Output complex numbers in polar (exponential) form?

[Mr.CRC]

Hi:

For ex:

In[2]:= ToExp[z_] := Abs[z] E^(I Arg[z])

In[4]:= ToExp[3 + 4 I]

Out[4]= 5 E^(I ArcTan[4/3])

However, this converts back to rectangular form when a float is involved:

In[5]:= ToExp[3 + 4. I]

Out[5]= 3.+ 4. I

Any idea how to make a consistent exponential complex number output?

Thanks.

--
_____________________
Mr.CRC
crobc...@REMOVETHISsbcglobal.net
SuSE 10.3 Linux 2.6.22.17

[Bill Rowe]

On 9/2/11 at 3:29 AM, crobc...@REMOVETHISsbcglobal.net (Mr.CRC)
wrote:

>For ex:

>In[2]:= ToExp[z_] := Abs[z] E^(I Arg[z])

>In[4]:= ToExp[3 + 4 I]

>Out[4]= 5 E^(I ArcTan[4/3])

>However, this converts back to rectangular form when a float is
>involved:

>In[5]:= ToExp[3 + 4. I]

>Out[5]= 3.+ 4. I

>Any idea how to make a consistent exponential complex number output?

Use Rationalize to convert the machine precision values to exact
values, i.e.,

In[5]:= toExp[z_] := Module[{x = Rationalize[z]},
Abs[x] E^(I Arg[x])]

In[6]:= toExp[3. + 4. I]

Out[6]= 5*E^(I*ArcTan[4/3])

[Simon]

Here's the answer I gave to a similar question (http://stackoverflow.com/q/3928489/421225) on stackoverflow.

If you only need to do it occasionally, then you could just define a function like

ComplexToPolar[z_] /; Element[z, Complexes] :=
Interpretation[
Grid[{{Abs[z], Superscript[E, Row[{I, Arg[z]}]]}}, Spacings -> .2],
z]

If always want objects with Head Complex, then something like

Unprotect[Complex];
Complex /: MakeBoxes[Complex[a_, b_], StandardForm] :=
With[{abs = Abs[Complex[a, b]], arg = Arg[Complex[a, b]]},
RowBox[{MakeBoxes[abs, StandardForm],
SuperscriptBox["\[ExponentialE]",
RowBox[{"\[ImaginaryI]", MakeBoxes[arg, StandardForm]}]]}]]
Protect[Complex];

For complex numbers such as 3 + 4 Pi I, the first option works, but the second doesn't. This is clear from the FullForm.

[Mr.CRC]

Thanks for the response.

I think the first option works.