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11: 8.9 Continued Fractions
8.9.1 Γ ( a + 1 ) e z γ ( a , z ) = 1 1 z a + 1 + z a + 2 ( a + 1 ) z a + 3 + 2 z a + 4 ( a + 2 ) z a + 5 + 3 z a + 6 , a 1 , 2 , ,
8.9.2 z a e z Γ ( a , z ) = z 1 1 + ( 1 a ) z 1 1 + z 1 1 + ( 2 a ) z 1 1 + 2 z 1 1 + ( 3 a ) z 1 1 + 3 z 1 1 + , | ph z | < π .
12: 8.6 Integral Representations
8.6.2 γ ( a , z ) = z 1 2 a 0 e t t 1 2 a 1 J a ( 2 z t ) d t , a > 0 .
8.6.3 γ ( a , z ) = z a 0 exp ( a t z e t ) d t , a > 0 .
8.6.5 Γ ( a , z ) = z a e z 0 e z t ( 1 + t ) 1 a d t , z > 0 ,
8.6.7 Γ ( a , z ) = z a 0 exp ( a t z e t ) d t , z > 0 .
13: 28.2 Definitions and Basic Properties
The standard form of Mathieu’s equation with parameters ( a , q ) is …
28.2.2 ζ ( 1 ζ ) w ′′ + 1 2 ( 1 2 ζ ) w + 1 4 ( a 2 q ( 1 2 ζ ) ) w = 0 .
28.2.3 ( 1 ζ 2 ) w ′′ ζ w + ( a + 2 q 4 q ζ 2 ) w = 0 .
28.2.19 q c 2 n + 2 ( a ( ν + 2 n ) 2 ) c 2 n + q c 2 n 2 = 0 , n .
14: 8.12 Uniform Asymptotic Expansions for Large Parameter
8.12.3 P ( a , z ) = 1 2 erfc ( η a / 2 ) S ( a , η ) ,
8.12.4 Q ( a , z ) = 1 2 erfc ( η a / 2 ) + S ( a , η ) ,
8.12.6 z a γ ( a , z ) = cos ( π a ) 2 sin ( π a ) ( e 1 2 a η 2 π F ( η a / 2 ) + T ( a , η ) ) ,
8.12.21 Q ( a , x ) = q
8.12.22 x ( a , 1 2 ) a 1 3 + 8 405 a 1 + 184 25515 a 2 + 2248 34 44525 a 3 + , a .
15: 25.1 Special Notation
k , m , n nonnegative integers.
a real or complex parameter.
16: 12.1 Special Notation
x , y real variables.
a , ν real or complex parameters.
17: 8.13 Zeros
8.13.1 1 + a 1 < x ( a ) < ln | a | , 1 < a < 0 .
For information on the distribution and computation of zeros of γ ( a , λ a ) and Γ ( a , λ a ) in the complex λ -plane for large values of the positive real parameter a see Temme (1995a). …
18: 31.14 General Fuchsian Equation
31.14.1 d 2 w d z 2 + ( j = 1 N γ j z a j ) d w d z + ( j = 1 N q j z a j ) w = 0 , j = 1 N q j = 0 .
The three sets of parameters comprise the singularity parameters a j , the exponent parameters α , β , γ j , and the N 2 free accessory parameters q j . With a 1 = 0 and a 2 = 1 the total number of free parameters is 3 N 3 . …
31.14.3 w ( z ) = ( j = 1 N ( z a j ) γ j / 2 ) W ( z ) ,
31.14.4 d 2 W d z 2 = j = 1 N ( γ ~ j ( z a j ) 2 + q ~ j z a j ) W , j = 1 N q ~ j = 0 ,
19: 8.18 Asymptotic Expansions of I x ( a , b )
8.18.1 I x ( a , b ) = Γ ( a + b ) x a ( 1 x ) b 1 ( k = 0 n 1 1 Γ ( a + k + 1 ) Γ ( b k ) ( x 1 x ) k + O ( 1 Γ ( a + n + 1 ) ) ) ,
8.18.4 a F k + 1 = ( k + b a ξ ) F k + k ξ F k 1 ,
8.18.8 x 0 = a / ( a + b ) .
8.18.9 I x ( a , b ) 1 2 erfc ( η b / 2 ) + 1 2 π ( a + b ) ( x x 0 ) a ( 1 x 1 x 0 ) b k = 0 ( 1 ) k c k ( η ) ( a + b ) k ,
8.18.18 I x ( a , b ) = p , 0 p 1 ,
20: 8.1 Special Notation
x real variable.
a , p real or complex parameters.