Manfred Börgens
Mathematics on stamps
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deutsche Flagge  deutsche Version
Bessel stamp   Germany 1984

  Scott 1422

Friedrich Wilhelm Bessel (1784 - 1846), Bessel functions

The Federal Republic of Germany issued this stamp on the occasion of Bessel's 200th birthday. It shows his portrait and the graphs of the Bessel functions (cylinder functions) of the 1st kind with orders  0  and  1 .

The labels in the graphics are difficult to read. In strong magnification one can read (or rather guess):

At the ends of the axes is written  x  (right) and  J0(x), J1(x)  (top). The horizontal scaling goes from  -12  to  14  in steps by  2 , the vertical scaling from  -0.6  to  1.0  in steps by  0.2 ; all minus signs and all decimal points are omitted, also the zero. The curve going through  (0,1)  is labelled  J0(x) , the curve through  (0,0)  is labelled  J1(x) .

J0(x)  and  J1(x)  are basic solutions of the Bessel differential equation

Bessel differential equation
(here the order  α  is  0  resp.  1 )

J0(x)  and  J1(x)  are the basic solutions of the 1st kind (the solutions of the 2nd kind are  Y0(x) ,  Y1(x) , they are not shown on the stamp).

The Bessel differential equation has numerous applications in physics, e.g. vibrations of membranes like drumheads.

As can be seen on the stamp,  J0(x)  and  J1(x)  are weakly damped vibrations. Both functions have a representation as power series:

formula for J0

formula for J1

last update 2004-10-10
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