Mathematics on stamps
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| Germany 1984
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
(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: