# how to take complex conjugate of a real variable

I have a function defined as follows:

g[θ_] := 1/k (Exp[-I δ0] Sin[δ0] + 3 Exp[I δ1] Sin[δ1 Cos[θ] +
5/2 Exp[I δ2 (3 (Cos[θ])^2 - 1))


and I compute the product of g with its complex conjugate as follows:

f[θ] Conjugate[f[θ]]


which gives me:

4.18986*10^-30 ((0.453154 - 0.288691 I) + (0.443562 - 0.0670825 I) Conjugate[Cos[θ]] +
(2.49994 - 0.0174532 I) (-1 + 3 Conjugate[Cos[θ]]^2)) ((0.453154 + 0.288691 I) +
(0.443562 + 0.0670825 I) Cos[θ] + (2.49994 + 0.0174532 I) (-1 + 3 Cos[θ]^2))


However I want theta to be real so that Conjugate[Cos[θ]] is just Cos[θ] and likewise for Conjugate[Cos[θ]]^2.

How can I do this?

EDIT

I should say my end goal is to plot and integrate the product of g with its product. I've tried what Rashid Zia suggested like so:

Plot[
Simplify[g[θ] Conjugate[g[θ]], θ ∈ Reals], {θ, 0, 2 Pi},
AxesLabel -> {rad, m^2/sr}
]


but the plot doesn't look like what I'm told it should, and the integral:

Integrate[2 Pi Simplify[g[θ] Conjugate[g[θ]], θ ∈ Reals], {θ, 0, Pi}]


doesn't seem to return the expected value.

Is there something I am missing here?

• The parentheses and brackets are not balanced in the definition of g[θ]. Please correct the definition. Also, f[θ] Conjugate[f[θ]] should read g[θ] Conjugate[g[θ]] – Bob Hanlon Apr 21 '16 at 4:08
• If all the parameters are real, just use ComplexExpand. – murray Apr 21 '16 at 14:59

## 3 Answers

You can use Simplify with the assumption that θ is an element of the Reals.

f = Conjugate[Cos[θ]]
Simplify[f, θ ∈ Reals]

• Note that using things like Simplify may perform manipulations you do not want, like replacing x^2 + 2 x y + y^2 with (x+y)^2. The minimal method is to use Refine as so: Refine[Conjugate[a+I b], θ ∈ Reals]. Also, in place of the assumption θ ∈ Reals you can use the assumption _Symbol ∈ Reals to assume that all explicit variables are real. This allows you to encapsulate this in a function without having to manually pass a list of arguments. For an exhaustive discussion, see here. – Jess Riedel Sep 27 '17 at 19:01
• Thanks @JessRiedel. I appreciate the suggestions and great linked discussion. – Rashid Oct 3 '17 at 17:52

If your aim is numerical and graphical I would not "worry" about the cosmetics of the form. In the following I have made k=1 (no values for $\delta$'s given). to avoid numerical problems just due to extreme scales (v precision). I have tried to "correct" the unbalanced parentheses referred to by Bob Hanlon. I may have made an error. If so, I apologize:

fun[δ0_Real, δ1_Real, δ2_Real, θ_,k_Real] :=
ExpToTrig[
1/k (Exp[-I δ0] Sin[δ0] +
3 Exp[I δ1] Sin[δ1 ] Cos[θ] +
5/2 Exp[I δ2 ] (3 (Cos[θ])^2 - 1))];
g[r_, s_, t_, u_, v_] :=
Chop[fun[r, s, t, u, v] Conjugate[fun[r, s, t, u, v]]]


I hope this be a kick start to the desired result. Apologies for any misunderstanding on my part.

You could also use

gc=Conjugate[g[θ]]/.Conjugate[cos_Cos]->cos


and h=g*gc and use h as the integrand or plotted function. One should avoid calling Simplify inside Plot or Integrate because this will be much slower than necessary.

Also, your integrand contains a factor 10^-30. So it is practically zero and nothing is plotted and the integral is zero.