Page 534 - Electromagnetics

π −z 3π
K ν (z) ≈ e , | arg(z)| < (E.64)
2z 2
1
nπ
j n (z) ≈ sin z − , | arg(z)| <π (E.65)
z 2
1
nπ
n n (z) ≈− cos z − , | arg(z)| <π (E.66)
z 2
e jz
n+1
(1)
h (z) ≈ (− j) , −π< arg(z)< 2π (E.67)
n
z
e − jz
n+1
(2)
h (z) ≈ j , −2π< arg(z)<π (E.68)
n
z
Recursion relationships
zZ ν−1 (z) + zZ ν+1 (z) = 2νZ ν (z) (E.69)
Z ν−1 (z) − Z ν+1 (z) = 2Z (z) (E.70)

ν
zZ (z) + νZ ν (z) = zZ ν−1 (z) (E.71)

ν

zZ (z) − νZ ν (z) =−zZ ν+1 (z) (E.72)
ν
zL ν−1 (z) − zL ν+1 (z) = 2νL ν (z) (E.73)

L ν−1 (z) + L ν+1 (z) = 2L (z) (E.74)
ν

zL (z) + νL ν (z) = zL ν−1 (z) (E.75)
ν

zL (z) − νL ν (z) = zL ν+1 (z) (E.76)
ν
zz n−1 (z) + zz n+1 (z) = (2n + 1)z n (z) (E.77)
nz n−1 (z) − (n + 1)z n+1 (z) = (2n + 1)z (z) (E.78)

n
zz (z) + (n + 1)z n (z) = zz n−1 (z) (E.79)

n

−zz (z) + nz n (z) = zz n+1 (z) (E.80)
n
Integral representations
1 π
J n (z) = e − jnθ+ jz sin θ dθ (E.81)
2π −π
1 π
J n (z) = cos(nθ − z sin θ) dθ (E.82)
π 0
1 −n π jz cos θ
J n (z) = j e cos(nθ) dθ (E.83)
2π −π
1 π z cos θ
I n (z) = e cos(nθ) dθ (E.84)
π 0
∞ π

K n (z) = e −z cosh(t) cosh(nt) dt, | arg(z)| < (E.85)
0 2
z n π 2n+1
j n (z) = cos(z cos θ) sin θ dθ (E.86)
2 n+1 n! 0
(− j) n π jz cos θ
j n (z) = e P n (cos θ) sin θ dθ (E.87)
2 0
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