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1: 28.20 Definitions and Basic Properties
§28.20(ii) Solutions Ce ν , Se ν , Me ν , Fe n , Ge n
For other values of z , h , and ν the functions M ν ( j ) ( z , h ) , j = 1 , 2 , 3 , 4 , are determined by analytic continuation. …
§28.20(iv) Radial Mathieu Functions Mc n ( j ) , Ms n ( j )
§28.20(vi) Wronskians
§28.20(vii) Shift of Variable
2: 10.29 Recurrence Relations and Derivatives
With 𝒵 ν ( z ) defined as in §10.25(ii),
𝒵 ν - 1 ( z ) - 𝒵 ν + 1 ( z ) = ( 2 ν / z ) 𝒵 ν ( z ) ,
𝒵 ν - 1 ( z ) + 𝒵 ν + 1 ( z ) = 2 𝒵 ν ( z ) .
𝒵 ν ( z ) = 𝒵 ν - 1 ( z ) - ( ν / z ) 𝒵 ν ( z ) ,
𝒵 ν ( z ) = 𝒵 ν + 1 ( z ) + ( ν / z ) 𝒵 ν ( z ) .
3: 10.36 Other Differential Equations
§10.36 Other Differential Equations
The quantity λ 2 in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by - λ 2 if at the same time the symbol 𝒞 in the given solutions is replaced by 𝒵 . …
10.36.1 z 2 ( z 2 + ν 2 ) w ′′ + z ( z 2 + 3 ν 2 ) w - ( ( z 2 + ν 2 ) 2 + z 2 - ν 2 ) w = 0 , w = 𝒵 ν ( z ) ,
10.36.2 z 2 w ′′ + z ( 1 ± 2 z ) w + ( ± z - ν 2 ) w = 0 , w = e z 𝒵 ν ( z ) .
4: 10.44 Sums
§10.44(i) Multiplication Theorem
If 𝒵 = I and the upper signs are taken, then the restriction on λ is unnecessary. …
§10.44(ii) Addition Theorems
The restriction | v | < | u | is unnecessary when 𝒵 = I and ν is an integer. …
§10.44(iv) Compendia
5: 28.27 Addition Theorems
§28.27 Addition Theorems
They are analogous to the addition theorems for Bessel functions10.23(ii)) and modified Bessel functions10.44(ii)). …
6: 11.12 Physical Applications
§11.12 Physical Applications
7: 10.25 Definitions
Its solutions are called modified Bessel functions or Bessel functions of imaginary argument. …
Branch Conventions
Symbol 𝒵 ν ( z )
Corresponding to the symbol 𝒞 ν introduced in §10.2(ii), we sometimes use 𝒵 ν ( z ) to denote I ν ( z ) , e ν π i K ν ( z ) , or any nontrivial linear combination of these functions, the coefficients in which are independent of z and ν . …
8: 10.43 Integrals
Let 𝒵 ν ( z ) be defined as in §10.25(ii). …
z ν + 1 𝒵 ν ( z ) d z = z ν + 1 𝒵 ν + 1 ( z ) ,
z - ν + 1 𝒵 ν ( z ) d z = z - ν + 1 𝒵 ν - 1 ( z ) .
e ± z z ν 𝒵 ν ( z ) d z = e ± z z ν + 1 2 ν + 1 ( 𝒵 ν ( z ) 𝒵 ν + 1 ( z ) ) , ν - 1 2 ,
e ± z z - ν 𝒵 ν ( z ) d z = e ± z z - ν + 1 1 - 2 ν ( 𝒵 ν ( z ) 𝒵 ν - 1 ( z ) ) , ν 1 2 .
9: 11.8 Analogs to Kelvin Functions
§11.8 Analogs to Kelvin Functions
10: 10.39 Relations to Other Functions
Elementary Functions
Parabolic Cylinder Functions
Confluent Hypergeometric Functions
10.39.7 I ν ( z ) = ( 2 z ) - 1 2 M 0 , ν ( 2 z ) 2 2 ν Γ ( ν + 1 ) , 2 ν - 1 , - 2 , - 3 , ,
Generalized Hypergeometric Functions and Hypergeometric Function