@Hullzie16 The command discont gives the set of discontinuity points. I took the three points (say a,b,c) in the interval 0..5.
The functions were plotted in the intervals  e .. a-e,  a+e .. b-e,  b+e .. c-e,  c+e .. 5-e  (e=0.001) and then displayed together.

@Hullzie16 I took the real parts of the functions. If you want the gaps (to see where the functions have complex values), just remove Re.

@oggsait The equality you want is true only for alpha=1.

@oggsait After expand(simplify(%));  ==>

Download fdiff.mw

@dharr It depends on conventions. E,g. in complex analysis, z > 7 means: Im(z) = 0 and Re(z)>7,  but z > 2+3i is not accepted (nonsense).

@nm  Excellent, thanks!

@Anthrazit Almost. I saw your post and I wanted to answer there, but later I decided to make a separate one, including some code. (Actually I think that your post should have been a question.)

@Carl Love Thank you. Yes, sprintf is definitely easier to read, I should have used it.

- You have alias(f=f(w)) and then D(f)(w). 

- D@@4  is not the same as D^4

- You must be sure that the exprssion you want to substitute appears as a subexpression (e.g. a*b is not subexpression in a*b*c).

@student_1 

int((f_exact(x,t)-f_numeric(x,t))^2, [x=0..2,t=0..2], numeric)^(1/2);

You may want to divide by sqrt(Area) (=2 in this case).

@Federiko On my old computer:

xx:=CodeTools:-Usage(LinearSolve(AA, bb)):
memory used=411.13MiB, alloc change=70.00MiB, cpu time=3.53s, real time=3.43s, gc time=1.22s

(For symbolic computations using floats is usually not a good idea).

@ecterrab Maybe I miss something (I use Physics very seldom) but the Wirtinger derivatives seem to be used only at a formal level:

diff(conjugate(z)+z*conjugate(z)^2, conjugate(z));  # ok

but, diff(Re(z), conjugate(z)) and diff(abs(z), conjugate(z)) are both 0.
(probably a convert(..., conjugate) is mandatory).

Edit. Actually it is not complicated to compute the Wirtinger derivatives; more work would be to integrate them in the system.

diffW := proc(f::algebraic, z::{And(symbol,Not(constant)), specfunc(And(symbol,Not(constant)),conjugate)})
  local g:= convert(f,conjugate),zb;
  g:=subs(conjugate(op(z))=zb,g);
  subs(zb=conjugate(op(z)),diff(g,`if`(type(z,symbol),z,zb)))
end proc:

@Axel Vogt Yes, the derivative is real. As mentioned, the modified `diff/abs` implements d/dx abs(f(x+I*y)). But as well known, if f is analytic, d/dz f(z) = d/dx f(x+I*y), where z = x + I*y. The complex derivative d/dz abs(z) does not exit at any point (not even at real points). The same for abs(GAMMA(2*t + I)): the complex derivative does not exist.

Unfortunately, the Wirtinger derivatives d/dz, d/dzbar  are not implemented in Maple (or other CASes AFAIK).