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11: 10.76 Approximations
§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions
Real Variable; Imaginary Order
12: 11.3 Graphics
See accompanying text
Figure 11.3.1: 𝐇 ν ( x ) for 0 x 12 and ν = 0 , 1 2 , 1 , 3 2 , 2 , 3 . Magnify
See accompanying text
Figure 11.3.2: 𝐊 ν ( x ) for 0 < x 16 and ν = 0 , 1 2 , 1 , 3 2 , 2 , 3 . Magnify
See accompanying text
Figure 11.3.3: 𝐇 ν ( x ) for 0 x 12 and ν = 3 , 2 , 3 2 , 1 , 1 2 . Magnify
See accompanying text
Figure 11.3.4: 𝐊 ν ( x ) for 0 < x 16 and ν = 4 , 3 , 2 , 1 , 0 . … Magnify
§11.3(ii) Modified Struve Functions
13: 10.66 Expansions in Series of Bessel Functions
10.66.1 ber ν x + i bei ν x = k = 0 e ( 3 ν + k ) π i / 4 x k J ν + k ( x ) 2 k / 2 k ! = k = 0 e ( 3 ν + 3 k ) π i / 4 x k I ν + k ( x ) 2 k / 2 k ! .
14: 11.1 Special Notation
§11.1 Special Notation
For the functions J ν ( z ) , Y ν ( z ) , H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) , I ν ( z ) , and K ν ( z ) see §§10.2(ii), 10.25(ii). The functions treated in this chapter are the Struve functions 𝐇 ν ( z ) and 𝐊 ν ( z ) , the modified Struve functions 𝐋 ν ( z ) and 𝐌 ν ( z ) , the Lommel functions s μ , ν ( z ) and S μ , ν ( z ) , the Anger function 𝐉 ν ( z ) , the Weber function 𝐄 ν ( z ) , and the associated Anger–Weber function 𝐀 ν ( z ) .
15: 10.74 Methods of Computation
In the case of the modified Bessel function K ν ( z ) see especially Temme (1975). …
Kontorovich–Lebedev Transform
§10.74(viii) Functions of Imaginary Order
16: 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).
17: 11.2 Definitions
§11.2 Definitions
§11.2(i) Power-Series Expansions
Particular solutions: … Particular solutions: …
18: 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 .
19: 28.22 Connection Formulas
§28.22 Connection Formulas
The joining factors in the above formulas are given by …
28.22.13 M ν ( 1 ) ( z , h ) = M ν ( 1 ) ( 0 , h ) me ν ( 0 , h 2 ) Me ν ( z , h 2 ) .
20: 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. …