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modified Bessel equation

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1: 10.25 Definitions
§10.25(i) Modified Bessel’s Equation
§10.25(ii) Standard Solutions
§10.25(iii) Numerically Satisfactory Pairs of Solutions
Table 10.25.1: Numerically satisfactory pairs of solutions of the modified Bessel’s equation.
Pair Region
2: 10.36 Other Differential Equations
§10.36 Other Differential Equations
3: 10.72 Mathematical Applications
§10.72(i) Differential Equations with Turning Points
Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. …
4: 10.73 Physical Applications
10.73.3 4 W + λ 2 2 W t 2 = 0 .
5: 10.34 Analytic Continuation
10.34.1 I ν ( z e m π i ) = e m ν π i I ν ( z ) ,
10.34.2 K ν ( z e m π i ) = e m ν π i K ν ( z ) π i sin ( m ν π ) csc ( ν π ) I ν ( z ) .
10.34.4 K ν ( z e m π i ) = csc ( ν π ) ( ± sin ( m ν π ) K ν ( z e ± π i ) sin ( ( m 1 ) ν π ) K ν ( z ) ) .
10.34.5 K n ( z e m π i ) = ( 1 ) m n K n ( z ) + ( 1 ) n ( m 1 ) 1 m π i I n ( z ) ,
10.34.6 K n ( z e m π i ) = ± ( 1 ) n ( m 1 ) m K n ( z e ± π i ) ( 1 ) n m ( m 1 ) K n ( z ) .
6: 10.39 Relations to Other Functions
10.39.2 K 1 2 ( z ) = K 1 2 ( z ) = ( π 2 z ) 1 2 e z .
10.39.5 I ν ( z ) = ( 1 2 z ) ν e ± z Γ ( ν + 1 ) M ( ν + 1 2 , 2 ν + 1 , 2 z ) ,
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 , ,
7: 10.30 Limiting Forms
10.30.1 I ν ( z ) ( 1 2 z ) ν / Γ ( ν + 1 ) , ν 1 , 2 , 3 , ,
10.30.2 K ν ( z ) 1 2 Γ ( ν ) ( 1 2 z ) ν , ν > 0 ,
10.30.3 K 0 ( z ) ln z .
10.30.5 I ν ( z ) e ± ( ν + 1 2 ) π i e z / 2 π z , 1 2 π + δ ± ph z 3 2 π δ .
8: 10.31 Power Series
10.31.1 K n ( z ) = 1 2 ( 1 2 z ) n k = 0 n 1 ( n k 1 ) ! k ! ( 1 4 z 2 ) k + ( 1 ) n + 1 ln ( 1 2 z ) I n ( z ) + ( 1 ) n 1 2 ( 1 2 z ) n k = 0 ( ψ ( k + 1 ) + ψ ( n + k + 1 ) ) ( 1 4 z 2 ) k k ! ( n + k ) ! ,
10.31.2 K 0 ( z ) = ( ln ( 1 2 z ) + γ ) I 0 ( z ) + 1 4 z 2 ( 1 ! ) 2 + ( 1 + 1 2 ) ( 1 4 z 2 ) 2 ( 2 ! ) 2 + ( 1 + 1 2 + 1 3 ) ( 1 4 z 2 ) 3 ( 3 ! ) 2 + .
10.31.3 I ν ( z ) I μ ( z ) = ( 1 2 z ) ν + μ k = 0 ( ν + μ + k + 1 ) k ( 1 4 z 2 ) k k ! Γ ( ν + k + 1 ) Γ ( μ + k + 1 ) .
9: 10.32 Integral Representations
10.32.1 I 0 ( z ) = 1 π 0 π e ± z cos θ d θ = 1 π 0 π cosh ( z cos θ ) d θ .
10.32.6 K 0 ( x ) = 0 cos ( x sinh t ) d t = 0 cos ( x t ) t 2 + 1 d t , x > 0 .
10.32.10 K ν ( z ) = 1 2 ( 1 2 z ) ν 0 exp ( t z 2 4 t ) d t t ν + 1 , | ph z | < 1 4 π .
10: 10.45 Functions of Imaginary Order
With z = x , and ν replaced by i ν , the modified Bessel’s equation (10.25.1) becomes …
10.45.3 I ~ ν ( x ) = I ~ ν ( x ) , K ~ ν ( x ) = K ~ ν ( x ) ,