Therefore, we get
Γ(1 + v) = vΓ(v)
Substitute v=v+1
Γ(2 + v) = (v + 1) ∗ v ∗ Γ(v) Γ(3 + v) = (2 + v) ∗ (v + 1) ∗ v ∗ Γ(v) Γ(n + 1 + v) = (n + v) ∗ (n − 1 + v) ∗ … .∗ (2 + v) ∗ (v + 1) ∗ v ∗ Γ(v)
Substitute the above equation into 2
1 2 𝑣 ∗ Γ(1 + v)
(−1)
2
=
Γ(n + 1 + v) v ∗ Γ(v)
2 2 ∗ ! ∗
1 2 𝑣 ∗ vΓ(v)
(−1)
2
=
Γ(n + 1 + v) v ∗ Γ(v)
2 2 ∗ ! ∗
(−1) 2 2+ ∗ ! ∗ Γ(n + 1 + v) (−1) 2 2− ∗ ! ∗ Γ(n + 1 − v)
=𝑣 =
2
=−𝑣 =
2
Since 2+1 = 0 in the case r=v , then we get
+
= ∑
∞
+𝑣
= ∑
𝑁=0
∞
∞
(−1) 2 2+𝑣 ∗ ! ∗ Γ(n + 1 + v)
2+𝑣 = ∑
2+𝑣
= ∑ 2
𝑁=0
𝑁=0
2
∞
(−1) ( ) 2+𝑣 ! ∗ Γ(n + 1 + v)
𝑣
= ∑
𝑁=0
Let this equation be J v In the case = −
2
∞
(−1) ( ) 2−𝑣 ! ∗ Γ(n + 1 − v)
−𝑣
= ∑
𝑁=0
Let this equation be J −v Therefore, the above two equations J v
and J −v
are called the Bessel equation of
the first kind of the order v and –v respectively y = 1 𝑣 () + 2 𝑣 ()
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