Laplace transform of converging analytic functions

I am wondering whether the following statement is true or not.

A Laplace transform of a converging analytic function has finite number of poles in the left half plane. (Converging means it converges to a constant.)

Any partial help will be appreciated.

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Converging to a constant where? Please be more specific about what you mean by a "converging analytic function", giving an example. –  Captain Oates Sep 23 '10 at 2:15
For instance, converging to a constant 0(zero). f(t)=e^(-5t) t>=0 is one example. However, for an exponentially decaying case like this example, my statement is rather easy to prove. Here is another example; f(t)=1/log(t) t>0 In this case, it still converges to zero but really slow. I want to know how the Laplace transform of this function becomes. One book says laplace transform of analytic functions has countable many poles. I want to know how this statmenet will change if I have the additional condtion, which is, a functions is also converging to a constant as well as it's analytic. –  mim Sep 23 '10 at 4:00