Bessel equation

Theorem 2.2.2 Poisson type representations Let γ be a contour satisfying
(1 − t 2 1 m+
) 2e izt
= 0.
Then
is a solution of the Bessel equation.
2.3 Bessel function
The Bessel function is defined as
z m
m iπ
Jm(z) = e 2 Im(−iz)
m −iπ
= e 2 Im(iz)
=
∞
n=0
= 1
iπ
γ
γ(1)
γ(0)
(1 − t 2 1 m−
) 2e izt dt
2n+m
(−1) n z
2
n!Γ(m + n + 1)
−iπ
e m
m
2 iπ
K(−iz) − e 2 K(iz)
Jm(e ±iπ z) = e ±imπ Jm(z).
Theorem 2.3.1 If Rez > 0, then
Jm(z) = 1
2πi ]−∞,0 +
z
exp
,−∞[ 2 (t − t−1
dt
)
tm+1 = 1 z m
exp s − z2 ds
m+1 .