Digital Library of Mathematical Functions
The Gamma Function -- Asymptotic Expansions

Asymptotic Expansions

Contents

Poincaré-Type Expansions

As z → ∞ in the sector | ph z | ≤ π - δ ( < π ) ,

ln Γ ( z ) ∼ ( z - 1 2 ) ln z - z + 1 2 ln ( 2 π ) + ∑ k = 1 ∞ B 2 k 2 k ( 2 k - 1 ) z 2 k - 1
ψ ( z ) ∼ ln z - 1 2 z - ∑ k = 1 ∞ B 2 k 2 k z 2 k

For the Bernoulli numbers B 2 k , Also,

Γ ( z ) ∼ ⅇ - z z z ( 2 π z ) 1 2 ( ∑ k = 0 ∞ g k z k )
g 0 = 1 , g 1 = 1 12 , g 2 = 1 288 , g 3 = - 139 51840 , g 4 = - 571 24 88320 , g 5 = 1 63879 2090 18880 , g 6 = 52 46819 7 52467 96800
g k = 2 ( 1 2 ) k a 2 k ,

where a 0 = 1 2 2 , and

a 0 a k + 1 2 a 1 a k - 1 + 1 3 a 2 a k - 2 + … + 1 k + 1 a k a 0 = 1 k a k - 1
k ≥ 1 .

Wrench(1968) gives exact values of g k up to g 20 . Spira(1971) corrects errors in Wrench's results and also supplies exact and 45D values of g k for k = 21 , 22 , … , 30 . For an asymptotic expansion of g k as k → ∞ see Boyd(1994) .

With the same conditions

Γ ( a z + b ) ∼ 2 π ⅇ - a z ( a z ) a z + b - ( 1 2 )

where a ( > 0 ) and b ( ∈ ℂ ) are both fixed, and

ln Γ ( z + h ) ∼ ( z + h - 1 2 ) ln z - z + 1 2 ln ( 2 π ) + ∑ k = 2 ∞ ( - 1 ) k B k ( h ) k ( k - 1 ) z k - 1

where h ( ∈ [ 0 , 1 ] ) is fixed.

Also as y → ± ∞ ,

| Γ ( x + ⅈ y ) | ∼ 2 π | y | x - ( 1 2 ) ⅇ - π | y | 2

uniformly for bounded real values of x .

Error Bounds and Exponential Improvement

If the sums in the expansions (Equation 1) and (Equation 2) are terminated at k = n - 1 ( k ≥ 0 ) and z is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. If z is complex, then the remainder terms are bounded in magnitude by sec 2 n ( 1 2 ph z ) for (Equation 1), and sec 2 n + 1 ( 1 2 ph z ) for (Equation 2), times the first neglected terms.

For the remainder term in (Equation 3) write

Γ ( z ) = ⅇ - z z z ( 2 π z ) 1 2 ( ∑ k = 0 K - 1 g k z k + R K ( z ) )
K = 1 , 2 , 3 , … .

Then

| R K ( z ) | ≤ ( 1 + ζ ( K ) ) Γ ( K ) 2 ( 2 π ) K + 1 | z | K ( 1 + min ( sec ( ph z ) , 2 K 1 2 ) )
| ph z | ≤ 1 2 π

Ratios

If a ( ∈ ℂ ) and b ( ∈ ℂ ) are fixed as z → ∞ in | ph z | ≤ π - δ ( < π ) , then

Γ ( z + a ) Γ ( z + b ) ∼ z a - b
Γ ( z + a ) Γ ( z + b ) ∼ z a - b ∑ k = 0 ∞ G k ( a , b ) z k

Also, with the added condition ℜ ( b - a ) > 0 ,

Γ ( z + a ) Γ ( z + b ) ∼ ( z + a + b - 1 2 ) a - b ∑ k = 0 ∞ H k ( a , b ) ( z + 1 2 ( a + b - 1 ) ) 2 k

Here

G 0 ( a , b ) = 1 , G 1 ( a , b ) = 1 2 ( a - b ) ( a + b - 1 ) , G 2 ( a , b ) = 1 12 ( a - b 2 ) ( 3 ( a + b - 1 ) 2 - ( a - b + 1 ) )
H 0 ( a , b ) = 1 , H 1 ( a , b ) = - 1 12 ( a - b 2 ) ( a - b + 1 ) , H 2 ( a , b ) = 1 240 ( a - b 4 ) ( 2 ( a - b + 1 ) + 5 ( a - b + 1 ) 2 )

In terms of generalized Bernoulli polynomials we have for k = 0 , 1 , …

G k ( a , b ) = ( a - b k ) B k ( a - b + 1 ) ( a )
H k ( a , b ) = ( a - b 2 k ) B 2 k ( a - b + 1 ) ( a - b + 1 2 )
Γ ( z + a ) Γ ( z + b ) Γ ( z + c ) ∼ ∑ k = 0 ∞ ( - 1 ) k ( c - a ) k ( c - b ) k k ! Γ ( a + b - c + z - k )