The problem is as follows: find a closed-form expression, as a function of $k$, for the following integral. (Common special functions are acceptable and expected).

$$\int_{0}^{\infty} \frac{k^x}{\Gamma(x+1)} dx$$

I've tried quite a few substitutions on it, but sadly to no avail. I really doubt that a simple answer, in terms of elementary functions, exists, but this prompts me to ask whether this is a known integral and if it can be expressed in terms of other special or nonstandard functions.

This is, as might be evident from the integral itself, not a homework problem or exercise in any book, so I don't expect a solution to necessarily exist. I stumbled upon the integral through other work, and was curious to see if it has a "solution," so to speak.


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