Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 189

where Γ ( a ,y) is defined by the integral

∞

t a − 1 e − t dt . Γ ( a ,y) = (2.403)

y

Thus, the incomplete gamma function can be expressed as

γ ( a ,y )
Γ

P ( a ,y) =

( a ) ,

(2.404)

or,

− Γ

( a ,y)
Γ

P ( a ,y) = 1

( a ) .

(2.405)

The evaluation of either γ( a ,y)or ( a ) allows cal-

culation of the incomplete gamma function. A comprehensive account of numerical
methods for the calculation of these and related functions is given by Press et al.
(1992, pp. 206-213).
For su

Γ ( a ,y) and the gamma function Γ

ciently small y, the power series expansion of

a ∞ a ∞

Γ ( a ) 1

e − y y

n

e − y y

n

γ( a ,y) =

n ) y

=

n ) y

(2.406)

Γ + + + ··· +

( a 1 a ( a 1) ( a

n = 0 n = 0

can be used (Erdelyi et al. , 1953, p. 135), while for larger y the continued fraction
development

e − y y

a

Γ ( a ,y) = (2.407)

−

1 a

y +

1

1 +

2 − a

y +

2

1 +

3 − a

y +

1 +···

can be employed (Erdelyi et al. , 1953, p. 136). For calculation of the gamma func-
tion itself we use the seven-term approximation of Lanczos (1964),

a a +

2

2 ) √ 2π

1
2

e − ( a + 5 +

1

Γ ( a + 1) = + 5 +

c 0

c 1
a +

c 2
a +

c 6
a +

× +

1 + 2 +···+ 6 +

, (2.408)