QUARTERLY OF APPLIED MATHEMATICS
Vol. IX April, 1951 No. 1
A CONVENIENT GENERAL SOLUTION
OF THE CONFLUENT HYPERGEOMETRIC EQUATION,
ANALYTIC AND NUMERICAL DEVELOPMENT*
BY
T. S. KUHN**
Harvard University
1. Introduction. The standard forms for the general solution of the Confluent Hyper-
geometric Equation prove too unwieldy for application to many physical problems,
particularly in the field of Quantum Mechanics. The two standard power series solu-
tions,1 Mn,±m(y), reduce to a single regular polynomial solution whenever 2m is an
integer (the standard case for quantum mechanics), and in this case the two integral
solutions2, W*n,m (±y), must be computed with an asymptotic expansion which is
cumbersome for most physically interesting values of y.
The utility of all of these solutions is further limited because their form necessitates
undertaking a complete recomputation for every physically significant value of n. This
makes the labor of computation almost prohibitive in the physically important case
where both n and y must be treated as continuous variables.
The possibility of achieving a more manageable form of the solutions was first
indicated by the work of Wannier3 and Jastrow.4 Wannier showed that in theory the
function Mn,m(y) could be developed as a series in descending powers of n with coeffi-
cients given in terms of Bessel functions. Jastrow actually exhibited the first two terms
of an asymptotically similar series for the solution Wn,m(y). This paper completes the
above treatments by producing analytically a general solution of the differential equation
as a power series in 1/n2 with coefficients readily calculable in terms of known functions.
This treatment differs from those noted above not only in the generality of its results,
but also in the ease with which successive terms of the series may be explicitly generated.
The method employed here makes it possible to exhibit the two particular solutions of
the equation which go to zero as y —> 0 and as y —> + » and to relate these analytically
to the earlier solutions Wn,m(y) and Mn,m(y). Finally this paper will exhibit analytic and
numerical values for the coefficients of several of the series of greatest physical interest.
2. The general series solution. For physical applications Whittaker's standard form
of the Confluent Hypergeometric Equation,
*Received January 25, 1950.
**Junior Fellow, Society of Fellows.
JE. T. Whittaker and G. N. Watson, Modern analysis, (Cambridge University Press, 1940), fourth
edition, §16.1.
mid. §16.12.
SG. H. Wannier, Phys. Rev. 64, p. 358 (1943).
4R. Jastrow, Phys. Rev. 73, p. 60 (1948).
tfu
dy2'
T. S. KUHN [Vol. IX, No. 1
4= (d
is conveniently transformed by the substitutions
r = | ny, m = I + | (2)
to the form
dlU
2^+r_ ^+i-i«+iiy..,o,
ar L n r r J
(3)
which, with e = —1/n2, is just the hydrogenic radial wave equation in Rydberg units.
The form (3) will be taken as standard in this paper.
The further substitutions
z = (8r)1/2; [/<'■->= I (4)
z
reduce (3) to the form
- n~2(|z)4 F"1" = 0, (5)
where V< is the Bessel operator of index 21 + 1, i.e.
V, = z2 J + z | + z2 - (2Z + l)2. (6)
It will be assumed here and proven in the Appendix that the general solution of (5)
may be written in the form of a power series in 1/n2, i.e.
r'-n\z) = Zn2kVil\z), (7)
k" 0
where the functions V[l) (z) are analytic functions of z and the series (7) converges
absolutely and uniformly for all real I and for all real n in the region \ n\ > nn, n0 being
an arbitrary positive number. Corresponding solutions of (3) may then be written in
the form
U«-n\z) = Zn2kU(k'\z), (8)
k-0
in which the U[l)(z) are given in terms of the Vi'\z) by the equation
Uil\z) = \zV[l\z). (9)
Since (7) converges uniformly and absolutely it may be inserted in the differential
equation (5), and differentiated term by term. The terms may then be rearranged and
the coefficients of the various powers of 1/n2 equated to zero, which is necessary and
sufficient to make (7) a solution of (5). This procedure yields an infinite set of simul-
taneous differential equations for the coefficients V[0 (2):
ViK" = 0 (10a)
ViF'° = (l2)4^"1 • (10b)
1951] THE CONFLUENT HYPERGEOMETRIC EQUATION 3
Now (10a) is just Bessel's equation of index 21 + 1 whose general solution is the
cylindrical functions e2i+1(2), an arbitrary linear combination of the Bessel function
Jii+iiz), and the Weber function, F2i+1(z). These cylindrical functions obey the usual
recursion formulas for Bessel functions:
e„_i(z) + en+1(z) = — e„(z)
z
(11)
ze'n(z) + nen(z) = zen-,(z).
By utilizing these relations and the definition (8) of the Bessel operator Vi , it can be
shown straightforwardly that
^
This equation permits solutions of (10b) to be generated directly. For set q = 0 in
(12). Then this equation becomes identical with (10b) in the case k = 1, provided that
V[l)(z) is defined by the bracket on the left side of (12), i.e. by
Till __ ^ "t-
(22) ®2,+3 12 (2 s) ®2,+4 ' (13)
Fi =
Thus a, solution has been found for the first of equations (10b).
For k = 2, the right hand side of (10b) is just (iz)4Fi which, by (13), must contain
two terms of the form (^2)a+4e2i+1+u with q = 2 and q = 3. These are both of the form
found on the right of equation (12), so that F2"(z) can be generated by two applications
of the procedure outlined above. Higher terms are generated successively by the same
process. For example, the first four coefficients of the series for U{0'n\z) are
U,
(0)
= (r)e'
U[0) =i(iz]e3-^Uz)e4
• -m
(14)
W " k (I - eio (I ■)'«■ + ^0 (1 •)'«. - Io^68 (1 •)"«» ■
It has been assumed, and will be proven in the Appendix, that the series generated
above and illustrated in (14) is itself analytic and uniformly convergent in z so that it
may be differentiated term by term with respect to z to yield a series for the derivatives
of the solutions of equation (3). Such a series is discussed further in section 5.
3. Some important particular solutions. The series generated above yields a general
solution of equation (3) because the linear combination aJm(z) + f3Ym(z) to be inserted
for the cylindrical function Gm remains entirely arbitrary, so that the coefficients U[l)(z)
are in fact ambiguously defined. For physical application of the series it is necessary to
examine the effect of removing this ambiguity by particular choices of the constants
4 T. S. KUHN [Vol. IX, No. 1
a and /?. This examination is facilitated by defining two sets of coefficients,
and 1Ui'\z), which are gained from the ambiguous coefficients U'k1' (z) by substituting
Jm(z) and Ym(z) respectively for the cylindrical functions Gm(z). Two particular inde-
pendent solutions of (3), °t/<! ,n) (z) and 1Uu'n)(z), may then be defined as the result of
applying the summation (8) to the coefficients °Uil)(z) and 1 U'kl) {z), respectively.
The most general solution of (3) may now be written in the form
f(l, n) °Ua-n\z) + g(I, n) lUa'n\z) = £ n2k[f(l, n) °U[l\z) + g(l, nYUil\z)], (15)
A: = 0
where f(l, n) and g(l, n) are entirely arbitrary. Important particular solutions of (3) are
gained by specifying these arbitrary functions in (15).
Since the Bessel functions all have zeros and the Weber functions all have poles at
the origin, it can be shown that °U"'n)(z) is the only particular solution of (3) with a
zero at the origin. It must therefore be identical, except in amplitude, with the particular
solution M„,m(y) of (1), and a comparison of the leading terms (in z) of the expansions
of the two series yields
Mn,i+1/2(z2/4n) = nl~lT(2l + 2) aUa'n\z). (16)
A second solution of physical interest, the only particular solution which goes to
zero as z goes to infinity, may be discovered by a comparison of the series developed
above with a series solution given by Wannier. In the paper previously noted, Wannier
defines two solutions of (5), Jh+i(z) and Nh+1(z), by the formulas
A^«+i(z) - 1
sin (21 + 1 )t
nl+1
Ju+iiz) = (1/2 z)T(2l + 2) Mn'1+1/2(z /4n)
r[n - Iv'" cos (2Z + 1)t ~ J"-2l-l(z)
]•
(17)
These solutions are shown to be independent and well defined for all values of I and n.
Wannier further proves that the particular solution which goes to zero at infinity may
be written5
IF„,i+1/2(22/4n) = (z2/4n)1/2[r(n + I + 1 )nl~1/2r2l+1(z) cos (n - I - 1 >
(18)
+ r(n - T)nl+x/2Nh+i(z) sin (n - I - 1)tt].
Since the series expansion of ilfn>!+1/2 is well known, equations (17) and (18) com-
pletely determine the expansion of Wn,i+i/2 for any value of I. It follows that if, for a
given fixed value of I, the first m + 1 coefficients "U'k" and have been developed
by the generating procedure, the functions f(l, n) and g(l, n) may be determined (to
terms in ri~2m) by explicit comparison of the series expansions (in z) of (15) and (18).
If 21 + 1 is not an integer, it is convenient to compare the coefficients of the terms in
(}z)~21 and in (Jz)2!+2 in the two series; if 21 + 1 is an integer the coefficient of the terms
(%z)2l+2 and (%z)2l+2 log (fz) are most conveniently compared. In the latter case, 21 + 1
an integer, these two terms of (18) are given by
6Wannier's paper has — r(n — l)nl+1,2 • • • , a discrepancy which I assume to be due to a misprint in
the original.
1951] THE CONFLUENT HYPERGEOMETRIC EQUATION
TT7■ t 2 H \ I r(w —Z —f— 1) (1/2 z)2l+2 / , J !\
Wn.l+1/2(z /4n) = • • • + — ^ ^, jcos (n - I - 1)tt
+ - sin (n — I — 1)J 2 log (| z) + 2y — E — + — 0 — log (n)
■k \Z / m_i m
(2Z - r)!
(19)
+ r(»- Z)(2Z + 1)! E(-i)r
22'+1~r(2Z + 1 - r)!r(» + I + 1 - r)J
+
in which ¥(x) = d/dx log T(x) and 7 = Euler's constant.
This procedure has been carried out for the two most important cases, 1 = 0 and
I = 1. In both cases the manipulation yields (to terms in n~10)
Wn.l+W2(z2/4n) = n-'-'Tin + I + l)[cos (n - I - 1)t°U"'n\z)
(20)
+ sin (n - I - l)v 1C7"<,,n,(af)].
There are additional theoretical reasons for supposing the equation (20) is in fact
valid for all values of I, integral and non-integral, but a general proof of this result has
not yet been given. Until such a proof is produced the method outlined above may be
used to produce an equivalent result for any value of I for which the coefficients °Uil)
and 1Ukl> have been developed.
4. An alternate generating procedure. Since the Weber functions have not been tabu-
lated for large indices, it is convenient to develop the formulas for U[l\z) so that they
involve only G0(z) and &i(z). This may be accomplished by repeated application of the
first of the recursion formulas (11) to the functions Vil\z) generated by the method of
section 2, but this reduction is arduous and may conveniently be replaced by the pro-
cedure sketched below.
The function Vo \z) is just G2i+1(z), and this may, by application of the recursion
formulas, be rewritten in the form
M /\2i If /-. \2. + l
Vil\z) = E a,(J z) e„ + E mJz) e, , (21)
where the constants a,- and bt are known rational numbers and the constants to, n, M,
and N are known integers.
The function V[l\z) must be expressible in the form
M + 2 /-. \2* N+2 /-, \ 2* +1
Vi'\z) = Z a^z) e0 + E ©! , (22)
where the constants and & are unknown rational numbers which can be determined
by applying the differential equation (10b) to (21) and (22). This application of the
differential equation is facilitated by the use of the equations
V,
V,
(|z)°e„ = -49(i2)a+1ei + [q2 - (21 + 1)2](| s)"c„
(| 2=)<'e1] = 4g(!*)a+1e„ + [(ff - l)2 - (21 + i)l(^) e, ,
(23)