# Removing Abs from Abs[a + Exp[I*c]b]^2

Title says it all. Mathematica produces sometimes expressions of the form

Abs[a + Exp[I*c]b]^2


where all three quantities a, b and c are positive real numbers. That expression of course reduces simply to the law of cosines

a^2 + b^2 + 2a b Cos[c]


however, I cannot get Mathematica to display it in this form. Any ideas?

## Update

Not sure why all responses here are not simple and direct, maybe in older versions this did not work, but now it does:

Abs[a+Exp[I*c] b]^2//ComplexExpand//FullSimplify


a^2+b^2+2 a b Cos[c]

(Abs[a+Exp[I*c] b]^2+Abs[a-Exp[I*c] b]^2)/(2 Abs[a+I b]^2)//ComplexExpand//FullSimplify


1

\$Version


14.0.0 for Mac OS X x86 (64-bit) (December 1, 2023)

Until someone figures out how to do it in a better way, here is a work around. Define a function (which is true only for your specific case of parameters):

RemoveAbs[x_] := FullSimplify[ComplexExpand[Sqrt[x Conjugate[x]]]]


Then

Abs[a + Exp[I*c] b]^2 /. Abs -> RemoveAbs


a^2 + b^2 + 2 a b Cos[c]

There is a simpler version inspired by @Nasser answer. Define a function

RemoveAbs[x_] := ComplexExpand[Abs[x]]


and now, for the sake of variety, apply simplification at the end:

Abs[a + Exp[I*c] b]^2 /. Abs -> RemoveAbs // FullSimplify


a^2 + b^2 + 2 a b Cos[c]

Point is, all the above will work for more complicated cases because /. is a ReplaceAll. Even for cases that are convoluted via things like, for example, TraditionalForm - the only thing you need is that InputForm still has Abs in it. For instance, imagine a beast of expression, here rather a simpler one but in reality they could go for pages:

(Abs[a + Exp[I*c] b]^2 + Abs[a - Exp[I*c] b]^2)/(2 Abs[a + I b]^2) // TraditionalForm


and now, voila, you get your simplification:

% /. Abs -> RemoveAbs // FullSimplify


1

ClearAll[a, c, b];
expr = a + Exp[I*c] b;
Simplify@Expand@(ComplexExpand[Abs@expr]^2)


Or in non @ friendly way

Simplify [ Expand [ ComplexExpand[Abs[expr]]^2 ] ]

• you are placing ^2 outside of ComplexExpand, he may want to deal with the expression as whole starting from Abs[a + Exp[I*c]b]^2 . Just noting the difference, but this may work for him too. Commented Sep 1, 2013 at 5:11
• @VitaliyKaurov I do not see why these are not the same. The above applies ComplexExpand on something before squaring, vs. squaring then applying ComplexExpand (which will not have simplified). Mathematically these seem to be the same to me since ComplexExpand does not add or take anything, but modifies the shape of the expression. But may be I am missing something. Commented Sep 1, 2013 at 5:29
• You are right, mathematically they are the same. But there could be a rather technical issue of order of operations. Imagine you have more complicated expression as I added in my answer below. It is given as whole. Now you need to apply some functions to it to simplify it. Though your assumption is absolutely correct, one still would need to add a few tricks to get this working. I was talking about "tricks", not about the math of it ;) Commented Sep 1, 2013 at 5:46

I like very much the solution of Nasser, especially since it is direct, i.e. it does not use any workaround, but just the Mathematica technique. Not to leave such a basic exercise with only one solution I would like to offer a very simple one:

    Clear[exprA, exprB];
exprA = a + Exp[I*c] b
exprB = exprA /. c -> -c (* This produces the conjugated expression *)
(* a + b E^(I c)    *)
(*  a + b E^(-I c)  *)


Now let us recall that Abs[z]^2=z z* and just multiply

    ComplexExpand[exprA*exprB] // Simplify

(*   a^2 + b^2 + 2 a b Cos[c]  *)


It should be noted that I used the replacement c->-c to obtain the conjugated value using that it is known all parameters to be positive, including c. It can be done in a more general way using Conjugate[]. In this case the final result is the same, but the intermediate results (if one chooses to look at them) are more cumbersome. Anyway,

    ComplexExpand[exprA*Conjugate[exprA]] // Simplify
(* a^2 + b^2 + 2 a b Cos[c]  *)