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Confluent Hypergeometric Functions

​
confluent hypergeometric function
of first kind
of second kind
argument
x
x
-x
- x
a
2
c
3
The confluent hypergeometric differential equation
xy''+(c-x)y'-ay=0
has a regular singular point at
x=0
and an essential singularity at
x=∞
. Solutions analytic at
x=0
are confluent hypergeometric functions of the first kind (or Kummer functions):​
1
F
1
(a,c,x)=
∞
∑
n=0
(a)
n
(c)
n
n
x
n!
=1+
a
c
x+
a(a+1)
c(c+1)
2
x
2!
+
a(a+1)(a+2)
c(c+1)c+2)
3
x
3!
+…
,where
(α)
n
​
are Pochhammer symbols defined by
(α)
0
=1
,
(α)
1
=α
,
(α)
n
=α(α+1)(α+2)…(α+n-1)=Γ(α+n)/Γ(α)
. For
c=0,-1,-2,…
, the function becomes singular, unless
a
is an equal or smaller negative integer (
|a|≥|c|
), and it is convenient to define the regularized confluent hypergeometric
1
F
1
(a,c,x)=
1
Γ(c)
F
1
(a,c,x)
, which is an entire function for all values of
a
,
c
and
x
.The second, linearly independent solutions of the differential equation are confluent hypergeometric functions of the second kind (or Tricomi functions), defined by
U(a,c,x)=
-a
x
2
F
0
(a,1+a-c,-
-1
x
)
, where the generalized hypergeometric function
2
F
0
represents a formal asymptotic series.If the hypergeometric function with argument
±ix
is complex, both the real and imaginary parts are plotted (black and red curves).For certain choices of the parameters
a
and
c
, the hypergeometric functions are related to various transcendental and special functions. Several illustrations are given in the snapshots.
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