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