I changed your code to

f := "this:///Images/Maple.jpg";
img := Read(f);

and it works - you then get some warnings later that rotation angles should be in radians, which I'm sure you can fix.

Although

diff(ln(GAMMA(x)), x)=Psi(x)

Using the chain rule we find

diff(ln(GAMMA(1/x)), x)=-Psi(1/x)/x^2;

which explains the missing -x^2.

I don't think there is a builtin command, but there an implementation of a heap, which allows it to be done easily. If you wanted the whole list partially sorted, then just sort the selected ones and follow with the rest. (The smallest ones are in decreasing order; the largest are in increasing order.)

Download partselect.mw

One solution to this error is to supply an approximate solution, and since you said tanh(x) was a known solution, I tried that. But then I realized tanh(x) goes to -1, not 0, as x->-infinity. If I change the boundary condition to z(-15)=-1 it works. But if you really wanted z(-infinity)=0, then you can try a better approximate solution.

dsolve.mw

See

https://www.mapleprimes.com/questions/235168-How-Do-I-Generate-Magic-And-Semi-Magic

for some solutions.

You can add a constant to each cell to get other ones, but not sure what exactly you mean by random.

For your first case, the initial conditions are specified as 

dsolve({DE, R(0) = 1, D(R)(0) = 1}, numeric, range = 0 .. 20)

I made up a value for the derivative in the second condition; you will no doubt have a better value. Or perhaps you wanted a boundary condition as your second condition.

And a similar problem applies for your second problem.

PS: In your second problem in defining F you probably wanted an explicit multiplication after the first ), so (...)*(...)

Your second problem doesn't seem well-posed at theta=0.

Here some progress:

DIFFERENTIAL_EQUATION.mw

Download pade2.mw

The degree 3 result here is also a problem - I'll submit an SCR.

Not as general as @acer's solutions. It seems evalhf handles the Gamma function but not factorial so in this particular case converting to GAMMA works.

[Edit: This explanation is incorrect - see below]

failed_plot.mw

[Edit: the OP has now completely changed this post, so my responses here and below are no longer relevant.]

The equations in your post outline a laplace transform analytical solution and you seem to want to follow that, but in your worksheet you set up for a numerical solution of the original untransformed equations, abandoning the laplace transform method.

So forgetting laplace transforms and following your worksheet, you have two errors in formulating a consistent problem. You had strange parentheses after alpha0 in your Cond, which I guessed as a correction (in red). [Edit] What about initial conditions? 

new_paper_dust.mw

If you want the analytical solution then you can start by transforming the pdes to odes with intrans:-laplace, gving you Eq (16) etc, then solve the ode system - but then it is not a good idea to give numerical values to your parameters.

Maple tries to figure out the file format you want from the file extension, but doesn't recognize .dat. You can use the format option to specify which format you want, for example CSV:

Download testfile.mw

Edit: for saving Maple objects in a .m file for reading in later into Maple use the save command

@sursumCorda Vote up. I could not beat your code, which as you no doubt realize is very efficient because (1) CharacterFrequencies uses external code, (2) it works directly on the strings and (3) anyway most of the time is taken working out the primes.

I tried converting the integer to an Array of digits, and then invoking a compiled routine to search the Array. But the additional conversion time offsets potential efficiency gain, and the prime generation time still dominates. (Actually using numboccur 10 times, dopping out once you find 4, is more efficient and is close to your result)

Given that manipulating digits in integers is probably a common operation of interest I'm surprised that there isn't a better way to extract them. convert(p,base,10) is shockingly slow compared to the methods that go via sprintf("%d",p). Of course the sprint code could probably be easily modified to produce an Array or List.

I'm confused about the relative merits of nextprime over multiple calls to ithprime, and there my timings weren't consistent - perhaps there is an internal list that is added to as another prime is found, or there is a huge table anyway. 

Download Aseq.mw

(worksheet not displaying right now)

contours.mw

Replace G0 with

G0:=proc(y) if not type(y,numeric) then return 'procname'(args) end if;
    if y<0 then exp(y)/(1+exp(y)) else 1/(exp(-y)+1) end if;
   end proc; 

Use prev[1..-1] := curr[1..-1]; to copy the contents regardless of size (or the for loop), rather than prev:=curr; You cannot assign to a formal parameter.

This has nothing to do with the size - your first attempt had prev=curr, not prev:=curr so did not work either.

Here's one way to do it. Of course you could do it all in one loop.

Download framepng.mw

Test_prn_files.zip

if the singularity is replaced by the limit of the function at the origin, then you can make some progress, though I'm being careless about checking consistency of different limit directions. What do we know about the signs of the parameters?

Download mtaylor.mw