Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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 atx= 0 and an essential singularity atx=∞.

This equation can be obtained by merging two of the singularities of the ordinary

hypergeometric equation (18.136). The parametersaandcare given real numbers.

Show that settingx=z/bin the hypergeometric equation, and lettingb→∞, yields the

confluent hypergeometric equation.

Substitutingx=z/binto (18.136), withd/dx=bd/dz, and lettingu(z)=y(x), we obtain

bz

(

1 −

z

b

)d (^2) u
dz^2
+[bc−(a+b+1)z]
du
dz
−abu=0,
which clearly has regular singular points atz=0,band∞.Ifwenowmergethelasttwo
singularities by lettingb→∞,weobtain
zu′′+(c−z)u′−au=0,
where the primes denoted/dz. Henceu(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 atx= 0 and an essential singularity atx=∞.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 second-
order 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 theconfluent 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 lettingx→x/band
carrying out the limiting processb→∞. Thus, from (18.141) and (18.143), two
linearly independent solutions of (18.147) are (whencis not an integer)
y 1 (x)=1+
a
c
x
1!

• a(a+1)
  c(c+1)
  z^2
  2!
  +···≡M(a, c;x), (18.148)
  y 2 (x)=x^1 −cM(a−c+1, 2 −c;x), (18.149)
  whereM(a, c;x) is called theconfluent hypergeometric function(or Kummer
  function).§It is worth noting, however, thaty 1 (x) is singular whenc=0,− 1 ,− 2 ,...
  andy 2 (x) is singular whenc=2, 3 , 4 ,.... Thus, it is conventional to take the
  §We note that an alternative notation for the confluent hypergeometric function is 1 F 1 (a, c;x).