Confluent hypergeometric functions

18.11 CONFLUENT HYPERGEOMETRIC FUNCTIONS
18.11 Confluent hypergeometric functions
The confluent hypergeometric equation has the form
xy + (c − x)y − ay = 0;
(18.147)
it has a regular singularity at x = 0 and an essential singularity at x = ∞.
This equation can be obtained by merging two of the singularities of the ordinary
hypergeometric equation (18.136). The parameters a and c are given real numbers.
Show that setting x = z/b in the hypergeometric equation, and letting b → ∞, yields the
confluent hypergeometric equation.
Substituting x = z/b into (18.136), with d/dx = bd/dz, and letting u(z) = y(x), we obtain
z d2 u
du
bz 1 −
+ [bc − (a + b + 1)z] − abu = 0,
b dz 2
dz
which clearly has regular singular points at z = 0, b and ∞. If we now merge the last two
singularities by letting b → ∞, we obtain
zu + (c − z)u − au = 0,
where the primes denote d/dz. Hence u(z) must satisfy the confluent hypergeometric
equation. In our discussion of Bessel, Laguerre and associated Laguerre functions, it was
noted that the corresponding second-order differential equation in each case had a
single regular singular point at x = 0 and an essential singularity at x = ∞. From
table 16.1, we see that this is also true for the confluent hypergeometric equation.
Indeed, this equation can be considered as the ‘canonical form’ for secondorder differential equations with this pattern of singularities. Consequently, as we
mention below, the Bessel, Laguerre and associated Laguerre functions can all be
written in terms of the confluent hypergeometric functions, which are the solutions
of (18.147).
The solutions of the confluent hypergeometric equation are obtained from
those of the ordinary hypergeometric equation by again letting x → x/b and
carrying out the limiting process b → ∞. Thus, from (18.141) and (18.143), two
linearly independent solutions of (18.147) are (when c is not an integer)
a(a + 1) z 2
ax
+
+ · · · ≡ M(a, c; x),
c 1!
c(c + 1) 2!
y2 (x) = x1−c M(a − c + 1, 2 − c; x),
y1 (x) = 1 +
(18.148)
(18.149)
where M(a, c; x) is called the confluent hypergeometric function (or Kummer
function).§ It is worth noting, however, that y1 (x) is singular when c = 0, −1, −2, . . .
and y2 (x) is singular when c = 2, 3, 4, . . . . Thus, it is conventional to take the
§
We note that an alternative notation for the confluent hypergeometric function is 1 F1 (a, c; x).
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SPECIAL FUNCTIONS
second solution to (18.147) as a linear combination of (18.148) and (18.149) given
by
M(a − c + 1, 2 − c; x)
M(a, c; x)
π
− x1−c
U(a, c; x) ≡
.
sin πc Γ(a − c + 1)Γ(c)
Γ(a)Γ(2 − c)
This has a well behaved limit as c approaches an integer.
18.11.1 Properties of confluent hypergeometric functions
The properties of confluent hypergeometric functions can be derived from those
of ordinary hypergeometric functions by letting x → x/b and taking the limit
b → ∞, in the same way as both the equation and its solution were derived. A
general procedure of this sort is called a confluence process.
Special cases
The general nature of the confluent hypergeometric equation allows one to write
a large number of elementary functions in terms of the confluent hypergeometric
functions M(a, c; x). Once again, such identifications can be made from the series
expansion (18.148) directly, or by transformation of the confluent hypergeometric
equation into a more familiar equation for which the solutions are already
known. Some particular examples of well known special cases of the confluent
hypergeometric function are as follows:
M(a, a; x) = ex ,
M(−n, 1; x) = Ln (x),
M(−n, 12 ; x2 ) =
ex sinh x
,
x
n!m!
Lm (x),
M(−n, m + 1; x) =
(n + m)! n
(−1)n n! H2n+1 (x)
,
M(−n, 32 ; x2 ), =
2(2n + 1)!
x
√
π
M( 12 , 32 ; −x2 ) =
erf(x),
2x
M(1, 2; 2x) =
(−1)n n!
H2n (x),
(2n)!
x
M(ν + 12 , 2ν + 1; 2ix) = ν!eix ( )−ν Jν (x),
2
where n and m are integers, Lm
n (x) is an associated Legendre polynomial, Hn (x) is
a Hermite polynomial, Jν (x) is a Bessel function and erf(x) is the error function
discussed in section 18.12.4.
Integral representation
Using the integral representation (18.144) of the ordinary hypergeometric function, exchanging a and b and carrying out the process of confluence gives
1
Γ(c)
etx ta−1 (1 − t)c−a−1 dt,
M(a, c, x) =
Γ(a)Γ(c − a) 0
(18.150)
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