Dear researchers,

1) I have the integral simbol applied to a(x).exp(i.lambda.phi(x)) d(Floor(x)) ,

between x=1 to Infinite or written in LaTEX:

\[ \int_{1}^{\infty} a(x) e^{i \lambda \phi(x)} \, d(\lfloor x \rfloor), \]

where \( a(x) =\frac{3^{-x}}{x} \) is a given function and \( \phi(x) = arccos(x.ln(2+sin(x)))\)

Here, \( d(\lfloor x \rfloor) \) indicates integration with respect to the floor function of \( x \), which takes integer steps.

I have read about methods of phase stationary and non stationary and others.

2) Additionally, I am concern about sums like:

[ \sum_{\lambda=-\infty}^{\infty} i^{\lambda} J_{\lambda}(i) \lambda^{-k}, \]

where \( J_{\lambda}(i) \) denotes the Bessel function of the first kind of order \( \lambda \), \( i \) is the imaginary unit, and \( k \) is a fixed integer.

Best regards,

Carlos