Isolate imaginary/real part of simple algebraic expression?

Questions
1

Using Mathematica can be often very frustrating. Either I’m doing something wrong here, or Mathematica is just dumb.

So, I have the below expression:

-1 + ((2*I*(1 + Log[Pi] - (I*Pi)/2))/Pi)*(r + I*t)

And I am trying to put it in form a+i*b, nothing complicated at all, but Mathematica chokes on it.
I tried to use the below, but it’s ignoring my attempts to tell it that r and t are reals:

In[298]:= Clear[r, t]; x1 = -1 + ((2*I*(1 + Log[Pi] - (I*Pi)/2))/
     Pi)*(r + I*t); 
  FullSimplify[Abs[x1]*Cos[Arg[x1]], Assumptions -> 
      r*In*Reals && t*In*Reals]

Out[298]= -1 - Im[t] + Re[r] - (2*(1 + Log[Pi])*(Im[r] + Re[t]))/Pi

Where am I going wrong? Please help, I really appreciate.

2

Since r and t are both Real

ComplexExpand[-1 + ((2*I*(1 + Log[Pi] - (I*Pi)/2))/Pi)*(r + I*t)]

returns

-1 + r - (2*t)/Pi - (2*t*Log[Pi])/Pi + I*((2*r)/Pi + t + (2*r*Log[Pi])/Pi)
3

Thanks, that’s the function I needed.

4

Wow, the result is absolutely horrible. Mathematica doesn’t simplify a lot of the things it should:

ComplexExpand[(Cos[r v] Cosh[t v] - 
    I Sin[r v] Sinh[t v]) (Cos[t Log[Cos[v]]] - I Sin[t Log[Cos[v]]])]

Cos[v] Cos[1/2 t Log[Cos[v]^2]] Cosh[t v] Cosh[t Arg[Cos[v]]] - 
 Cosh[t Arg[Cos[v]]] Sin[v] Sin[1/2 t Log[Cos[v]^2]] Sinh[t v] + 
 Cos[v] Cos[1/2 t Log[Cos[v]^2]] Cosh[t v] Sinh[t Arg[Cos[v]]] - 
 Sin[v] Sin[1/2 t Log[Cos[v]^2]] Sinh[t v] Sinh[t Arg[Cos[v]]] + 
 I (-Cos[v] Cosh[t v] Cosh[t Arg[Cos[v]]] Sin[1/2 t Log[Cos[v]^2]] - 
    Cos[1/2 t Log[Cos[v]^2]] Cosh[t Arg[Cos[v]]] Sin[v] Sinh[t v] - 
    Cos[v] Cosh[t v] Sin[1/2 t Log[Cos[v]^2]] Sinh[t Arg[Cos[v]]] - 
    Cos[1/2 t Log[Cos[v]^2]] Sin[v] Sinh[t v] Sinh[t Arg[Cos[v]]])
5

Can you show the simplified version including all those things?

Are you assuming anything about the values of those variables?

Thanks.

6

Maybe something like this?

Assuming[Element[{v, t, r}, Reals] && Cos[v]^2 != 0, 
 FullSimplify@PiecewiseExpand[ReIm@ComplexExpand[
     (Cos[r v] Cosh[t v] - I Sin[r v] Sinh[t v])*
      (Cos[t Log[Cos[v]]] - I Sin[t Log[Cos[v]]])]]]