Inspired by a great video on the Laplace transform, I was looking for an easy way to visualize complex functions of time, i.e., functions of time where the output is a complex number. This is fairly easy to do with Guile Scheme and Gnuplot, though the method could be polished more into a proper application, which I didn’t do.

This is the core function for plotting the data:

```
(define (complex-plot t0 tmax step f)
(unless (> t0 tmax)
(let ((z (f t0)))
(display t0)
(display " ")
(display (real-part z))
(display " ")
(display (imag-part z))
(display "\r\n")
(complex-plot (+ t0 step) tmax step f))))
```

We just pass in the starting value (`t0`

), the maximum time value, a step value (usually some fraction of Pi), and a lambda with our complex function of time. Writing complex functions of time is pretty easy in Guile Scheme because all the numbers are represented internally as complex numbers anyway. E.g.:

```
scheme@(guile-user)> (complex-plot 0 25 (/ 3.1415 32) (lambda (t) (exp (* 0+i t))))
0 1.0 0.0
0.098171875 0.9951850104692727 0.09801425884672994
0.19634375 0.9807864101254524 0.19508464243304188
0.294515625 0.9569428571883648 0.2902763649975122
0.3926875 0.9238839645735444 0.3826727322450213
...etc...
```

We have to output this to a file:

`(with-output-to-file "out.txt" (lambda () (complex-plot 0 25 (/ 3.1415 32) (lambda (t) (exp (* 0+i t))))))`

Then we start `gnuplot`

and use a 3d splot with lines. (Credit goes to this tutorial.)

```
gnuplot> set xrange [0:25]
gnuplot> set yrange [-2:2]
gnuplot> set ticslevel 0
gnuplot> splot "out.txt" u 1:2:3 with lines
```

Another interesting complex function of time is the addition of two complex functions of time which leave only a signal on the real axis:

`(lambda (t) (+ (exp (* 0+i t)) (exp (* 0-i t))))`

And here is my hacker emacs screen, where the action is happening!