Force an Im-Re separation in a complex number with parameter

Question

I was doing a computation with Mathematica (summing a series) and a part of the result was the following:

$$\frac{\Gamma \left(\frac{1}{4}\right)}{3 \sqrt{2} \sqrt[4]{-\frac{\sqrt[3]{r^3}}{2}-\frac{1}{2} i \sqrt{3} \sqrt[3]{r^3}+1}}$$

I would like to take the denominator, and (considering that $r$ is POSITIVE, that is $r > 0$) I would like to separate the denominator into an Imaginary and a Real part, in such a way that I can then take the complex conjugate and rewrite that fraction as $A + iB$.

The problem is that the commands I tried, do not work... Is there a way to force that separation, considering that $r > 0$ always?

I tried with

ExpToTrig[
Simplify[ComplexExpand[...], Assumptions -> {r > 0}]]

where the dots stands for the expression above. I also tried only with the denominator and it does not work.

Any smarter / stronger command?

Thank you so much!!

Answer 1

I found this way to proceed:

Simplify[ComplexExpand[
3 Sqrt[2] (1 - 1/2 (r^3)^(1/3) - 1/2 I Sqrt[3] (r^3)^(1/3))^(1/4), 
TargetFunctions -> {Re, Im}], r > 0]

With the following output

$$3 \sqrt{2} \sqrt[8]{r^2-r+1} \left(\cos \left(\frac{1}{4} \tan ^{-1}\left(2-r,-\sqrt{3} r\right)\right)+i \sin \left(\frac{1}{4} \tan ^{-1}\left(2-r,-\sqrt{3} r\right)\right)\right)$$

Could this be ok?

Answer 2

To answer, "Could this be OK?" in your answer

expr = 3 Sqrt[2] (1 - 1/2 (r^3)^(1/3) - 1/2 I Sqrt[3] (r^3)^(1/3))^(1/4);

Note that with the assumption that r > 0 then this expression can be expressed more compactly as

expr2 = Assuming[r > 0, expr // Simplify]

(* 3 (4 + (-2 - 2 I Sqrt[3]) r)^(1/4) *)

Your approach provides the same result with this form

expr3 = Assuming[r > 0, 
  expr2 // ComplexExpand[#, TargetFunctions -> {Re, Im}] & // Simplify]

(* 3 Sqrt[2] (1 - r + r^2)^(
 1/8) (Cos[1/4 ArcTan[2 - r, -Sqrt[3] r]] + 
   I Sin[1/4 ArcTan[2 - r, -Sqrt[3] r]]) *)

To get the form A + B I, you must Expand this expression

expr4 = expr3 // Expand

(* 3 Sqrt[2] (1 - r + r^2)^(1/8) Cos[1/4 ArcTan[2 - r, -Sqrt[3] r]] + 
 3 I Sqrt[2] (1 - r + r^2)^(1/8) Sin[1/4 ArcTan[2 - r, -Sqrt[3] r]] *)

Verifying that this is separated into the real and imaginary parts of the original expression:

Assuming[r > 0, (expr4 // First) == Re[expr] // ComplexExpand // FullSimplify]

(* True *)

Assuming[r > 0, (expr4 // Last)/I == Im[expr] // ComplexExpand // 
  FullSimplify]

(* True *)