) SF,.
(5)
In expanded form, Eq. (5) becomes a transcendental equation in t in the general case, or a quadratic equation in the case when Im v=jZ=O:
t2 ( zZcos cp+zi sin cp)-2t ( (j,+l> cp+F,) -
(zZ cos q-zi
(6)
sin cp) =2tjz In (t) .
By bringing the contour l”, into coincidence with the contour of steepest descent, we eliminate the oscillations of the modulus of the integrand in Eqs. (1) and (2). Then, Eqs. (1) and (2) take the form 1 efWh=exp 1’S =exp (iF,) + i exp (i&J
(iF,)
1 eRe f@) (dt + it dq) exp (cps)
1’s s eRcf(t, Q) (cos cp dt / dq+t rs 1 eRef(t, v) (sin ‘pdt /dqft 1,
sin ‘p)dq cos (p) dep.
The two integrals in Eq. (7) can be evaluated directly, since they are real. At all points of the contour other than the saddle-point, the values of the derivative dt/dq are found from the equation of the contour of fastest variation of the modulus of Eq. (6) by differentiation:
(7)
~omputution of Hunkel (Bessel) functions of complex index and argument
61
The derivative dt/dq can be evaluated at the saddle-point by using the relationships (61 dt
drp
wherem=
ctg
=t
((p-arg
h,),
1,2,3,4.
In these expressions, cp signifies the angle at which the contour of fastest variation leaves the saddle-point.
The Caseof Real Index and Avant
Zi>O, z2=j2=0
The case of a red index Tm v=O is distinctive, since, as we mentioned above, Eq. (6) transforms in this case to a quadratic equation in t, which is solvable analytically:
t=
(il+l) v+F, z2 co.7 cp+ z1 sin (p
The different signs correspond to the contours of descent and rise; on moving through the saddle-point, the sign reverses for the chosen type of contour. Simple analysis of Eq. (8) for the case”of a real argument yields the topology of the contours of descent and rise illustrated in Fig. 2, u--c (z2 = jZ = 0). Arrows going away from the saddle-point indicate a contour of descent, and arrows towards the saddle-point, contours of rise. With z1 > j1 + 1, the contours of steepest descent can be taken directly as the contours of integration in Eqs. (1) and (2). In the case jl + 1 2 zl, as the contour we took the intervla of the real axis from zero to the saddle-point (the interval UC,* Fig. 2, c), then moved from this along the contour of descent into the upper or lower h~f-pIane.
(8)
G. F. Remenets
62
a FIG. 2
a) for
zi>fi+i,
b) for
zi=fi+l,
b) for
Im (hl) =Im @,)=O,
c)for
zicfi+i
a
FIG. 3 a) for
Im (A,)=-Im
(h,)>O,
-1m
c) for
Im (h,)=
(hz) CO
FIG. 4
FIG. 5
. a) for
Im f(h) =-Im
b) for f(L) X-4 -1m f(L) CO
Im j(L) =O,
c) for
Im f(h) H
Computation of Hankel (Bessel) functions of complex index and argument
63
The Case of Real Index and Complex Argument In the case of a real index and complex argument, the equation for the contours of fastest variation of the modulus of the phase functionf(X) is, as before, a second-order equation in t (see Eq. (6)). However, the asymptotes to these contours, in the neighbourhood of zero and infinity of the X plane, are turned relative to the real axis through an angle $=-arctg (z,/z~) .We denote them in the figures by broken lines. If the imaginary part of the argument is relatively small, 1zz 1Czi, ~(0, the topology of the contours illustrated in Fig. 2, a-c respectively, is modified into the asymmetric topology shown in Fig. 3, a-c. The version of Fig. 3, b is intermediate between the versions of Fig. 3, a and c. If the computations are performed along the contour of descent, we obtain, in the case shown in Fig. 3, a, Imf&) = F1 > 0,the values of the Hankel functions 13$1sz)(z) , whereas in the case of Fig. 3, c, Im f(Xi ) = F1< 0,we get the doubled value of the Bessel functions 22, (z) and the value of the Hankel function of the 2nd kind H(Y2)(2). The value of the Hankel function of the 1st kind is found from the definition: Ilt”(5)=21,(2)
-@‘(Z).
If the modulus of the imaginary part of the argument z is further increased, z2 < 0, we move from the versions shown in Fig. 3 to those of Fig. 4 respectively ( 1zz1>zi,jz=O). As before, the version of Fig. 4, b is intermediate, Im f(L) =F,=O. A change of sign of the imaginary part of the argument (z+O) leads to the first saddle-point 1A, I > 1 being in the lower half-plane, and the second saddle-point I hz I ==c1 being in the upper half-plane. The topology of the contours illustrated in Figs. 3 and 4 is correspondingly modified. Notice also that, in the cases illustrated in Fig. 3, b and 4, b, which are encountered when the imaginary parts of the phase function vanish at the saddle-points F, we get the doubled value of the Bessd function or the value of the Hankel function when integrate along the contour passing through the first saddle-point, depending on whether the numerical integration is performed after reaching the second saddle-point (zero or infinity).
64
G. F. Remenets
The Case of a Real Argument and Complex Index (i2 > 0) With a complex index Im v=jzfO Eq. (6) is transcendental with respect to both its variables t and cp.We solve it by Newton’s method by an iterative process with respect to one variable, t or cp, depending on which of them is used as the variable of integration. The decision as to which variable to use for the integration is based on the value of the ratio dt/tdq along the contour. If this is much greater than unity, the integration should be performed with respect to the modulus oft. Otherwise, the integration should be with respect to the second variable cp. Notice also that the integration step is selected according to the curvature of the contour. The fact that the index Y is complex does not affect the asymptotes to the contours of descent and rise, so that, in the case of a real argument and the conditions Im y=jz>O, j2Kjl we have the types of behaviour of the constant phase contours illustrated in Fig. 5 (zr >jr + 1, z2 = 0), obtained by complex “disturbance” of the versions of Fig. 2. For a large complex disturbance jzB ji we pass from the versions of Fig. 5 to those of Fig. 6 (zr 1 is in the lower half-plane.
The General Case of Complex Index and Complex Argument (Re z > 0) When the index and argument are both complex, the number of possible types of topology of the contours is increased, since the saddle-points may then be in the left-hand half-plane (if (vi-l)/z)>r71/2, see Eq. (4)). Here, if (~+l)/z) +x and hz-+eAin. A more detailed analysis of the present case 1(v+l)lzI + 1, then A,+@, does not reveal any essentially new features as compared with the cases discussed above. To conclude our description of the algorithm, notice that, in order to avoid overflow and conversion to the machine zero at the saddle-points h. of the integrands of Eqs. (1) and (2), it is desirable to normalize the integrands (to unity with respect to their modulus at the saddle-points). For this, we only have to take the factor exp {Re f(k)} outside the integral sign. The function exp {f(k) -f(L)} is then integrated numerically.
65
Computation of Hankel (BesseI) functions of complex index and argument
TABLE 1
ZP
il
h
16
0
0
4
16
0
0
12
16
0
0
20
16
0
20
4
16
0
20
12
16
0
20
20
16
-5
0
4
16
-5
0
20
16
-5
20
4
16
-5
20
20
1600
0
0
400
1600
0
0
1200
1600
0
0
2000
1600
0
2000
1200
1600
0
2000
2000
9600
-50
0
400
1600
-50
0
2000
1600
-50
2000
400
1600
-50
2000
2000
Re and Im H(‘) Y (2)
Re and Im Hc2) Y (I)
-0.51043 *102 0.88380.102 -0.27690.107 -0.27256.108 -0.51558.10’3 -0.46487.10’3 0.29327 0.63223
-0.19892~10-3 13.96 -0.30821.10-3 -0.11745.10-9 10.14 0.11560.10-* -0.26594.10-'4 3.11 0.23978.10-14 -0.35400 -3.16 -0.63563 0.26111.10-2 - 10.7 -0.10385.10-' O.27168.1O-s -20.26 0.19796.10-4 -0.95793.10-Q 13.93 -0.18714.10-6 -0.25877.10-18 3.59 0.14115.10-17 -0.58060.10-2 -7.46 0.25071.10-1 0.51390.10-* -24.8 -0.16924.10-S -0.20579~10-Q74 1548.7 0.16137.10-274 0.13873.10-82" 1166.7 0.39691.10-820 0.62021.10-1366 464.5 -0.20727.10-'366 -0.11753.10-121 -1007.6 0.34876.10-'22 -0.5635.10-389 -1931.8 -0.5965.10-39" -0.2008~10-498 1552.95 ~-0.1419.10-4Q' 536.07 --0.90194'10-17~ -0.13640.10-'706 --0.87971.10-87 -680.7 0.36337.10-87 -0.67452.10-7~~ -2370.9 u.11900.10-~43
-0.13955.10' 0.23369.10' 0.66643.103 -0.88685.103 -0.12161.10~ 0.12785.106 -0.99226.1015 -0.17360.10" 0.92986 -0.95620 0.20681.10' O.31212.107 -0.11582.10~"J -0.90829.102" 0 24752.10817 -0:70808~10~17 0,35081~10'"3 0.11725.10~3~ -0.11140*10"9 -0.20554.101QO -0.2820*103~~ -0.2477.103*" -0.1071.I0496 -0.2348.i04Q6 -0.26345.1017"3 0.13346*1017oQ -0.59896.1083 -0.34786.1084 0.22249.107~0 -0.13998.10'4'
Im f WI)
Some results of computations according to a program based on our algorithm will be quoted. Let us first give the values of the Hankel functions and their derivatives with respect to index and argument for the real argument .a=346873716 and the complex index v=3420.91585+i10.1484583. The absolute accuracy of the integration, and the relative accuracy of determining the points of the contour of steepest descent, were respectively e=10m4 and 10b7.The digits printed in italic are the true digits, as was established by a comparison with results kindly loaned to us by S. T. Rybachek and obtained by Iris algorithm [2] :
G. F. Remenets
66
H:“(z)=-0.044619--
0.17444,
H”‘(z)=(-0.21023+i0.57065) 4
H($)(z)+-0.28483+i
IO+ 0.096&O-‘,
$-@‘(a)=(-0.10085-i
O.2268)1O-2,
-$
H;1’(+-(0.28366-~
0.0951) 10-l,
&
H’,2’(~)=(0.10038+i
0.02262) 10-2.
From this result it is clear, in particular, that, to an accuracy of one percent, the Hankel function derivatives in the region v-z are connected by g
HP’ 2)(z)= -
& fp)
(2).
This fact played an important part in the author’s paper [8] q To illustrate the range of application of the program, Table 1 shows the values of the Hankel functions of complex index with small arguments ( 1z I-16) and large arguments (]z]-16.10’) and diff erent ratios of the real to imaginary parts of the index v. In conclusion, I thank G. I. Makarov for useful discussions. Translated by
D. E. Brown
REFERENCES 1.
WALTERS, L. C., and JOHLER. J. R., On the diffraction of spherical radio waves by a finitely conducting spherical earth, J. Res. NBS, 660, l, lOl-106,1962.
2.
RYBACHEK, S. T., and GYUNNINEN, E. M., Propagation of long and super-long radio waves in the Earth-ionosphere waveguide channel, in: Problems of wavediffrction and propagation (Probl. difraktsii i rasprostraneniye vom), 6, 115-123, LGU, Leningrad, 1966.
3.
RYBACHEK, S. T., Super-long wave propagation in the Earth-ionosphere Geomagnetizm i Aeronomiya, 8,3.
4.
REMENETS, G. R., and RYBACHEK, S. T., Two methods of complex eigenvalues evaluation in the waveguide VLF propagation problem, International symposium, Theory of electromagnetic waves(abstracts), Tbilisi, 9-15 September 1971, Nauka, Moscow, 761, 1971.
5.
WATSON, G. N., A treatiseon the theory of Bessel functions, Cambridge U.P., 1944.
6.
SMIRNOV, V. I., Course of higher mathematics (Kurs vysshei matematiki), 3, Gostekhixdat, Moscow, 1953.
waveguide channel,
Computation of Hankel (Bessel) functions of complex index and argument 7.
PETRASHEN’, G. I., SMIRNOVA, N. S., and MAKAROV, G. I., On the asymptotic forms of the cylindricalfunctions, Uch. zap. LGU, 170, 27, 7-95, 1953.
8.
REMENETS, G. F., The variational method of moments and some iterative processes, in: Problems of wave diffraction and propagation (Probl. difraktsii i rasprostraneniya voln), 11,56-77, LGU, Leningrad, 1972.
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