# How can I convert a complex number into an exponent form

When I have an expression such as

(1/4 + I/4) ((1 - 2 I) x + Sqrt y)


it is hard to get an intuition of the number.

So I want to convert it to the complex exponent notation:

$$\frac{1}{2} \sqrt{\frac{5}{2}} e^{-\frac{i \pi }{12}} x+\frac{1}{2} \sqrt{\frac{3}{2}} e^{\frac{i \pi }{4}} y$$

and I also want to convert $(-1)^{1/4}$ to $\exp \left(\frac{i \pi }{4}\right)$.

How can I do it with Mathematica?

This explicitly converts any numeric quantities in the expression to the desired form:

polarForm = Expand[# /. z_?NumericQ :> Abs[z] Exp[I Arg[z]]] &;


e.g.

(1/4 + I/4) ((1 - 2 I) x + Sqrt y) // polarForm


$\frac{1}{2} \sqrt{\frac{5}{2}} e^{\frac{i \pi }{4}-i \text{ArcTan}} x+\frac{1}{2} \sqrt{\frac{3}{2}} e^{\frac{i \pi }{4}} y$

(-1)^(1/4) // polarForm


$e^{\frac{i \pi }{4}}$

• sorry if I'm wrong because I'm not sure about these terms in English, but isn't this an exponential form, not polar? – user907860 Feb 3 '15 at 23:45
• @user907860, both terms are valid I believe. – Simon Woods Feb 4 '15 at 6:23

You can use the definitions together with ComplexExpand :

z = (1/4 + I/4) ((1 - 2 I) x + Sqrt y);
abs = ComplexExpand[Abs[z]]
arg = ComplexExpand[Arg[z], TargetFunctions -> {Re, Im}]
(* Sqrt[4 x^2 + (x + Sqrt y)^2]/(2 Sqrt) *)
(* ArcTan[3 x + Sqrt y, -x + Sqrt y] *)

(* check *)
z - abs Exp[I arg] // FullSimplify
(* 0 *)


The requisite conversion functions are already included in David Park's Presentations add-on:

<<Presentations
PolarToExp @ ComplexToPolar[(1/4 + I/4) ((1 - 2 I) x + Sqrt y)]


(The Presentations actually includes a polar form for a complex number of modulus r and argument theta that displays in the form r \[Angle] theta. And ComplexToPolar yields that form. Hence the need for the outside function PolarToExp`. Thus there is a distinction made between "polar form" and "exponential form" -- as there should be.)