Where
() − −𝑣
cos(𝜋) 𝑣
()
() =
𝑣
sin(𝜋)
For example 2 ′′ + ′ + ( 2 − 1 3 ) = 0 v=-3 or v=3 then all you have to do is to substitute v= 1 3
and v= − 1 3
into 𝑣
, −𝑣
and
y = 1
𝑣
() + 2
𝑣
()
1 3
(−1) 𝑛 ( 𝑥 2
) 2𝑛+
∞ =0
E.g.: Bessel function of first kind of order 3 1 3 Bessel function of first kind of order -3
= ∑
!∗Γ(n+ 4 3
)
1 3
2
∞
) 2−
(−1) (
−1 3
= ∑
2 3
)
! ∗ Γ (n +
=0
Bessel function of second kind
y = 1
𝑣
() + 2
𝑣
()
1 3
() −
cos (
𝜋) 1 3
()
1 3
−
() =
1 3
1 3
sin(
𝜋)
1 2
() −
() − 2
1 3
()
1 3
()
1 3
1 3
−
−
=
=
√3 2
√3
Therefore,
1
−
1 3 ∗ Γ(
1 3
4 3
2
∞
) 2+
(−1) (
2
)
∑
=
4 3
16 3
)
! ∗ Γ (n +
=0
as 1
= 0
Bibliography K.F.RILEY, M. a. (2006). Mathematical Methods for physcs ad Engineering. Keiser, G. (1991). Optical Fiber Communications.
22
Made with FlippingBook - professional solution for displaying marketing and sales documents online