Classical Bessel Functions
The classical theory of Bessel functions is closely connected with the investigation of the integral (see(7.3.16)) $$\frac{1} {2\pi }\int \nolimits \nolimits _{0}^{2\pi }\cos (k\sin (u) - ku)\ \mathrm{d}u$$ (7.0.1) by Bessel(1824). He took k as an integer
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Classical Bessel Functions
The classical theory of Bessel functions is closely connected with the investigation of the integral (see (7.3.16)) 1 2
Z
2
cos.k sin.u/ ku/ du
(7.0.1)
0
by Bessel (1824). He took k as an integer and obtained many results. After the time of Bessel, investigations of these integrals, which by then bore his name, became numerous. G.N. Watson consolidated these results in a monograph entitled “A Treatise on the Theory of Bessel Functions” (Watson (1944) is the second edition). In what follows, we give a brief insight into the classical theory, thereby concentrating on specific features such as integral and series expressions, recurrence relations, and orthogonality relations. The exercises of Sect. 7.4 present applications of Bessel functions to discontinuous integrals and the modeling of electrons in a magnetic field. The theory is extended in Chap. 8 to the Helmholtz equation in q dimensions. For further properties and details in classical Bessel functions we refer to Abramowitz and Stegun (1972), Lebedev (1973), Watson (1944), and Whittaker and Watson (1948).
7.1 Derivation and Definition of Bessel Functions We consider vibrations of a membrane fixed in a circular frame, i.e., a disk B21 of radius 1. Let Z be the amplitude of the membrane which follows the wave equation x Z D
1 @2 Z c 2 @t 2
in B21 ;
Z D 0 on @B21 D S1 ;
W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 7, © Springer Basel 2013
(7.1.1)
347
348
7 Classical Bessel Functions
p where c D = is the phase-velocity. Here, is the surface tension of the membrane and is its mass-density. We are interested in time-harmonic vibrations, i.e., Z.x; t/ D Z.x/ exp.i!t/
(7.1.2)
which gives us .x Z.x// exp.i!t/ D
1 .! 2 /Z.x/ exp.i!t/; c2
(7.1.3)
or equivalently, x Z.x/ C
!2 Z.x/ D 0 c2
in B21
(7.1.4)
and Z D 0 on S1 . Thus, Z has to fulfill the Helmholtz equation in B21 , i.e., x Z C k 2 Z D 0 in B21 ;
ZD0
on S1
(7.1.5)
with k D !=c being the wave number. The spherical geometry leads us to polar coordinates and we use separation of variables, i.e., Z.x/ D U.r/˚.'/;
where x D Œr cos.'/; r sin.'/T :
(7.1.6)
We get 0 D x Z C k 2 Z D
1 @2 1 @ @2 Z C Z C Z C k2Z @r 2 r @r r 2 @' 2
1 1 D U 00 .r/˚.'/ C U 0 .r/˚.'/ C 2 U.r/˚ 00 .'/ C k 2 U.r/˚.'/ r r D
rU 0 .r/ ˚ 00 .'/ r 2 U 00 .r/ C C C r 2k2: U.r/ U.r/ ˚.'/
(7.1.7)
This can only be true for all r 2 Œ0; R and for all ' 2 Œ0; 2/ if ˚ 00 .'/ D ˚.'/
(7.1.8)
r 2 U 00 .r/ rU 0 .r/ C C r 2 k 2 D : U.r/ U.r/
(7.1.9)
and
7.1 Derivation and Definition of Bessel Functions
349
The first ordinary differential equation, ˚ 00 .'/ C ˚.'/ D 0;
(7.1.10)
possesses a solution if and only if D n2 , n 2 N0 . The solution is given by ˚.'/ D A cos.n'/ C B sin.n'/:
(7.1.11)
Inserting D n2 into the second ordinary differential equation gives us r 2 U 00 .r/ C rU 0 .r/ C .r 2 k 2 n2 /U.r/ D 0:
(7.1.12)
Substitute now x D kr and we find x 2 U 0
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