Page 533 - Electromagnetics

n
N −n (z) = (−1) N n (z) (E.44)
I −n (z) = I n (z) (E.45)
K −n (z) = K n (z) (E.46)
n
j −n (z) = (−1) n n−1 (z), n > 0 (E.47)

Power series

∞ n+2k
k (z/2)
J n (z) = (−1) (E.48)
k!(n + k)!
k=0
∞ n+2k
(z/2)
I n (z) = (E.49)
k!(n + k)!
k=0
Small argument approximations |z| 1.

1
n
z
J n (z) ≈ (E.50)
n! 2
z
1
ν
J ν (z) ≈ (E.51)
(ν + 1) 2
2
N 0 (z) ≈ (ln z + 0.5772157 − ln 2) (E.52)
π
n

(n − 1)! 2
N n (z) ≈− , n > 0 (E.53)
π z
ν

(ν) 2
N ν (z) ≈− , ν > 0 (E.54)
π z
1
n
z
I n (z) ≈ (E.55)
n! 2
1
ν
z
I ν (z) ≈ (E.56)
(ν + 1) 2
n
2 n! n
j n (z) ≈ z (E.57)
(2n + 1)!
(2n)! −(n+1)
n n (z) ≈− n z (E.58)
2 n!
Large argument approximations |z| 1.

2
π νπ
J ν (z) ≈ cos z − − , | arg(z)| <π (E.59)
πz 4 2

2
π νπ
N ν (z) ≈ sin z − − , | arg(z)| <π (E.60)
πz 4 2

2 j(z− − )
νπ
π
H (1) (z) ≈ e 4 2 , −π< arg(z)< 2π (E.61)
ν
πz

2 − j(z− − )
νπ
π
(2)
H ν (z) ≈ e 4 2 , −2π< arg(z)<π (E.62)
πz

1 z π
I ν (z) ≈ e , | arg(z)| < (E.63)
2πz 2
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