They too can be computed as sums over series:
I α x
ð Þ ¼
X 1
m¼0
1
m!Γ m þ α þ 1
ð
Þ
x
2
2mþα
ðD:4Þ
and using:
K α x
ð Þ ¼
π
2
I Àα x
ð Þ À I α x
ð Þ
sin απ
ðD:5Þ
Since we only need two modified Bessel functions of the second kind, K 1/3 (x) and
K 2/3 (x), we note with Kim that they can also be expressed as follows (Kim, 3.10):
K1
3
x
ð Þ ¼
ffiffi ffi
3
p Z 1
0
τ sin
3
2
x τ þ
1
3
τ
3
h
i
dτ
ðD:6Þ
and
K2
3
x
ð Þ ¼
ffiffi ffi
3
p Z 1
0
τ cos
3
2
x τ þ
1
3
τ
3
h
i
dτ
ðD:7Þ
322
Appendix D: Special Functions for Synchrotron Radiation