What i want is to know if these are artifacts of our mathematical systems or if they are a consequence of the nature of this particular region of space-time; what applications do these have, how do they work as starshades?.... and a nice pic wouldn't hurt.@vv 

@rcorless 

This is facinating stuff, most beautiful pictures i've ever seen. I was woundering can we make these 3D, add a density parameter on z, mirror the other side to give it clarity?

• Γ(z)Γ(1−z)=π/sin(πz)

so cant we interchange the greeks and do a little algebra and get 

sqr(pi*csc(alpha*pi){a*s^(-beta)+b*s^(-alpha)}

, which is what i get by hand, but my old Schaum's outline does not have these exponents on s interchanged with the corresponding multiples. i.e. it has

sqr(pi*csc(alpha*pi){a*s^(-alpha)+b*s^(-beta)}

Thank you Acer for your time, that was driving me mad. If it's not to much to ask, for the love of mathematics, can you see my other post on Laplace Transforms?

https://www.mapleprimes.com/questions/232588-How-Do-Solve-This-Laplace-Transform?sq=232588

I havn't used Maple since 1995, my old 3.5 wouldn't read, so I bought new Maple. I'll post more on this Laplace thread, my programming failures that is.

Now I will study this last code you posted. Thanks again.

restart;
with(inttrans);
with(gfun);
assume(alpha + beta = 1);
fnn := a*t^(-alpha) + b*t^(-beta);
inttrans[laplace](fnn, t, s)

inttrans[:-laplace](fnn, t, s)

simplify(inttrans[laplace](fnn, t, s))

Only yields

        a*laplace(t^(-alpha), t, s) + b*laplace(t^(-beta), t, s)

Red: ℜ(z^3+3^z)>17

Blue: Im(z^3+3^z)>0