6.1 Confluent Hypergeometric Function of 1st and 2nd Kind
93
Computational Aspects
The single terms in Eq. (6.5) can be rewritten as
t (0) = 1 , t(1) =
a
b · 1
z
1
· 1 =
a
b · 1
z · t (0)
t (2) =
a
b · 1
a + 1
(b + 1) · 2
z
2
=
a + 1
(b + 1) · 2
z · t (1) · · ·
t (n) =
a + n − 1
(b + n − 1) · n
z · t (n − 1),
and thus by computing
a + n − 1
(b + n − 1) · n
z , n > 0,
and building a cumulative product, all single terms of the sum are created. To
minimize cancellation errors due to the finite precision of the numbers, we split
the sum in its real and imaginary part and each part in its positive and negative
contributions. Finally, each of this four subseries will be sorted in descending,
respectively, ascending order before summing up. In case the computation does
not converge, the first step is to manipulate the series based on the Kummer
transformation.
The Kummer transformation reads
1 F 1 (a; b; z) = exp(z) 1 F 1 (b − a; b; −z) ,
(6.6)
and it turns out that for not too large arguments z the new series representation
numerically converges in most cases. In case of no successful evaluation either
an asymptotic series expansion or the path integration technique will be used, if
necessary in combination with a Kummer transformation.
The path integration is based on differential equation (6.3). The first derivative
for the initial condition is computed by
d
dz
1 F 1 (a; b; z) =
a
b
1 F 1 (a + 1; b + 1; z) .
(6.7)
For large arguments |z| > 30 the asymptotic series [1]
1 F 1 (a; b; z) = exp(±iπa)
Γ (b)
Γ (b − a)
n=0
(a) n (1 + a − b) n
Γ (n + 1)
(−z)
n
+ exp(z)z
a−b Γ (b)
Γ (a)
n=0
(b − a) n (1 − a) n
Γ (n + 1)
z
n
(6.8)
93
Computational Aspects
The single terms in Eq. (6.5) can be rewritten as
t (0) = 1 , t(1) =
a
b · 1
z
1
· 1 =
a
b · 1
z · t (0)
t (2) =
a
b · 1
a + 1
(b + 1) · 2
z
2
=
a + 1
(b + 1) · 2
z · t (1) · · ·
t (n) =
a + n − 1
(b + n − 1) · n
z · t (n − 1),
and thus by computing
a + n − 1
(b + n − 1) · n
z , n > 0,
and building a cumulative product, all single terms of the sum are created. To
minimize cancellation errors due to the finite precision of the numbers, we split
the sum in its real and imaginary part and each part in its positive and negative
contributions. Finally, each of this four subseries will be sorted in descending,
respectively, ascending order before summing up. In case the computation does
not converge, the first step is to manipulate the series based on the Kummer
transformation.
The Kummer transformation reads
1 F 1 (a; b; z) = exp(z) 1 F 1 (b − a; b; −z) ,
(6.6)
and it turns out that for not too large arguments z the new series representation
numerically converges in most cases. In case of no successful evaluation either
an asymptotic series expansion or the path integration technique will be used, if
necessary in combination with a Kummer transformation.
The path integration is based on differential equation (6.3). The first derivative
for the initial condition is computed by
d
dz
1 F 1 (a; b; z) =
a
b
1 F 1 (a + 1; b + 1; z) .
(6.7)
For large arguments |z| > 30 the asymptotic series [1]
1 F 1 (a; b; z) = exp(±iπa)
Γ (b)
Γ (b − a)
n=0
(a) n (1 + a − b) n
Γ (n + 1)
(−z)
n
+ exp(z)z
a−b Γ (b)
Γ (a)
n=0
(b − a) n (1 − a) n
Γ (n + 1)
z
n
(6.8)