92
6 Confluent Hypergeometric Function
tion argument is within the radius of convergence, this does not guarantee that the
numeric computation based on the series representation will converge due to the
finite precision of the numbers.
In the following we will concentrate on the confluent hypergeometric function
of first kind 1 F 1 (a; b; z) and second kind U(a, b, z), the hypergeometric function
2 F 0 (a 1 , a 2 ; ; z), the confluent hypergeometric limit function 0 F 1 (; b; z), and the
Whittaker functions M a,b (z) and W a,b (z).
Function Overview
The SPECFUNPHYS class conhyp supports the evaluation of the functions listed
above.
6.1
Confluent Hypergeometric Function of 1st and 2nd Kind
Confluent hypergeometric functions are of importance for a wide variety of
applications, such as Coulomb wave functions (which we will discuss in the next
chapter), photon scattering, but also in computational finance, to mention only a
few.
The confluent hypergeometric functions are solutions of Kummer’s equation
z
d 2 F
dz 2 + (b − z)
dF
dz
− aF = 0
(6.3)
and also called Kummer functions. Kummer’s equation has two linearly independent
solutions
1 F 1 (a; b; z) and z
1−b
1 F 1 (a − b + 1; 2 − b; z),
(6.4)
or alternative the confluent hypergeometric function of 2nd kind U , Eq. (6.10a).
6.1.1 The Function 1F 1
The series expansion of the confluent hypergeometric function 1 F 1 is given by
1 F 1 (a; b; z) = 1 +
a
b · 1
z +
a(a + 1)
b(b + 1) · 1 · 2
z
2
+
a(a + 1)(a + 2)
b(b + 1)(b + 2) · 1 · 2 · 3
z
3
+ · · ·
= 1 +
a
b
z
1!
+
(a) 2
(b) 2
z 2
2!
+
(a) 3
(b) 3
z 3
3!
+ · · · ,
(6.5)
with (x) n the Pochhammer Symbol. This series is convergent in C, and becomes a
finite polynomial of degree |a| for a negative integer and b ≥ a no negative integer,
but diverges for b negative integer and a ≥ b not a negative integer.
6 Confluent Hypergeometric Function
tion argument is within the radius of convergence, this does not guarantee that the
numeric computation based on the series representation will converge due to the
finite precision of the numbers.
In the following we will concentrate on the confluent hypergeometric function
of first kind 1 F 1 (a; b; z) and second kind U(a, b, z), the hypergeometric function
2 F 0 (a 1 , a 2 ; ; z), the confluent hypergeometric limit function 0 F 1 (; b; z), and the
Whittaker functions M a,b (z) and W a,b (z).
Function Overview
The SPECFUNPHYS class conhyp supports the evaluation of the functions listed
above.
6.1
Confluent Hypergeometric Function of 1st and 2nd Kind
Confluent hypergeometric functions are of importance for a wide variety of
applications, such as Coulomb wave functions (which we will discuss in the next
chapter), photon scattering, but also in computational finance, to mention only a
few.
The confluent hypergeometric functions are solutions of Kummer’s equation
z
d 2 F
dz 2 + (b − z)
dF
dz
− aF = 0
(6.3)
and also called Kummer functions. Kummer’s equation has two linearly independent
solutions
1 F 1 (a; b; z) and z
1−b
1 F 1 (a − b + 1; 2 − b; z),
(6.4)
or alternative the confluent hypergeometric function of 2nd kind U , Eq. (6.10a).
6.1.1 The Function 1F 1
The series expansion of the confluent hypergeometric function 1 F 1 is given by
1 F 1 (a; b; z) = 1 +
a
b · 1
z +
a(a + 1)
b(b + 1) · 1 · 2
z
2
+
a(a + 1)(a + 2)
b(b + 1)(b + 2) · 1 · 2 · 3
z
3
+ · · ·
= 1 +
a
b
z
1!
+
(a) 2
(b) 2
z 2
2!
+
(a) 3
(b) 3
z 3
3!
+ · · · ,
(6.5)
with (x) n the Pochhammer Symbol. This series is convergent in C, and becomes a
finite polynomial of degree |a| for a negative integer and b ≥ a no negative integer,
but diverges for b negative integer and a ≥ b not a negative integer.