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1: 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
2: 10.76 Approximations
§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions
Real Variable; Imaginary Order
3: 10.44 Sums
§10.44(i) Multiplication Theorem
§10.44(ii) Addition Theorems
Graf’s and Gegenbauer’s Addition Theorems
§10.44(iii) Neumann-Type Expansions
§10.44(iv) Compendia
4: 28.27 Addition Theorems
They are analogous to the addition theorems for Bessel functions10.23(ii)) and modified Bessel functions10.44(ii)). …
5: 10.74 Methods of Computation
In the case of the modified Bessel function K ν ( z ) see especially Temme (1975). … For applications of generalized Gauss–Laguerre quadrature (§3.5(v)) to the evaluation of the modified Bessel functions K ν ( z ) for 0 < ν < 1 and 0 < x < see Gautschi (2002a). …
Kontorovich–Lebedev Transform
6: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
10.28.1 𝒲 { I ν ( z ) , I - ν ( z ) } = I ν ( z ) I - ν - 1 ( z ) - I ν + 1 ( z ) I - ν ( z ) = - 2 sin ( ν π ) / ( π z ) ,
10.28.2 𝒲 { K ν ( z ) , I ν ( z ) } = I ν ( z ) K ν + 1 ( z ) + I ν + 1 ( z ) K ν ( z ) = 1 / z .
7: 10.73 Physical Applications
§10.73(i) Bessel and Modified Bessel Functions
Consequently, Bessel functions J n ( x ) , and modified Bessel functions I n ( x ) , are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. … … On separation of variables into cylindrical coordinates, the Bessel functions J n ( x ) , and modified Bessel functions I n ( x ) and K n ( x ) , all appear. …
8: 10.1 Special Notation
The main functions treated in this chapter are the Bessel functions J ν ( z ) , Y ν ( z ) ; Hankel functions H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) ; modified Bessel functions I ν ( z ) , K ν ( z ) ; spherical Bessel functions j n ( z ) , y n ( z ) , h n ( 1 ) ( z ) , h n ( 2 ) ( z ) ; modified spherical Bessel functions i n ( 1 ) ( z ) , i n ( 2 ) ( z ) , k n ( z ) ; Kelvin functions ber ν ( x ) , bei ν ( x ) , ker ν ( x ) , kei ν ( x ) . For the spherical Bessel functions and modified spherical Bessel functions the order n is a nonnegative integer. … For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
9: 10.26 Graphics
§10.26(i) Real Order and Variable
See accompanying text
Figure 10.26.7: I ~ 1 / 2 ( x ) , K ~ 1 / 2 ( x ) , 0.01 x 3 . Magnify
See accompanying text
Figure 10.26.8: I ~ 1 ( x ) , K ~ 1 ( x ) , 0.01 x 3 . Magnify
See accompanying text
Figure 10.26.9: I ~ 5 ( x ) , K ~ 5 ( x ) , 0.01 x 3 . Magnify
See accompanying text
Figure 10.26.10: K ~ 5 ( x ) , 0.01 x 3 . Magnify
10: 10.37 Inequalities; Monotonicity
§10.37 Inequalities; Monotonicity
10.37.1 | K ν ( z ) | < | K μ ( z ) | .