Bessel equation

If m ∈ Z, then I−m(z) and Im(z) are linearly independent and span the space of solutions of the
modified Bessel equation.
We have
Im(e ±iπ z) = e ±iπm Im(z).
1.4 Integral representations of modified Bessel function
We start with Bessel-Schläfli-type representations. If Rez > 0, then the basic contour integral representation
of Im is
Im(z) = 1
2πi ]−∞,0 +
z
exp
,−∞[ 2 (t + t−1
) t −m−1 =
dt (1.4.5)
1
z
m
exp s +
2πi 2
z2
s
4s
−m−1 ds. (1.4.6)
]−∞,0 + ,−∞[
To see this note that by (1.4.5), Im is holomorphic around zero. Besides, by the Hankel identity
1 1
= exp (s) s
Γ(m + 1) 2πi
−m−1 ds.
We also have
]−∞,0 + ,−∞[
I−m(z) = 1
2πi [(0−0) +
z
exp
] 2 (t + t−1
) t −m−1 dt. (1.4.7)
Here, the contour starts at 0 from the negative side on the lower sheet, encircling 0 in the positive
direction and ends at 0 from the negative side on the upper sheet,
To see this we make a substitution t = s −1 in (1.4.5) noting that dt = −s −2 ds, and then we change
the orientation of the contour.