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complex variables

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1: 1.1 Special Notation
x , y

real variables.

z

complex variable in §§1.2(i), 1.91.11, real variable in §§1.51.6.

w

complex variable in §§1.91.11.

2: 32.12 Asymptotic Approximations for Complex Variables
§32.12 Asymptotic Approximations for Complex Variables
3: 5.1 Special Notation
j , m , n

nonnegative integers.

z = x + i y

complex variable.

a , b , q , s , w

real or complex variables with | q | < 1 .

4: 32.1 Special Notation
m , n

integers.

z

complex variable.

5: 21.8 Abelian Functions
An Abelian function is a 2 g -fold periodic, meromorphic function of g complex variables. In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. …
6: 4.34 Derivatives and Differential Equations
4.34.7 d 2 w d z 2 - a 2 w = 0 ,
4.34.8 ( d w d z ) 2 - a 2 w 2 = 1 ,
4.34.12 w = ( 1 / a ) sinh ( a z + c ) ,
4.34.13 w = ( 1 / a ) cosh ( a z + c ) ,
4.34.14 w = ( 1 / a ) coth ( a z + c ) ,
7: 4.20 Derivatives and Differential Equations
4.20.9 d 2 w d z 2 + a 2 w = 0 ,
4.20.10 ( d w d z ) 2 + a 2 w 2 = 1 ,
4.20.12 w = A cos ( a z ) + B sin ( a z ) ,
4.20.13 w = ( 1 / a ) sin ( a z + c ) ,
4.20.14 w = ( 1 / a ) tan ( a z + c ) ,
8: 6.1 Special Notation
x

real variable.

z

complex variable.

9: 25.1 Special Notation
k , m , n

nonnegative integers.

s = σ + i t

complex variable.

z = x + i y

complex variable.

10: 4.14 Definitions and Periodicity
4.14.4 tan z = sin z cos z ,
4.14.5 csc z = 1 sin z ,
4.14.6 sec z = 1 cos z ,
4.14.8 sin ( z + 2 k π ) = sin z ,
4.14.9 cos ( z + 2 k π ) = cos z ,