Appendix D: Special Functions for Synchrotron
Radiation
Synchrotron radiation and spectroscopy employ a wide variety of special functions
that are often unfamiliar to chemists and biologists. Here we limit ourselves to a few
functions critical for description of synchrotron radiation. The more expansive
descriptions in Wikipedia are so well done that it is worth making an exception to
the “no-links” rule: https://en.wikipedia.org/wiki/Bessel_function. There are also
excellent descriptions in Wolfram MathWorld: http://mathworld.wolfram.com/
BesselFunctionoftheFirstKind.html and so on.
Appendix D.1: Bessel Functions
There are two regular Bessel functions, J n (z) and Y n (z), and they are independent
solutions of the differential equation:
z
2 y
00
þ zy
0
þ z
2
À n
2
À
Á
y ¼ 0
ðD:1Þ
Bessel functions of the first kind, J n (z), are needed to describe the properties of
cylindrical waveguides and undulators, and they also appear in the description of
nuclear forward scattering. In Mathematica they are defined by BesselJ[n,z]. For
what it’s worth, such Bessel functions are the radial part of the modes of vibration of
a circular drum (Fig. D.1).
For integer n, they can be computed as sums over an infinite series:
J n x
ð Þ ¼
x
2
n X 1
k¼0
À1
ð Þ
k
k! n þ k
ð
Þ!
x
2
2k
ðD:2Þ
© Springer Nature Switzerland AG 2020
S. P. Cramer, X-Ray Spectroscopy with Synchrotron Radiation, Biological and Medical
Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-28551-7
319