deutsche Version

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 1^st 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 1^st kind (the solutions of the 2^nd 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: