@Adam Ledger 

Note that Whittaker's functions do not appear here; they are included only in the second edition of  "A Course of Modern Analysis" written with G.N. Watson.

@Abdoulaye 

You can define the intervals and the expressions (this can be done automatically in a procedure):

intervals:=-infinity..0, 0..1, 1..2, 2..3, 3..infinity:
expressions:= 0, t^2, 1, 3-t, 0:
max( seq( maximize(expressions[i],t=intervals[i]),i=1..nops([intervals])) );
    1

@Markiyan Hirnyk 

My point was that a person interested in Maple and minimal surfaces should be informed about the existence of this book (not very easy to find); probably I was completely wrong.

@Markiyan Hirnyk 

Your question is not answered in the book, but some other special surfaces are computed symbolically using Maple (V.5 and 6).

@Markiyan Hirnyk

It's the site of a Chinese university; I had no problems with it.

https://www.researchgate.net/publication/268347899_Constructing_Minimal_Triangular_Mesh_Based_on_Discrete_Mean_Curvature

or directly to the pdf:

http://www.math.zju.edu.cn/webpagenew/uploadfiles/attachfiles/2011112721225969.pdf

Edit.
Oprea J. - The mathematics of soap films_Explorations with Maple, AMS, 2000

@Adam Ledger 

Looking closer to your F, it is easy to see that actually F(s) = GAMMA(s), all the terms in the sum being 0 except for k=0.

If you are interested in such problems I'd recommend the book

Bressoud D., Wagon S. - A course in computational number theory, Wiley, 2000

It also contains Mathematica code but you could try to convert it into Maple.

@Adam Ledger 

@aamirkhan 

Yes, it seems that there exists an elementary integral wrt s. Your f_3 is a sum of >6000 terms, and each can be integrated simbolycally (but I did not verify all of them). Maple seems to refuse to integrate globally. You could use map(int,...)  but I don't think that the resulting HUGE expression will be useful.
Note also that using floats in symbolic computations is not usually a good idea.

This post reminds me of the following interesting formula for the nth prime due to Willan:

nthprime := n ->
1 + sum(floor(n^(1/n)*sum(floor(cos(Pi*((x-1)!+1)/x)^2),x=1..m)^(-1/n)),m=1..2^n);

It is correct but cannot be practically used for n>9 say.

@Markiyan Hirnyk 

Let's come back to our horses :-)

A:=plot3d(1, x=0..4, y=0..3, view = [0..4, 0..3, 0..2],
      image =cat(kernelopts(mapledir),"/data/images/fjords.jpg")):
with(plots):with(plottools):
f := transform( (x,y,z) -> [x^2-y^2, 2*y*x]):
display(f(A), axes = none);

@Mac Dude 

To avoid the interpolation one may take the inverse transform (provided it is bijective).
So, let A be a subset of the rectangle R(A), and f:A-->B  the transform (bijective).
B subset of a rectangle R(B).
Simply take the color(b) := color(g(b)), b in B, where g is the inverse mapping.

@Markiyan. Unortunately plottools:-transform can be applied only to the plot structure, not to the background.
One may try plot3d(...,image=filename).

@Mac Dude 

Conformal mappings are easy to construct:

(x,y) |--> (u,v) = (Re f(x+iy), Im f(x+iy)),

f being holomorphic such as f(z)=z^2, f(z)=1/z, f(z)=sin(z) etc.

[ The example is essentially (x,y) |--> (x, y^3) which is of course not conformal;
a conformal example would be e.g.  (x, y) |-->  (x^2-y^2, 2*x*y) ]
The origin does not matter, a translation does not affect the conformality.

Of course the image will have to be clipped (into a rectangle); if f(z_0) = oo, it wil be of course clipped.
The example can be easily adapted.

Thanks, I have inserted.

@Preben Alsholm 

@rlopez 

I think that the (global) bivariate limits for rational functions were introduced in Maple 2015 only.
I remember this because I was really impressed by the new algorithm.