Poincare's theorem and 2nd order ode

[jimmy neutron]

This is a request about the second order differential equation
y'' + (k^2 + f(r))y(r) = 0 (1)
where k is a (real) constant and f(r) is a real valued function of r that has some constraints regarding integratability.

According to the following reference:
"Lectures in Scattering Theory", A. G. Sitenko, Pergamon (1971) Chapter 7 pp 86-87:

"According to Poincare's theorem if a coefficient occurring in a second-order differential equation not only depends on the co-ordinate but is also an entire function of some parameter, the solution of the equation, satisfying boundary conditions which are independent of that parameter, is an entire function of the parameter, is an entire function of the parameter for any value of the co-ordinate. Therefore, the function phi(k,r), which is a solution of (1) satisfying the boundary conditions phi(k,0) = 0 and phi'(k,0) = 1 is an entire function of the parameter."

The request is: Can anyone give a reference for this?

 

[gtoguy97]

A reference for the above statement can be found in the following book: "The Theory of Functions of a Real Variable" by E. T. Copson, Dover Publications (1986), Chapter 8, pp. 325-326.