||И — lOg U = Z -f С,
but the functions и thus obtained cannot be expressed in terms of z by performing algebraic operations and taking exponentials and logarithms. Can one be sure in advance that Bessel's equation is not susceptible to similar treatment?
Investigations on the possibility of integrating differential equations of the first order by elementary operations performed upon both the dependent variable and the independent variable were carried out by Lorenz, Steen, Hansen and Petersen,* and by Mordukhai-Boltovskoi.f Lagutinski has studied systems of equations of the first order. J
In the present paper, we prove a general theorem (§3) which shows that the solutions of Bessel's equation do not satisfy elementary equations, § except for the given values of v.
Our theorem states that if any solution of equation (A) satisfies an elementary equation in w and z, then the general solution of (A) is an elementary function of z.
The procedure in the present paper is purely formal. The many questions of a function-theoretic nature which will arise will be found treated in our papers on elementary functions published in the Transactions of this Society. ||
2. The I-Functions. We shall call any algebraic function of w and z an l-function of order zero, and the variables w and z monomials of order zero.