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The function `C` is called `Binomial` in Mathematica. The function `P` does not appear to be pre-installed, but you can define it yourself: P[n_, r_] := n!/(n - r)!
I know very little about PDEs, but your equations seem singular at `r==0`. Also, they do not seem compatible when `t==r==0`: {eqr,eqphi,Aric}/.{t->0,r->0} {-2 Ar[0,0,phi]+(Ar^(0,0,2))[0,0,phi] == 0, 2 (Ar^(0,0,1))[0,0,phi] == 0, ...
It is a nonlinear boundary value problem, unlikely to have a closed-form solution. Try `NDSolve`.
Then you can simply write the condition into your system: Solve[Simplify[{(-r + 9/40 b ((7 Sqrt[c] Sqrt[r] + Sqrt[q + b iy q + 49 c r])^2/(1 + b ix)^2 + (2 q^2)/(Sqrt[ q + b ix q + 49 c r]...
One way to do it is with pattern replacement: factorFromSum = Sum[a_*b_, {var_, lmts__}] /; FreeQ[b, var] :> b*Sum[a, {var, lmts}]; Sum[2*b0*b1*Subscript[x, i], {i, 1, n}] + Sum[b1^2*Subscript[x, i]^4, {i, 1, n}] /....
It seems that `Region` makes a rough plot to display quickly.
You can see from the plot that the primitive is discontinuous at the point where you evaluate it: Block[{\[Beta] = 1, \[ScriptCapitalD]LS = 1, Rs = 2}, ReImPlot[S, {R, Rs - \[Beta] \[ScriptCapitalD]LS - 1, Rs + \[Beta]...
At the first attempt the plot range was random, but I was lucky. Then I collected the coordinates of the intersection with the mouse, using the "get coordinates" tool. I used that value as seed for `FindRoot`.
Are you sure it is `F(lambda)` instead of `F[lambda]`? If you define `c1 = RandomReal[3,1]` you get a list instead of a number: is that really what you want? I would simply write `c1 = RandomReal[3]`.
As for the primitive of `Abs[ff[x,m,s]]`, I get back the input, not 0. The square of `ff` is a simple sum of Gaussians, there is no problem in finding a primitive in terms of the error function `Erf`, and in calculating the limits at infinity.