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There is the following important General Principle: if a parameter enters in a linear differential equation additively, for example $$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$ where the parameter is $\lambda$, then any solution $w(x,\lambda)$ normalized at a point, so that the normalization does not depend on $\lambda$, is an ENTIRE FUNCTION of $\lambda$, for any fixed $x$.

Normalization can be:

a) $w(x_0)=a,\; w'(x_0)=b$, if $x_0$ is non-singular, or

b) $w(x)\sim (x-x_0)^\rho(1+g(x)$, where $g$ is holomorphic, if $x_0$ is a regular singularity, and $\rho$ is the larger exponent at it. (Or smaller exponent when the difference is not an integer, or this can be modified when the difference is an integer).

c) I do not write the normalization for irregular singularity, but this can be done. (The result is proved in a special case in the book of Y. Sibuya, Global theory of a second order linear ordinary differential equation.... ).

d) Or whatever reasonable normalization you can think about, with the only condition that it does not depend on $\lambda$.

Example: $w''+\lambda w=0.$ Under normalization a) with $x_0=0$:

$$w(x,\lambda)=a\cos\sqrt{\lambda}x+b\frac{\sin \sqrt{\lambda} x}{\sqrt{\lambda}},$$

an enitre function of $\lambda$. But $e^{i\sqrt{\lambda}x}$ has a wrong normalization, and is not an entire function of $\lambda$.

I've seen this Principle stated many times with references to Poincaré. For example, V. de Alfaro and T. Regge, Potential scattering. North-Holland Publishing Co., Amsterdam; Interscience Publishers Division John Wiley & Sons, Inc., New York 1965,

In the Introduction, they discuss this a lot, and give a reference on Poincaré... but his reference is wrong: the paper they refer to contains no such results. They say that Poincare studied case b), which is more difficult than a).

My question: where did Poincaré write on this? (Poincaré wrote 400+ papers, so before examining all of them, I decided to ask here:-)

Of course, with the normalization a) this is easy to prove and is proved in many places. Any reference where this is proved under the normalization b)? (Poincaré or not). Any reference where more general statements of this sort are discussed? For higher order equations, for PDE's, for parameter entering in a different way?

Remark. I read the Resumé Analytique of Poincaré (his own survey of his work) and did not find this.

Edit. See also Dependence of a solution of a linear ODE on parameter

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  • $\begingroup$ Shouldn’t this follow from the contour integral representations discussed at hsm.stackexchange.com/a/5876 ? $\endgroup$ – Francois Ziegler Feb 21 at 22:10
  • $\begingroup$ @Francois Ziegler: they seem to discuss non-singular problems (my case a), but let me check the references on Poincare given there. $\endgroup$ – Alexandre Eremenko Feb 22 at 1:45
  • $\begingroup$ Weyl (1910, p. 222) still seems to only explicitly address your case (a), with reference to Picard (1908, p. 89). Hilb (1915, p. 501) credits Poincaré (1884). Did you mean to call the exponent in (b) something other than $\lambda$? $\endgroup$ – Francois Ziegler Feb 22 at 4:05
  • $\begingroup$ @Francois Ziegler: Thanks for the lambda and for the additional references. Regge writes that Poincare considered also case b), which is much more difficult than a). But his reference is wrong. $\endgroup$ – Alexandre Eremenko Feb 22 at 5:05
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    $\begingroup$ @Francois Ziegler: Interestingly, Regge also credits what you call Poincare (1884) in your second comment (Acta, 4, 213). I read this paper. There is no mentioning of the question in it. $\endgroup$ – Alexandre Eremenko Feb 22 at 14:23
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Poincaré Analyticity and the Complete Variational Equations refers to Poincaré's treatise "New Methods of Celestial Mechanics".

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    $\begingroup$ They refer to a different theorem, on LOCAL analyticity for non-linear equations. I need global analyticity for liner equations. $\endgroup$ – Alexandre Eremenko Feb 22 at 1:43

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