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I need to generate synthetic data to test an algorithm used to process data from an experiment. A synthetic wavelet is constructed using the following equations, but I am uncertain how to numerically evaluate the improper integral shown below.

[tex]

\[

u(t) = {\mathop{\rm Re}\nolimits} \left\{ {{\textstyle{1 \over \pi }}\int\limits_0^\infty {S\left( \omega \right)\exp \left[ {i\left( {\omega t - kr} \right)} \right]d\omega } } \right\}

\]

[/tex]

In the equation above, [tex]{\mathop{\rm Re}\nolimits} [/tex] indicates the real part, [tex]\pi[/tex] is the ubiquitous pi constant, [tex]i[/tex] denotes an imaginary number, and [tex]\omega[/tex] is the angular frequency (1/s).

Also,

[tex]

\[

S\left( \omega \right) = 4\sqrt \pi \frac{{\omega ^2 }}{{\omega _0^3 }}\exp \left[ { - \frac{{\omega ^2 }}{{\omega _0^2 }}} \right]

\]

[/tex]

In the equation above, [tex]\omega_0[/tex] is a reference angular frequency (1/s).

Also,

[tex]

\[

kr = \left( {1 - \frac{i}{{2Q}}} \right)\left( {\frac{\omega }{{\omega _0 }}} \right)^{ - \gamma } \omega t

\]

[/tex]

In the equation above, [tex]Q[/tex] is a constant real number, [tex]\[\gamma = (\pi Q)^{ - 1} \] [/tex], and [tex]t[/tex] is the time (s).

How would I numerically integrate the improper integral to obtain [tex]u(t)[/tex], also using the other formulas listed above?