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complex variable and parameters

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1: 32.1 Special Notation
m , n integers.
x real variable.
z complex variable.
k real parameter.
2: 25.1 Special Notation
k , m , n nonnegative integers.
x real variable.
a real or complex parameter.
s = σ + i t complex variable.
z = x + i y complex variable.
3: 28.19 Expansions in Series of me ν + 2 n Functions
28.19.2 f ( z ) = n = f n me ν + 2 n ( z , q ) ,
28.19.3 f n = 1 π 0 π f ( z ) me ν + 2 n ( z , q ) d z .
28.19.4 e i ν z = n = c 2 n ν + 2 n ( q ) me ν + 2 n ( z , q ) ,
4: 12.8 Recurrence Relations and Derivatives
12.8.1 z U ( a , z ) U ( a 1 , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 ,
12.8.2 U ( a , z ) + 1 2 z U ( a , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 ,
12.8.3 U ( a , z ) 1 2 z U ( a , z ) + U ( a 1 , z ) = 0 ,
12.8.4 2 U ( a , z ) + U ( a 1 , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 .
12.8.5 z V ( a , z ) V ( a + 1 , z ) + ( a 1 2 ) V ( a 1 , z ) = 0 ,
5: 33.13 Complex Variable and Parameters
§33.13 Complex Variable and Parameters
6: 12.4 Power-Series Expansions
12.4.1 U ( a , z ) = U ( a , 0 ) u 1 ( a , z ) + U ( a , 0 ) u 2 ( a , z ) ,
12.4.2 V ( a , z ) = V ( a , 0 ) u 1 ( a , z ) + V ( a , 0 ) u 2 ( a , z ) ,
12.4.3 u 1 ( a , z ) = e 1 4 z 2 ( 1 + ( a + 1 2 ) z 2 2 ! + ( a + 1 2 ) ( a + 5 2 ) z 4 4 ! + ) ,
12.4.5 u 1 ( a , z ) = e 1 4 z 2 ( 1 + ( a 1 2 ) z 2 2 ! + ( a 1 2 ) ( a 5 2 ) z 4 4 ! + ) ,
12.4.6 u 2 ( a , z ) = e 1 4 z 2 ( z + ( a 3 2 ) z 3 3 ! + ( a 3 2 ) ( a 7 2 ) z 5 5 ! + ) .
7: 12.1 Special Notation
x , y real variables.
z complex variable.
a , ν real or complex parameters.
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. …
8: 8.4 Special Values
8.4.5 Γ ( 1 , z ) = e z ,
8.4.7 γ ( n + 1 , z ) = n ! ( 1 e z e n ( z ) ) ,
8.4.10 Q ( n + 1 , z ) = e z e n ( z ) ,
8.4.11 e n ( z ) = k = 0 n z k k ! .
8.4.12 γ ( n , z ) = z n ,
9: 10.10 Continued Fractions
10.10.1 J ν ( z ) J ν 1 ( z ) = 1 2 ν z 1 1 2 ( ν + 1 ) z 1 1 2 ( ν + 2 ) z 1 , z 0 ,
10.10.2 J ν ( z ) J ν 1 ( z ) = 1 2 z / ν 1 1 4 z 2 / ( ν ( ν + 1 ) ) 1 1 4 z 2 / ( ( ν + 1 ) ( ν + 2 ) ) 1 , ν 0 , 1 , 2 , .
10: 10.33 Continued Fractions
10.33.1 I ν ( z ) I ν 1 ( z ) = 1 2 ν z 1 + 1 2 ( ν + 1 ) z 1 + 1 2 ( ν + 2 ) z 1 + , z 0 ,
10.33.2 I ν ( z ) I ν 1 ( z ) = 1 2 z / ν 1 + 1 4 z 2 / ( ν ( ν + 1 ) ) 1 + 1 4 z 2 / ( ( ν + 1 ) ( ν + 2 ) ) 1 + , ν 0 , 1 , 2 , .