## Problem. ##

Let $\{\lambda_n\}_{n\in\mathbb N}$  be a sequence of complex numbers .  Let's call a family of exponential functions $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ $F$-independent (where $F$ is either $\mathbb C$ or $\mathbb R$) iff whenever the series with complex coefficients

$$f(s)=\sum\limits_{n=1}^{\infty}a_n e^{\lambda_n s},\qquad s\in F,$$
converges to $f(s)\equiv 0$ uniformly on every compact subset of $F$, we have that 
$a_n=0$ for all $n\in\mathbb N$. 

>**Question.** Assume that a sequence $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ is $\mathbb C$-independent. Is it $\mathbb R$-independent?


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##  Background and motivation. ##

A particularly interesting case for applications is when $|\lambda_n|\sim n$. A.F. Lavrent'ev  (whose work was mentioned in a previous [MO question][1]) proved that if $n=O(|\lambda_n|)$ then the corresponding family of exponentials is $\mathbb C$-independent (see also [this note][2]). It is relatively easy to construct a sequence of exponentials which is not $\mathbb C$-independent (see, e.g., [here][3]).


The question is related to the problem of uniqueness of solutions to the so called gravity equation
$$f(x+h)-f(x-h)=2h f'(x),\qquad x\in \mathbb R,$$
where $h>0$ is fixed. The equation appears in the study of radially symmetric central forces (the long history of the gravity equation and some known results are presented in this [article][4] by S. Stein).
 
Titchmarsh proved that an arbitrary solution to the gravity equation has the form 
$$f(x)=Ax^2+Bx+c+\sum\limits_{n=1}^{\infty}a_n e^{\lambda_n x},\qquad x\in \mathbb R,$$
where $a_n\in\mathbb C$, $n\in \mathbb N$ and $\lambda_n$ are the solutions of the equation
$\sinh hz=hz$. Thanks to the Lavrentiev result, the sequence $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ is $\mathbb C$-independent. If the answer to the question above is positive, then every sufficiently smooth function satisfying the gravity equation with two different $h_1$ and $h_2$ is a quadratic polynomial.


  [1]: http://mathoverflow.net/questions/30975/dirichlet-series-expansion-of-an-analytic-function
  [2]: http://www.math.purdue.edu/~eremenko/dvi/expo.pdf
  [3]: http://algo.inria.fr/csolve/sstein.html
  [4]: http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=&s5=remarks%2520on%2520the%2520gravity%2520equation&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq