A treatise on the theory of Bessel functions, by G. N. Watson.

42

THEORY OF BESSEL FUNCTIONS

[CHAP. III

If J, (z) be defined by ~ 3'1 (8) when v =-r-, the solution now constructed is
Co 2-'-` r (2 - r) J_,_ (z) + C2,+, 2.+- r (r + ) J, (z).
It follows that no modification in the definition of J, (z) is necessary when
v= + (r + ); the real peculiarity of the solutio n in this case is that the
negative root of the indicial equation gives rise to a series containing two
arbitrary constants, co and c21.+, i.e. to the general solution of the differential
equation.
3'12. A funtdamental system of solutions of Bessel's equation.
It is well known that, if y, and y2 are two solutions of a linear differential
equation of the second order, and if yj' and y' denote their derivates with
respect to the independent variable, then the solutions are linearly independent if the WVonskian determinantt
iy/ Y2/
does not vanish identically; and if the Wronskian does vanish identically,
then, either one of the two solutions vanishes identically, or else the ratio of
the two solutions is a constant.
If the Wronskian does not vanish identically, then any solution of the
differential equation is expressible in the form cl y1 + c2 y2 where c, and c2 are
constants depending on the particular solution under consideration; the
solutions y1 and y2 are then said to form a fundamental system.
For brevity the Wronskian of y, and y, will be written in the forms
Z {yl y2}, A2 {Y, Y2},
the former being used when it is necessary to specify the independent variable.
We now proceed to evaluate. {(z), J-, (z)}.
If we multiply the equations
V, LJ^ () =, V= J, () =
by J, (z), J_- (z) respectively and subtract the results, we obtain an equation
which may be written in the form
d [z {J-(,), J_- ()}]W = 0,
* In connexion with series representing this solution, see Plana, 1Menm. della.R. Accad. delle
Sci. di Torino, xxvi. (1821), pp. 519-538.
t For references to theorems concerning Wronskians, see Encyclopedie des Sci. Math. ii. 16
(~ 23), p. 109. Proofs of the theorems quoted in the text are given by Forsyth, Treatise on
Differential Equations (1914), ~~ 72-74.